SYMMETRIES AND LAPLACIANS introduction to Harmonic Analysis, Group Representations and Applications
NORTH-HOLLAND MATHEMATICS STUDIES 174 (Continuation of the Notas de Matematica)
Editor: Leopoldo NACHBIN Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil and University of Rochester New York, U S A .
NORTH-HOLLAND - AMSTERDAM
LONDON
NEW YORK
TOKYO
SYMMETRIES AND LAPLACIANS Introduction to Harmonic Analysis, Group Representations and Applications
David GURARIE Department of Mathematics and StaYjstics Case Western University Cleveland, OH, U.S.A.
1992
NORTH-HOLLAND - AMSTERDAM
LONDON
NEW YORK
TOKYO
ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands
ISBN: 0 444 88612 5
0 1992 ELSEVIER SCIENCE PUBLISHERS B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V., Copyright & Permissions Department, RO.Box 521, 1000 A M Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U S A . All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher, Elsevier Science Publishers B.V. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands
In the memory of my parents, whom I owe so much to Valentina
Eli and Mark
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TABLE OF CONTENTS
Introduction .................................................................................................................. 1 Chapter 1. Basics of representation theory
51.1. Groups and group actions ............................................................................. $1.2. Regular and induced representations; Haar measure and convolution algebras .................................................................................... 51.3. Irreducibility and decomposition .................................................................. 51.4. Lie groups and algebras; the infinitesimal method ........................................
13 24 37 46
Chapter 2. Commutative Harmonic analysis
$2.1. Fourier transform: inversion and Plancherel formula .................................... 62 52.2* Fourier transform on function-spaces ........................................................... 71 52.3. Some applications of Fourier analysis ........................................................... 83 52.4. Laplacian and related differential equations .................................................. 92 52.5* The Radon transform ................................................................................. 120 Chapter 3 . Representations of compact and finite groups
53.1. The Peter-Weyl theory ............................................................................... 53.2. Induced representations and Frobenius reciprocity ..................................... 53.3* Semidirect products ...................................................................................
125 137 147
Chapter 4 . Lie groups SU(2) and Sq3) 54.1. Lie groups and SU(2) and Sq? and their Lie algebras ................................ 161 54.2. Irreducible representations of U(2)............................................................................. 164 &3* Matrix entries and characters of irreducible representations: Legendre and Jacobi polynomials .............................................................. 171 54.4. Representations of SO(3): angular momentum and spherical harmonics ..... 174 $4.5* Laplacian on the n-sphere .......................................................................... 184 Chapter 5 . Classical compact Lie groups and algebras
$5.1. Simple and semisimple Lie algebras; Weyl “unitary trick” .......................... 55.2. Cartan subalgebra, root system, Weyl group .............................................. $5.3. Highest weight representations ................................................................... 95.4* Tensors and Young tableaux ...................................................................... $5.5. Haar measure on compact semisimple Lie groups ....................................... $5.6. The Weyl character formulae ..................................................................... $5.7* Laplacians on symmetric spaces .................................................................
191 197 206 217 227 231 241
Chapter 6. The Heisenberg group and semidirect products
$6.1. Induced representations and the Mackey’s group extension theory .............. 257
...
Vlll
56.2. The Heisenberg group and the oscillator representation .............................. $6.3* The Kirillov orbit method .......................................................................... Chapter 7. Representations of SL,
274 290
.
305 57.1. Principal. complementary and discrete series .............................................. 57.2. Characters of irreducible representations .................................................... 313 57.3. The Plancherel formula on SL. (W) .............................................................. 317 67.4. Infinitesimal representations of SL. spherical functions and characters ....... 328 §7.5* Selberg trace formula ................................................................................. 334 $7.6* Laplacians on hyperbolic surfaces W/r....................................................... 348 57.7* SL. (C) and the Lorentz group .................................................................... 361 Chapter 8. Lie groups and hamiltonian mechanics 58.1. Minimal action principle; Euler-Lagrange equations; canonical formalism ... 369 58.2. Noether Theorem, conservation laws and Marsden-Weinstein reduction ..... 378 385 $8.3. Classical examples ...................................................................................... 58.4, Integrable systems related to classical Lie algebras..................................... 393 $8.5* The Kepler Problem and the Hydrogen atom ............................................. 408 Appendices: A: Spectral decomposition of selfadjoint operators .............................................. 423 B: Integral operators .......................................................................................... 427 C: A primer on Riemannian geometry: geodesics. connection. curvature ............. 430 Rderences ................................................................................................................. List of frequently used notations ................................................................................ Index .........................................................................................................................
439 447 449
Introduction Throughout this book word Group will be synonymous with transformations and symmetries. Groups are omnipresent in the mathematical and (possibly) physical universe. In Mathematics symmetries play several important functions. On the one hand they allow to reduce the “number of variables”, and often render problem soluble (like reducing the order, or separating variables in differential equations). On the other hand they allow to analyze and synthesize complex objects in terms of simple (elemental) blocks/constituents (eigenvalues of matrices and operators, expansion in Fourier modes, harmonic analysis in general). The prime example came from the very onset of the subject, the Galois theory of algebraic equations. The thrust of this early work was precisely to relate the symmetry (Galois) group of the equation to its solvability’ in radicals. Then Sophus Lie set up the task to develop the “Galois theory” for differential equations. He naturally came up with the concept of a continuous Lie group and its infinitesimal versions Lie algebra. Since its inception in the mid XIX-century the Lie theory rapidly grew and spread across the broad range of subjects, to occupy its present central position at the crossroads of Geometry, Analysis, Differential equations, Classical mechanics and Physics in general. Symmetries in physical systems2 are inextricably linked to conservation laws (via Noether’s Theorem), and the Galois chain repeats: “symmetries of hamiltonians”
+ “conserved integrals” + “solvability (via symmetry-reductions)”.
Our book sets more limited goals. The main objective is to introduce the reader to a wide range of concepts, ideas, results and techniques, that evolve around
. 1-
[symmetry-groups\, Irepresentations], and
We strived to stress the diversity
and versatility of the subject, but at the same time to unravel its common roots. More specifically, our main interest lies in geometrical objects and structures {X}, discrete or continuous, that possess sufficiently large symmetry-group G, like regular graphs (Platonic solids); lattices; symmetric Riemannian manifolds, and other. All such objects have a natural Laplacian A, a linear operator on functions over X, invariant under the group action. There are many problems associated with Laplacians on X. Typically one is interested in certain “continuous or discrete-time” evolutions, on
X, random walks, diffusion processes, wave-propagation. All these problems require to ‘Thence came the terminology of solvable groups (abelian ext,ensions of abelian groups), as the abelian ones were directly associated with solvability. ‘Symmetries acquire yet greater significance in the murky world of subatomic physics, where nothing could be observed, measured, sensed and compared to the everyday experiences of the macroworld. Here symmetries and conservation laws provide the only guiding light, and Lie groups enter in the very formulation of fundamental models of particles, fields and forces.
2
introduction
compute certain “functions of A”, L = f ( A ) , like powers { A ” } ; the semigroup, generated by A {e-“}; function {cos(tJ-6)}, etc. To explain and motivate the problems and the methods, based on symmetries, we shall start with a few simple prototypical examples. Fibonacci sequence is defined by a 2-term recurrence relation: Ixn+l = x, (or more general, xnS1 = a x ,
+b
+ .z,-~]
~ ~ - and ~ ) the , two initial values {zo;xl}.One is asked
to find all {z”}. To solve the problem, one first converts the 2-term numerical recurrence to a one-term recurrence for vectors: y, = Frobenius-type matrix L =
and we need to compute all iterates {L“}. The latter could be accomplished by diagonalizing matrix L , i.e. solving the eigenvalue problem: finding spectrum {A,$,}, and the matrix of eigenvectors U = {&;$,}.
The latter serves to diagonalize L,
L = U A U - I , where A=diag{A,;X,}. Once L is brought into the diagonal form, its iterates {Ln= UA”U-’} come right away, as well as more general “function of L”, f(L)= U f ( A ) U - ’ . So we immediately get an explicit solution
’.
Of course, no group-analysis was needed here, as the “eigenvalue problem” had an easy direct solution. However, similar “finite” problems could quickly become intractable, as we shall demonstrate in our next example.
Random w&
on graphs
I‘. The process consists of random jumps from any site
(vertex) of the graph to any one of its nearest neighbors with equal probabilities. Starting with the initial position at {xo}, or the initial probability distribution Po(,) (ZE
r),one is asked to find the probability distribution after the R time-steps, { P n ( x ) } .
Once again vector {P,(z)} is completely determined by the initial state Po and the transition-matrix L , which could be called the “Laplacian” of the graph, that sends any vectorffunction $(x) on
I‘ into L!b(x) = ?&b(Y),
sum over all m nearest vertices {y} of
Y
2.
Since the jumps are statistically independent
(the process has no memory of the past, when deciding to go from {x} to {y}), and
stationary in time (the “jump-rules” remains the same for all times n = 0; 1;...), we get a sequence of iterates, 3The classical Fibonacci numbers are given by
=c
I+&”
+
]-fin
, the generalized ones, z, = clX1” where coefficients, {cl;ez} depend on the initial data {zo;zl}. 2,
, ( ~ ) c2(-7_!
+ C,X,~;
3
Introduction
P, = Ln[P,],at time n. To compute powers
{L”}we
(2)
need once again to diagonalize
L, i.e. to solve the
eigenvalue problem. But this time the difficulty increases drastically with the size of
r,
if one tries a direct approach (imagine diagonalizing a 20 x 20 “dodecahedral” matrix!). There is not much to do in the general case, so we turn to a special, but important, class of regular graphs (like polygons, Platonic solids, a “Knight on a periodic
r are characterized by the abundance of symmetries. One can associate with any such r group G of vertex-transformations, that preserve all links (the incidence relations) between vertices. For regular r, group G takes any vertex x into any other4, so r becomes a homogeneow space o f G, r = H\G, where H denotes the stabilizer subgroup of some xo E r, H = {g:xog = xo}. For instance, the chessboard”, etc.). Regular graphs
regular m-gone is itself a finite abelian group Z , = {integers modulo m} (its symmetries are made of all finite rotations and reflections), whereas Platonic solids (cube, octahedron, dodecahedron, etc.) are homogeneous spaces H\G
of some well-known
symmetric or alternating groups (see 31.1 and 33.2). Given a homogeneous space X, group G acts on X by translations, g:x-+xg,and this action gives rise to the regular representation R = R X of G on the function-space
L = L(X ) over X , R g f ( 4= f ( 4 .
(3)
One easily verifies that map g+Rg, from G into operators on L, takes group multiplication into the product of operators, in other words defines a group-
L commutes with all operators {Rg},group G leaves L invariant, and we have to analyze the resulting “reduced
representation. Since Laplacian eigenspaces { E x } of
L
in
representations” of G on eigenspaces { E x } . Among all representations of the group one distinguishes the minimal ones, called irreducible, the ones that do not allow further reduction to smaller invariant subspaces (they play the role of “joint eigenvalues” for a g E G}).The typical problems in representation theory include: family of operators {Rg: (i) characterization all ineducible {n’s}for a given group G (analysis-problem); and
(ii) decomposition of an arbitrary (natural) representation R into the direct sum (or integral, to be explained below) of irreducibles (synthesis problem). The harmonic analysis on X suggests a possible approach to the spectral problem 41n fact, in many cases G takes any connected pair of vertices {qy} to any other such pair. Such graphs T’s could be called 2-point symmetric spaces, by analogy with their continuous counterparts.
4
Introduction
for operator L. Namely, we need
(I) to study the structure of the symmetry-group G; (11) to determine its elemental (irreducible) representations
{T};
(111) to interpret Laplacian L in terms of group-elements of G, to be able to assign operators { T ~ to} it for various irreducible { T } (IV) finally, to decompose the regular representation into the direct sum of irreducible components: RX = @ d .
i
The latter means that space L ( X ) is split into the direct sum of subspaces @ Ei, 3
invariant under all operators { R,:g E G}, irreducible relative to the G-action. Such decomposition, generalizes the notion of spectral/resolution (diagonalization) of a single operator. Steps (I-IV), when successfully accomplished, would lead to the desired spectral resolution of L. Namely, the reduced Laplacian L I E j r L becomes scalar5, N
So the “eigenwalue spectrum {X,(L)]” could be identified the “representation spectrum {d}of R”, the “eigenspaces” being “irreducible components” of R. Once L is diagonalized we immediately obtain all iterates {L”), and more general functions {f(L)}, needed to solve (3). { T = ~ X,(L);
for each irreducible
T}.
The details of the scheme could be found in 52.3 (Finite Fourier transform) for polygons, and in 53.3 for Laplacians on Platonic solids. After a brief excursion through the finite/discrete cases (graphs) we shall turn now to a completely different geometric setup: smooth Riemannian manifolds, Lie groups and their quotients, symmetric spaces X . All these objects will be introduced and studied at length in the due course (chapters 1,4,5,6,7).
Here for the sake of
introduction we consider 3 familiar examples: the Euclidian space R”, the n-sphere
S” c Rn+’, and the hyperbolic (Poincare-Lobachevski) plane‘ W2 = { z = z+iy:y 2 0). To motivate the continuous case we shall mention 3 typical problems in the integral geometry with some physical context. The setup for all 3 is basically the same. We are given a solid body D of constant (or variable) density p ( z ) . In the first case one considers all lines {y}, passing through
D,and computes integrals along
y: ?(y)
=
I,
pds. In the
’There is a general reason, why Laplacian L goes into the scalar operator nL under any irreducible A, known as Schur’s lemma. ‘the latter could be also interpreted, as a continuous limit of properly scaled i n space and time random walks on trees.
5
Introduction
second solid D (assumed to be convex, symmetric) is illuminated from all possible directions, and we measure the area of the shadows, cast by D. In the third case (somewhat simpler than the second) we take all cross-sectional areas of D by the family of planes, passing through its center. In all 3 cases one would like to recover either density p (case l), or the geometric shape of D (case 2 and 3) from such data. The first case is essentially the celebrated X-ray transform, used in tomography. The 2-nd and 3-rd give somewhat different versions of the Radon transform on the 2-sphere. Indeed, a (convex) body D can be represented as a graph of function f(z) on the 2-sphere, S2 = { I z I = 1). The cross-sectional areas of D are obtained by integrating :fz along all
great circles {y} in S2. So in all cases we are dealing with certain integral transforms of functions, either on the Euclidian space, or the sphere.
W have the natural Riemannian metric (§1.1), and possess G of distance-preserving transformations (isometries). The
All 3 spaces, R", S", large symmetry-group
Euclidian isometries consists of rigid motions: group En, generated by all translations and rotations in R"; the n-sphere S" has the orthogonal symmetry-group SO(n+l), made of all (n+l) x ( n + l ) - orthogonal matrices in Rn+' 3 S"; symmetries of
W coincide
with the ubiquitous unimodular group SL,(R) - all 2 x 2 real matrices of detg = 1, acting az+b by the fractional-linear transformations, g: z-+Cz+d. In fact, all 3 examples represent the of group G modulo the maximal compact
so called symmetric spaces, quotients K\G
subgroup K , that fixes a particular point in X , K = {g:z,g examples:
Ii' = S q n ) for Rn and S",
while the hyperbolic
' A
= z,}.
In the above
= SO(2).
The analog of the discrete-time random-walk on space X is played either by
spherical means:
(average value of function
f over the sphere of radius r , centered at
{z}), or by more
general integral kernel,
I
u(z)+K[u]= Ii'(z;y)u(y)dy. Transition-operator7 K takes any distribution p,(y) on X into pl(z) = li'[po], at a single time-step. As before we are interested in iterates
{I{"},
and once again have to
face the "diagonalization problem" for Ii'. In general, the problem is hardly tractable, so we turn to a special class of integral kernels, K ( z ; y ) ,that commute with the G-action, 7Strictly speaking, density K ( z ;y)dy represents transitional probabilities t o j u m p from point {y} to {z}, provided jK(z;y)dy = 1.
6
Introduction
(3) on X . Any G-invariant operator has kernel K , that depends only on the distance’ T = d(z;y), between z and y, 1-(z;y) = K [ T ) .Spherical means give one such example, the Radon and X-ray transforms also obey this condition, but the foremost of all Ginvariant operators is the natural Laplacian’ A (the Laplace-Beltrami operator) on X . Associated to A there are various evolution processes on X (continuous analogs of the “discrete-time random walks”), described by partial differential equations (pde). One of them is the standard heatdiffusion problem, ut
- AU= 0; ~ ( 0 =) f;
(4)
whose formal solution of (4) is given by the heat-semigroup of A, u = et”[fl.
Another important model is the wave equation: utt - Au = ...,whose solutions can also be represented by “functions of A”, {u = e *it-;
or cos t a ; sin tJ--Zij}.
Since A commutes with the G-action on X , all its functions { f ( A ) }do the same, including the heat and wave-propagators (another names for the “transition-matrices”, I( = etA; e i t f i ) . But any such I( has a radially symmetric kernel (I = K ( r ) . So a pdeproblem for I(, is often reduced to an ordinary differential equation (see chapters 2;4;5). Those in many cases, including R”, S”, W, could be solved explicitly in terms of the well-known special functions. Once again we are lead to analyze the G-action on X , and the resulting regular representation R,f(z) = f(zg),on suitable function-spaces over X.
So spectral decomposition of any radial operator K is reduced to the harmonic analysis of representation R. The latter involves among other a good understanding of the symmetry-group G itself, its ‘elemental (irreducible) representations” nature of the direct sum and/or integral decomposition,
R=
@ x j ; or 3
xSdp(s).
{T},
and the
(5)
To give some feeling of the issues involved, let us take the Euclidian space Rn, as an abelian group with translations only. Irreducible representations of R” are all 1‘This result is a direct consequence of the “double-transitive” action of G on rank-one symmetric spaces X,i.e. any pair {z;y} C X , is taken any other equidistant pair. ’In fact, any spherically symmetric kernel K on X (in either one of 3 examples) is given by a “function of the Laplacian”, K = f(A). This result, far from obvious, comes from the harmonic analysis of rank-one symmetric spaces (chapters 4-5). It has to do with the “multiplicity-free spectrum” of regular representation R, equivalently, with the fact that the commutator of R (all operators, that commute with it) forms a commutative algebra. “Functions of operators” {f(L)}will be introduced and discussed in chapter 2 and Appendix A via spectral resolution of operator L.
7
Introduction
dimensional (characters!), they consist of the family of exponentials {eit. €:[ E R”}. Decomposition (5), amounts to the Fourier-transform of f(z), expansion in the Fourierseries or integral. In chapter 2 we explore in detail various aspects of the commutative Fourier analysis and apply it to differential equations. The relevant analysis on the n-sphere, however, has a very different flavor. Here one has to start with “irreducible representations”
S q n t l ) , that would play the role of exponentials
{T}
of the orthogonal group
{ e i ” * < } on
R”. This, by itself a
challenging task, requires to develop a fair amount of the linear and multilinear (tensorial) algebra, as well as the relevant structure theory of Lie groups and algebras (chapters 4-5). Irreducibles
{T}
of S q n t l ) turned out to be finite dimensional, and the
decomposition of R has purely “discrete spectrum”lo,
R=
co @ rk.
k=O The corresponding irreducible subspaces { 36k} of L2(Sn), called spherical harmonics, have many remarkable features. Aside of being eigenspaces of the spherical
Laplacian, they proved to be closely connected to harmonic polynomials on Fin+’, and the classical Legendre and Jacobi functions. Our last example
W2 reveals even more striking difference. SL, is a
simple, not
compact group (in fact, the “smallest” of them). Unlike R” or S q n ) , its irreducible
unitary representations are infinite-dimensional, they make up certain continuous and discrete families {#:s E R} and {rrn:mE Z}. The decomposition problem for regular representations { R X } on homogeneous spaces, strongly depends on the “geometric nature” of quotient
X
continuous (direct integral), R = discrete subgroup of
+
= H\G. In some cases (Poincare half-plane W) it becomes purely
G) -
TSdp(s);in other cases (compact quotients r\G,
purely discrete (direct sum) R =
%0 r S j .It
r-
could also be a
combination of both, like the regular representation R on the entire group
G! The
relevant harmonic analysis becomes quite involved (see chapter 7). Its most spectacular application to the spectral theory of Laplacians appears in the context of quotients X = r\G, modulo discrete subgroup r c G. It turns out that “spectrum of such X” (either Laplacian A,, or representation Rx) is intimately connected with some fine “arithmetic properties” of via the celebrated Selberg-trace Theorem (a noncommutative version of the “Poisson summation” on R”). As an off-short we
r,
-Both results are general and hold for arbitrary compact (finite in size) groups (chapter 3). But specific examples, like S q n ) will carry a great deal more structure and information about {m} and
R.
8
Introduction
establish some interesting links between the “eigenvaluespectrum {A,}
of A,”, and the
“geometric length-spectrum of X” (length of all closed geodesics in X ) . We elaborate some aspects of spectral theory on compact quotients r \ G in 57.6-7.7. Our discussion of the “continuous cases” vs. “discrete cases”, although very different in technical terms, clearly demonstrates similarities in the basic procedures. Namely,
I) investigation of the group-structure of G, and its “elemental” irreducible representations { T } ;
11) study of G-invariant objects “Laplacians” on G, or on its quotients X = H\G; 111) decomposition of natural G-actions, regular representations R X , RG; its L on G and X ;
connection to Laplacians
IV) application of parts (1)-(111), particularly, the correspondence between “spectrum of L” and “spectrum of R” to various “junctions of L”,like iterates {L”}, semigroups { e-“; e - t f i } , and solutions of the associated differential equations. The development in the book largely will largely follow the general scheme (IIV), although not always step by step, as in the model examples. We may or may not start with a simple model problem. Often a significant effort has to be spent first to study the relevant groups, and their representation theory: problems of irreducibility and decomposition (cf. chapters 5-7). As we mentioned our main interest lies in the natural G-actions and representations, regular RG; RX,and their generalizations called
induced representations. The latter extend G-actions on scalar function { f(s)}, to vector-functions and sections of vector-bundles over X (so G-action combines translations in the X-space with “twisting” in fiber-spaces). The decomposition of the regular and induced representations makes up the content of the so called Plancherel Theorem. Although we follow steps (I-IV) throughout the book their relative “size” and “weight”, as well as the amount and depth of applications, varies from chapter to chapter. Thus chapter 2 is largely devoted to applications of the commutative Fourier transform, while chapters 5 or 7, deal mostly with the analysis of groups and represent at ions. An introductory chapter 1 brings in the principal players: basic examples of
9
Introduction
groups and geometric structures (51.1). Then we proceed to develop some fundamental concepts and results of the representation theory (§§1.2-3), and the Lie theory ($1.4). In chapters 2-7 we take on the main themes of the book with many facets of the general scheme (I-IV). Each chapter has its own LLfavorite” Laplacians and symmetry-groups: 0
standard Laplacian A on commutative groups Wn;Un and certain domains
D c Rn, in chapter 2 0
polyhedral Laplacians and the relevant finite groups in chapter 3
0
the spherical Laplacian A on S2 and S”, whose theory is based on Lie groups
SU(2);SO(3), in chapter 4 0Laplacians on more general compact Lie groups and symmetric spaces in chapter 5 0
the harmonic oscillator H = - A
+ I x I ’,and the Heisenberg group in chapter 6
0Laplacians on the Poincare plane
W, SL,,
and compact Riemann surfaces r \ H
in chapter 7. Chapter 8 stands somewhat aside from the mainstream, as it takes on the subject of “symmetries in nonlinear(!) hamiltonian systems”, and their role in integrability. But the very end of chapter 8 (58.5) brings us back to the main issues in grouprepresentations. We discuss the quantization problem of classical hamiltonians, particularly, the “quantized Kepler problem” - the hydrogen atom, one of the best studied models in quantum mechanics (see [LL]). The “Laplacian” pops in here in a quite interesting and unexpected form. A remarkable feature of the Kepler (hydrogen) in @, is that the “discrete hamiltonian, Schrodinger operator H = - A -1 Izl
(negative) part of H” is equivalent to the inverse Laplacian A - ’ on the 3-sphere. Along with the main themes (I-IV) a number of side issues and problems come to the discussion in different places. The reader will learn, for instance, why the geodesic flow is integrable on Sz (chapter 8), and ergodic on compact negative-curved Riemann surfaces X (chapter 7); what group-representations have to do with a topological problem of “counting linearly independent vector fields on spheres” (chapter 3); how Lie groups and representations explain some peculiar properties and rela.t,ionsof special functions (Legendre, Jacobi, Hermite, etc.). The book was designed as an introduction to harmonic analysis and group representations for graduate students in Mathematics and applications, or anybody
10
Introduction
(non-expert), interested in the subject, who would like to gain a broad perspective, but also learn some basic techniques and ideas. In the words of H. Weyl
“... it
is primarily
meant f o r the humble, who want to learn as new the things set forth therein, rather than f o r the proud and learned who are already familiar with the subject and merely look for
quick and ezact information...” The material of the book is based on the lectures and seminars, given by the author over the passed few years at UC Irvine, Caltech, and CWRU. Student comments and suggestions were helpful in preparing the manuscript. Our goal was to cover a wide range of topics, rather than to delve deeply into any particular one. The exposition is largely based on examples and applications, which either precede or follow the general theory. Some are important on their own, others serve to elucidate and motivate general concepts and statements. Of course, if the general approach seemed conceptually easy and directly leads to the main point, we do not hesitate to bring it forth (like the “Peter-Weyl theory” of chapter 3). But we never engage in “abstract” studies for their own sake. In order to keep the minimal prerequisites, and to shorten the background preparations, the book often appeals to intuition, example and analogy, rather than formal derivations. So the reader versed to some degree in the basic Riemannian geometry, functional analysis and operator theory, should be able to go through most of the topics without difficulty. The less prepared reader would be granted a paragraph (or two) of a footnote/appendix style explanations, to enable him to grasp quickly a new concept (or idea), and to follow the rest. Certain parts of the book (sections/paragraphs) are addressed to a better prepared reader, without detracting from the main themes.
As we wanted not to rely heavily on the standard (4-8 semester) staple of real analysis, algebra, topology, the book provides a number of ‘shortcuts’. So some basic concepts in algebra, geometry, topology are introduces “on fly”, as the need arises. We were somewhat more patient and systematic with the analysis of operators and differential equations: chapter 2 could serve as a brief introduction to PDE’s, mostly from the classical standpoint (cf. [CHI; [WW]). We also provided 3 appendices: on spectral decomposition of self-adjoint operators (A); on integral operators (B), and on basic Riemannian geometry (C). Finally, to cover a sizable material in a moderate-size volume required a departure from certain standards of mathematical exposition. We found it impossible
11
Introduction
(and undesirable) to try to maintain a uniform level of rigor and detail throughout the text. So some results are provided with fairly complete arguments, others are only outlined, relegated to problems, or just stated. The role of formal proofs is in general downplayed. The book contains sufficient material for a 1 or 2-semester course in the Harmonic analysis and Group representations. The instructor could make several choices, and follow different path in selecting topics. Aside of chapter 1, that provides a general core and background, all other parts are relatively independent. So the reader could start at practically any place in the book, going back and forth, as deemed necessary. The only exception are chapter 4 and 5, which should proceed in their natural order. In each chapter we marked with
* more
advanced topics, that could be
omitted in the first reading. In writing an introductory text on a fairly broad subject, one inevitably has to make certain choices, and put aside some important topics. Our selection and style reflected largely on the author’s personal experience and prejudices, rather than anything else. Among a few important topics, left outside the scope, let us mention the representation theory of infinite-dimensional Lie groups and algebras (Kac-Moody), which was actively pursued over the passed 20 years, and recently came to the focus in connection with the String theory (monographs [Kac] and [GSW] present the mathematical and the physical view on the subject). Another important topic is related to symmetries of differential equations, dynamical systems, integrable hamiltonians. Although we do touch upon integrability in chapter 8, our analysis is limited to finitedimensional systems. The exciting developments in the field of “infinite integrable hamiltonians” over the past 30 years were also left out (see [Per]; [Olv] for references). The book was composed on the IEXP-2 word processing system, with an additional help of the Microsoft Windows PBRUSH-graphics, responsible for the figures (the author takes entire responsibility for errors, misprints, omissions). After a somewhat bumpy initiation to the world of the modern information technology, the author had a highly rewarding experience working with both programs. Both became the most indispensable tools in the arduous enterprise of writing and organizing the manuscript.
He thought he saw a Garden-Door That opened with a key He looked again, and found it was A Double Rule of Three: ‘And all its mystery”, he said, “Is clear as day to m e !” He thought he saw an Argument That proved he was the Pope He look again, and found it was A Bar of Mottled Soap. “A fact so dread, ” he faintly said, “Eztinguishes all hope !” Lewis Carroll, “The Mad Gardener’s Song”
Chapter 1. Basics of representation theory. 51.1. Groups and group actions. We introduce basic examples of discrete and continuous transformation groups; classical matrix Lie groups; isornetries of symmetric spaces; rigid motions of the Euclidian, spherical and hyperbolic geometry, as well as symmetries of regular polyhedra.
1.1. Geometric transformations. Groups, discrete and continuous, typically arise as symmetries of “geometric” structures of different kinds, which includes both discrete objects (graphs, polyhedra), and the continuous, like manifolds, symmetric spaces. Important examples of discrete groups include:
i) permutations of a finite sets A = {l;...n}, G(A) = W,; ii) isomorphisms of graphs, lattices, regular polyhedra;
iii) matrix groups over finite fields. Some of them will be described at the end of the section (example 1.1). Throughout this book we shall be mostly interested in the continuous (Lie) groups. The latter typically arise as transformations (linear or nonlinear) of vector spaces,
or
manifolds
A, equipped
with
certain
geometric
structure.
Such
transformations q5:A+A preserve the structure (or transform it in a prescribed manner), in other words they represent symmetries of A. For instance, classical (matrix) Lie groups over vector spaces R”, c”, or quaternionic1 Q”, consists of linear transformations, that preserve certain bilinear/quadratic forms, like the general linear
group made of all non-singular n x n - matrices, Gf, = {A:detA# 0 } , its subgroup
Sl, = { A :detA = l}, called special linear (or unimodular) group, as well as their numerous subgroups (orthogonal, unitary, symplectic etc.). Lie groups also arise naturally as isometries of certain Riemannian (pseudoRiemannian) manifolds, or more general conformal transformations2. The foremost cases are symmetric spaces, Riemannian manifolds which possess large isometry groups. Here we shall briefly discuss 3 basic examples of symmetric spaces (flat, spherical and hyperbolic), and the related symmetry groups. a) Euclidian space
Rn, respectively Minkowski Mn,equipped with either positive-
+ + +
‘Quaternions Q = {( = a b i c j dk a, b, c,d € R} form a noncommutative field (division algebra), generated by 3 imaginary units:i2 = j 2 = k2 = -1; i j = - - j k k ; j k = -kj=i; t i = -ik = j . They can be also represented by complex 2-vectors: ( = z wk ( z , w E C), where k2 = -1, with the multiplication rule: kzu = ii~ t. Quat,ernions along with R and C are known to form the only 3 possible division algebras over reals (see J3.1 for further details).
+
§l,l.Groups and group actions.
14
definite product: z . y = Czjyj, or indefinite product3: (z I y) = zoyo-
n-1
zcjyj. The
isometries of the Euclidian space form a group E, of rigid motions of W", generated by
+ a}, and rotations (orthogonal matrices) {U:'UU = I } . So each the form 4(z) = Uz + b. The proof is outlined in problem 1. The tricky part is to show that any rigid motion q5 is affine, d(z) = A z + b, with some
all translations {a:z+z transformation
4 E En is of
matrix A. Then orthogonality of A follows fairly straightforward. Thus group becomes a subgroup of a larger affine group,
En
+
A f f , = {4(z) = AX b: A E G L ; b E W"}. The En-linear factors { A } are either general orthogonal matrices, A E q n ) , or special orthogonal, A E S q n ) = {U:detU = l}, if 4 preserves the orientation.
Mn the role of the orthogonal affine factors is taken by the
In Minkowski space
Lorentz (pseudoorthogonaZ) group Sql;n - l), which c nsists of matrices B, that preserve the indefinite product (z I y) = Jz y, where J = - matrix of Minkowski form,
41-11)
{ A : ( A zI A d = (z I Y)). So symmetries of the Minkowski space form the Poincare group P, of Special Relativity. The latter is generated by all translations and all Lorentz transformations, 4(z) = Az
+ b; b E Rn;A E S q 1 ; n ) .
Let us notice that in all three cases (affine, Euclidian, Poincare) elements
4 are
2We recall that a Riemannian manifold Ab carries a positive definite metric g = Cgijdtidzj, i.e. an inner product (( I tJg = Cgij
O, on tangent spaces of Ab {( E Tz}.Pseudo-riemannian refers t o the indefinite metric g, usually of the type (+; - ;- ;...). Any diffeomorphism (change of variables) z = #(y), on a Riemannian (pseudo-Riemannian) manifold Ab transforms the metric: g+T = g# = #'*(go #) #', the new entries being N
gkm(Y)
= C (9; j 4) 'I' '
azjwhere
aykaYm9
4' = (%) denotes the Jacobian (matrix) of #. aY
Map # defines an isometry of Ab, if the transformed metric g4 is equal to g. Such maps # obviously preserve the length of any path y = { y ( t ) : 0 5 1 5 T } ,
IY['
T
=
{ Cgij(~(t))i.ii.jdt,
hence the distance between points, d(+;y) = rnin{L[y]: y(0) = 2 ; y ( T ) = y}. Conversely, for any distance preserving diffeomorphism #:Ab-+Jb, the differential 42 (Jacobian matrix of 4), considered as the map of tangent spaces, #':T,+Ty (y = d(z)), preserves the metric (norm) on T,, (#:([) I#:(()) = ((I (), for any tangent vector (. Isometries of Ab clearly form a group. Conformal maps # do not preserve metric, but multiply it with a scalar (conformal) factor p(z), i.e. g4(z) = p(z)g. So they form a larger symmetry group of Ab. 31n Special Relativity R4 = {zo;
...;z3} represents a simplified version of space-time,
serves as time variable, while (zl; z2;z3)represent space coordinates.
where
I,,
51.1. Groups and group actions
15
identified with pairs (A; b), and the group multiplication takes the form
( A ;b) * (A';b') = (AA';Ab' t b).
(1.1)
It is easy to check that translations {b} form a normal subgroup
H
N
Rn of G,
while linear factors { A } , or { U } , form a subgroup I(
N GL,, or S q n ) ; S q 1 ; n ) . So group G is decomposed into a semidirect product4, G = H D K . The representation theory of
semidirect products will be analyzed in chapters 3 and 6. Two other examples of geometric symmetries arise on the sphere and the hyperbolic (Poincare-Lobachevski) space, the prototypes of the spherical and hyperbolic geometries. They also serve as the simplest prototypes of symmetric spaces of the compact and non-compact (hyperbolic) type. b) Sphere: Sn-' = {3::113:11~= l} in Rn with the natural (Euclidian) metric has the isometry group G = S q n ) - all orthogonal transformations (rotations) in R". In fact,
Sn-' can be identified
with the quotient (coset) space of group G = S q n ) , modulo the
stabilizer (isotropy subgroup)'
K = S q n - 1) of
fixed point zo in
a
Sn-' (the
North
Pole!), Sn N G / K . In other words (definition of stabilizer), I( = { U : U ( z 0 )= x0}. c) Hyperbolic space W, can be realized either as Poincare-Lobachevski complex half-plane { z = 3:
+ iy: y > 0 } , with metric
or as complex disk D = {IZI
< 1 ) with metric, ds2 =
the Mobius transformation,
One can easily check that
0
rn. dzdf
Both spaces are related by
takes W onto D, and transforms their Poincare
metrics, one into the other. The isometry group of 04 in both realizations is made of the fractional-linear transformations: az+c @z+w = bz+d'
U:Z-+X",
4A semidirect product G = H D U , of groups H and U , where U acts by automorphisms on H , (zE H ; u E U ) , consists of all pairs g = (z,u) witbthe multiplication rule (2,u)* (y, w) = (2 * y" ;uw). It is easily seen that H = {(z;e)} forms a normal subgroup of G, while K = {(e;u)} a
subgroup, so that H
nK = { e } , and the whole group G = H . K .
'Stabilizer (isotropy subgroup) of point c in a G-space X , consist of all elements {g E C}, that leave z fixed, K , = {g: zg = z}.
31.1. Groups and group actions.
16
5
1
In the half-plane case matrices {A = fi]:ad- bc = 1 belong to SL,(R) - the real unimodular group. In the disk realization they are given by complex matrices
detA=iaiZ-IPlZ=l A that preserve the indefinite hermitian form: J ( z ) = I z1 1' - / z 2 on C2 = {(q; 2,)). This group is called conformal and denoted SU(1;l), by analogy with the standard unitary group SU(2), that preserves the positive
in
other
words
matrices
I,,
hermitian product, l z l z = 1z11' +1zzl2. Once again the hyperbolic space coincides with the quotient of G = SL, (or SU(l;l)),modulo stabilizer I< of a fixed point z,, = i E W. The stabilizer I< is easily seen to coincide with an orthogonal group in R2,
I< = {A:
= i} =
SO(^), so w N SL,/SO(~).
Remark: The hyperbolic nature of the Poincare-Lobachevski geometry stems from its close connection to a hyperboloid r = { z 2 - (z2+y2) = 1) in R3. The former represents an orbit of the Lorentz group Sq2;1), acting by linear transformations of @ N h.03. So the Lorentz metric ds2 = ds2+dyZ - dz2, restricted on I?, remains invariant under Sq2;1). Furthermore, its restriction, ds2 I r becomes positive-definite (since all rtangent vectors [ = (z;y;z) are space-Me, 2 + y 2 - z 2 > O!). So r turns into a Riemannian manifold with a large symmetry group (symmetric space). In fact,
r N Sq2;1)/S0(2), where Sq2) acts as a stabilizer of the vertex (0;O;l)of I?. The connection between I' and D is established via a stereographic map @ (see fig.1). We parametrize
r and D by polar coordinates in the sy-plane, { ( r ; 8 ) : z= 2 1 +r2)for l?,and { ( p , 8 ) ) for D.
Then @:p+r = 2'2P So 1-P
the metric on dr' dsz _-_ -$
is taken into the Poincare metric in D,
r:
l+r2
ds2 = 4
dp2
r2dO2,
+ p2d@
(1 - p y
.
Fig. 1. Sfereographic m a p 9 takes a I': z2 - (z2 + yz) = 1, in hyperboloid R3 -M3, i n f o f h e unit disk D p2 = x 2 + y z < 1, in the xy-plane, and transforms the nafural ( S O ( 2 ; 1)-invariant metric on r into the Poincare metric on D.
51.1. Groups and group actions
17
The relation between I?, D and W suggests that their groups of isometries should be identical. Indeed, we shall see (chapters 5,7) that SL,@) 21 Swl;1) makes a two-fold cover of SO(2;1)(problem 5). Thus we have exhibited 3 classes of manifolds with rich symmetry groups: Euclidian/Minkowski spaces, spheres and the hyperbolic plane. In all 3 cases group G acts transitively on A (each point is moved to any other by an element of G). Hence space A is identified with the quotient G / K , where K denotes a stabilizer of a point xo E A, namely Rfl
21 E,/SO(n);
(I = S q n ) - stabilizer of {O};
Mn 21 P,/SO(l;n-l) Sn-'
N
- stabilizer of {0} in P,
SO(n)/So(n - 1) - stabilizer of xo = (1; 0; ... )
W IISL,(R)/So(2);
or D = SU(1;1)/SO(2); K = So(2)-stabilizer of {i} (or (0))
In fact in all three cases group G acts in a stronger double-transitive manner on A, meaning that any pair of points (2; y } can be transformed into any other equidistant pair {x';y'}, d(e; y ) = d ( d ; y'). This fact has important implications for the analysis on such manifolds. Let us remark that all the Riemannian examples above (except Minkowski) belong to a wide class of symmetric spaces. These manifolds can be generally described as quotients of Lie groups modulo maximal compact subgroups, A N G / K , and we
shall see many more examples in subsequent sections.
1.2. Finite groups. The easiest to describe are commutative finite group. The complete list includes cyclic groups: Z, 21 Z/nZ, which could also be written in the - the n-th primitive root of unity), and complex form: {eiEw:O 5 k 5 n - l}, (w = their direct sums: G = Zfll x ... x Z ,.
2) Symmetries of regular polyhedra and finite subgroups of SO(3). We shall start with regular polygons in the plane.
i)
Symmetries of the regular n-gon consist of rotations by angles EZ , - a finite subgroup of all planar rotations SO(2), or the larger dihedral group, D, = Z, Z, c O(2) - a semidirect product of rotations and a reflection
{gk:k = 0; l...}
about any symmetry axis. D, can be thought of as the symmetry group of the dihedral A,, a solid in R3, built of 2 pyramids based on a regular polygon R, in W2 and a pair of opposite vertices that project onto the center of R, (see figure ). Then Z, implements
18
$1.1. Groups and group actions.
axisymmetric rotations of the dihedral in the plane, while a generator of
Z, flips 2
opposite (spatial) vertices. In 3-D one also has 5 Platonic perfect solids, each one with a symmetry group of orthogonal transformations in
W3 that
preserve its vertices:
ii) tetrahedral symmetries: A, - alternating group (even permutations) of order 4. The generators of A, are cyclic permutations of 3 vertices in each tetrahedral face, that leave the opposite vertex fixed. iii) cubo-octahedral symmetries: the symmetric (permutation) group W,. Cube contains two opposite regular inscribed tetrahedra (fig.2), so its group contains an A, plus an element u that transposes both tetrahedra. Hence Gcube= A,
UaA, = W,.
Fig.:! shows cube with 2 opposit inscribed tetrahedra.
A, - alternating group of order 5 [Cox]. Indeed, a dodecahedron contains 5 inscribed tetrahedra {TI;...;T5}(the set of 20 vertices is evenly split into 5 quadruples). A rotation of order 3 about any pair of opposit verteces leaves a pair of tetrahedra (say T,;T,) fixed and cyclicly permutes the remaining triple (2'5 T,;T,). Obviously, even permutations, {(123);(234);(345);...} generate As! iv) Icoso-dodecahedral group:
Fig.3 demonstrates one of 5 regular tetrahedrons, inscribed inside the dodecahedron. Any pair of opposit vertices ( A ;A') selects a pair of tetrahedra, and any rotation of order 3 about the AA'-aixis, leaves the pair fixed, and cyclicly permutes the remaining triple.
We shall see now that (i-iv) completely describe all finite subgroups of the orthogonal group S q 3 ) . Classification Theorem 1: A n y finite subgroup G of S q 3 ) coincides with one of the
above polyhedral groups (i)-(iv).
19
51.1. Groups and group actions The proof involves several steps. 1) We observe that any rotation U E S q 3 ) has two fixed points on the unit sphere
S2 = {It t 11 = l}, indeed the eigenvalues of an orthogonal U in R2 are X = e
* ip; l!
2) Take set X of all G-fixed points on S2. Set X is G-invariant, hence splits it into the union of G-orbits:
X = w1
G, = { U : U z = z}, and call
u...Uwm.
For each point z E X we consider its stabilizer
I G, I = n2 - the
degree of
(2).
same orbit have equal degrees (stabilizers are conjugate!),
80
Obviously, all points on the nz
= n(w 1.) = n3., for I E w j .
3) Next we count the number of pairs { ( z ; U ) :z E X ; U E Gz} (the “total degree of C”) in two ways: #{fixed points of all U E G\{e}} = “sum of degrees of all {z}”.This yields,
c
2(IGI-1)=
c rn
(IG,l
-l)=
lwjl(nj-1).
(1.3)
+EX 1 4) Dividing both sides of (1.3) by I G I we derive the equation relating the order of G t o degrees of its fixed points:
I
I
The rest of analysis closely resembles a classification of Platonic solids. Namely, one can show that m in (1.4) could take on two values m = 2;3 only! Each case is analyzed separately. 5) In case
m,one easily shows n1 = n2 = n = I G I. So G has two 1-point orbits of
degree n, which implies G = Zn- cyclic!
6) In case1 Case
there are 3 possible subcases:
PI = n2 = 2; n3 = nl. Here G has two “n-point” orbits of deg = 2 and a “2-point”
orbit of deg = n, which implies G = Dn- the dihedral group!
-1
Case
yields 3 possibilities for
n3
3; I G I = 12; Gtetr ( 3 orbits of degree: 2;3;3) 4; I G I = 24; GCube(3 orbits of degree: 2;3;4) 5; I G I = 60; Gdodec(3 orbits of degree:2;3;5)
In a similar vein one can describe symmetries of regular polytopes in higher dimensions [Cox].
3. Other interesting classes arise as automorphisms of finite groups, and regular graphs. Among them we shall mention finite groups of Lie type: G = GLn(F), and
SLn(IF),made of n x n-matrices with entries in
a
finite field IF. There many similarities
51.1. Groups and group actions.
20
in the analysis and representation theory of classical and finite-type Lie groups, but our attention will be focused mostly on the continuous case.
1.3. Compact groups. These are topological groups with compact space G. The E [O;l]}ci Rn/Zn; the classical most important examples include t o w 8" = {(tl;.,.tn):tj compact Lie groups, like orthogonal - S q n ) ; unitary - SU(n); symplectic - Sp(n), et al. The general theory of the compact and finite groups will be developed in chapter 3. The classical compact Lie theory will be covered in detail in chapters 4-5. Other interesting examples arise as matrix groups over p-odic numbers (pprime): Q,, or integers Z,, i.e. the closure of rationals Q, or integers Z, in the padic norm: II a llP = p-". Here n denotes the largest power (positive or negative) of prime p in fraction a = p"a'.
The corresponding p - a d i c Lie groups: GLn(Qp); SLn(Qp), and their compact subgroups
GL,(Z,); SL,(Z,), consists of all Qp(Zp)-valued n x n matrices (respectively matrices of det
= 1). These groups find applications in the number theory and algebraic geometry,
but we shall not venture into the subject of the padic analysis (see [GGP]; [JL]).
1.4. Lie groups form the most important and interesting class, which plays the fundamental role in many areas of Mathematics and Physics: analysis, differential equations, geometry; classical, quantum, statistical mechanics. In general, Lie groups are defined as manifolds with smooth (differential) group structure: multiplication and inversion operations. A brief introduction to the Lie theory is provided in $1.4. The main bulk of the book will deal with the analysis and representations of Lie groups, emphasizing both general aspects of the Lie group theory as well as many specific examples and applications. Lie groups could be divided into two large and distinct classes: solvable and nilpotent (a subclass of solvable); simple and semisimple. The definitions of both classes will be given in 51.4, and the detailed analysis conducted in chapter 6 (nilpotent and solvable groups), and chapters 4,5,7 (simple and semisimple groups). Although both classes differ substantially in their structure, there are abundant connections, and parallels in the harmonic analysis and representation theory. The first class is exemplified by
+
b}- all affine transformations of W. i) 1-D affine group: A f f = {da,*:x-iax This group can be also realized by 2 x 2 matrices of the form { A = b E W;a E R*}. ii) the celebrated Heisenberg group 3 x 3 matrices
W, (and its higher-D
(" i);
cousins Hn), realized by
21
31.1. Groups and group actions
iii) The group B, of upper/lower triangular matrices in GL, (called often the
Bore1 subgroup),
{
B,= A =
r
A, b
1
Xj;a,b,cER,orC
Groups Aff, and B, are solvable, whereas
.
W, - nilpotent.
In $1.4 we shall see that any Lie group can be decomposed into a semidirect
product B D H , of the solvable normal subgroup B, and a semisimple subgroup H . Any semisimple group H in turn can be decomposed into the direct product of sample groups. The latter were completely classified in celebrated works of E. Cartan. They comprise 4 series of classical Lie groups, listed below, and a few exceptional groups. Many classical Lie groups arise as linear transformations of vector spaces over 02, C or quaternions Q, that preserve certain bilinear/quadratic forms, in other words as subgroups of GL, or SL,. These include,
i) Orthogonal groups: q n ) , S q n ) , preserve Euclidian inner product x .y , q n ) = {U:Ux . Uy = z . y } , and S q n ) = {U E q n ) :det U = 1). There are also indefinite orthogonal groups q p ; q ) and S q p ;q), that preserve indefinite products: P P+Q (z y ) = f: z j y j - C z j y j in P+l
I
so that
RPtQ,
or
CP+q,
( U z I Uy) = (z I y ) , for all z,y. The principal difference between the definite-type and indefinite-type groups in the real case is that the former are compact, while the latter are not, as exemplified by the S q n + l ) and the Lorentz S q l ; n ) , mentioned earlier. But in the complex case the difference disappears, so S q p ;q ) N Sqp+q), for p , q. ii) Unitary groups: U(n), U(k;m), preserve definite (or indefinite) hermitian inner product in C“, z . F , or k
( z I w) = ?zjwj
- c zjwj; in cktrn. ktm
ktl
A particular example is the conformal group SU(1;1). iii) SympIectic grou s: Sp(n) consist of all 2n x 2n matrices that preserves a skewsymmetric form J =
[-I IF (z
I z’) = a:.y’-z’.y
= J z .z‘; z = ( 2 , y ) .
$1.1. Groups and group actions.
22
In all cases the corresponding group consists of matrices, that satisfy
U*JU = J , where U* denotes the transpose (or hermitian adjoint) of U , and
J is either identity (for
the orthogonal and unitary groups), or the symmetric form Xkm with k pluses and m minuses on the main (for the indefinite orthogonal and unitary groups), or the skew symmetric form J in the symplectic case. The above list contains most of the classical examples, but it does not represent their classification scheme (see chapter 5 ) , our emphasis was mainly on construction. The reader will find (problems 5,6), that 3 series overlap, particularly in low dimensions. More examples of this nature will come in $1.4 and subsequent chapters. 1.5. Discrete groups. Those typically arise as discrete subgroups of continuous
(Lie) groups, like lattices Z' 2 Z" in W", or their noncommutative counterparts, lattices r c Gf,; SL,, and other Lie groups. Important examples of the noncommutative lattices are i) the unimodular group I' = SL,(Z), and some related subgroups of SL, ii) discrete Heisenberg group: matrices (1.5) with integer a , b,c E Z.
iii) discrete subgroups
r of the Euclidian motion group En.
The latter are called crystallographic gToups, as they describe all possible crystalline arrangements in Euclidian spaces. They were thoroughly investigated, and classified in dimensions 2 and 3. As E, itself such groups are decomposed into a
r =A
U , where A 21 Zn forms a lattice in R", while a finite group U c Gf, acts by automorphisms (linear transformations) of A. semidirect product,
In 2-Dthe complete list contains 17 groups, whose U-components could take only 10 possible values: {Zm;Dm:m = 1;2;3;4;6}. The complete list in 3-D includes 219 nonisomorphic and 230 nonconjugate crystallographic groups. It was worked out by Fedorov, Schoenfliess and Barlow at the turn of the last century. As a step towards classification one needs all finite orthogonal symmetries, derived in Theorem 1. For further details and relates issues we refer to [BH]; [HC]; [Sch].
$1.1. Groups and group actions
23
Problems and Exercises: 1. Show that any rigid motion on R" (Euclidian or Minkowski) is linear, d(z) = A z for some a E R", and matrix A. Follow steps:
i) The Jacobian q5'(z) is an orthogonal matrix, i.e.
+ a,
aid. ajq5 = 6ij
ii) Differentiate the orthogonality relation (i) in the k-th variable, ak(...), then dot,. Show the resulting 4-tensor tLJ = (8,,,+4 Bid) to be symmetric multiply with 84 in both row-indexes: tik = tki; tm3= tJm; and antisymmetric in column-indexes (changing i - j or k-m changes its sign). iii) Show that any such t must be at once symmetric and antisymmetric in the pair of indexes {ij}. Hence t = 0, for all quadruples ijkm, which implies 8'4 = 0, i.e. q5 is linear! 2. The isometry group of S"-' consist of rotations {V E Sqn)}. Hint: any isometry of s n - 1 extends to an isometry of R"\{O} N S"-'xR+. But the latter are linear by problem l!. 3. Show that fractional-linear transformations (1.2) of SL, on H, or SU(1;I) on D, are isometries. 4. Show that all finite subgroups of S q 3 ) are either polyhedral, or polygonal groups, described in Example 1.1.
5. Establish and find explicit form of the isomorphism SL,(R)+SU(l;I), using their fractional-linear actions on H and D, and the Caley transformation 4: z - + G , from M z+z
to D. Notice that both are subgroups of the larger complex unimodular group SL,(C), with a Caley element u E SL,(C). Find and can be obtained by conjugation, U+U-'UU, U!
6. Show that the symplectic group Sp(1) coincides with SL,. 7. Show that complex groups S O ( p ; q ) and SO(p+q) are isomorphic (Hint: the indefinite product ( z 1 w ) in Cp+* is equivalent to the definite product z . w by conjugation with a complex diagonal matrix).
$1.2. Regular and induced representation
24
51.2. Regular and induced representations; Haar measure and Convolution algebras. We introduce two basic concepts of regular and induced representations, discuss continuity and unitarity and develop some basic algebraic constructions: direct sum, direct integral, tensor product. In the process we introduce the (invariant) Haar measure on groups and homogeneous (coset) spaces H\G, define convolution (group) algebras L’(G), and find the links between representations of the group and those of its groupalgebra.
2.1. Regular representations: In the last section we have described some examples of transformation groups acting on various geometric structures (graphs, manifolds, symmetric spaces), including the action of group on itself by the right/left multiplication (translation), g: x-+xg; or x+g-’x. With any such action of group G on space X (denoted by x - + x g ) , we can associate a
linear
G-action on functions over
homomorphism of G into linear operators over
X, {f(x)} = e = e ( X ) , i.e. a
e,
In particular, the right/left translations, x-+xg; x+g-lx,
on G give
IR,f(x) = f ( g - ’ z ) (left), or f ( x g ) (right), on e(G)l Homomorphisms g+T,, vector space
Y,are
(2.2)
of group G into linear transformations/matrices on a
called representations, and formulae (2.1)-(2.2) provide important
examples of regular representations of G on X, or on itself. In the analysis of group actions and associated representations on various function-spaces, it is often necessary to “integrate” over G or X . So we need some measures on X and G . 2.2. The Haar measure. It turns out that all discrete and continuous (locally compact) groups G , as well as large classes of homogeneous spaces6 X = H\G have an invariant measure, dp(xg) = dp(x), for all x E X, g E G, called the Haar measure. The general proof will be outlined below. More important, however, will be to compute the Haar measure in a suitably chosen coordinate system on group G or space X. Here we shall list a few examples of Haar measures on groups and homogeneous spaces (many more will appear throughout the book). ‘homogeneous (quotient) space X = H\G, or G / H , consists of all right/left cosets {z = Hgo:gOE C } , or {z goH:go E C}. Group G acts on X by right/left translations, g:z+zg = ( H g o ) g , or 2-9(z).So X could be viewed as a G-space, with a transitive G-action, and subgroup H coincides with the stabilizer of a fixed point, H = {g:rog= zo}.
$1.2. Regular and induced representation
a) finite/discrete group G has dp(z) =
25
C C ~ ~sum ( Xof) &-functions over
all
Y
y E G.
b) for commutative groups Wn;Tn, dp coincides with the standard Lebesgue measure (volume element), dz = dxldz, ...dx,. We have shown in $1.1that Rn could be regarded as a homogeneous space of the Euclidian motion group G = En. The Lebesgue measure is clearly invariant under all translations and rotations on
W", hence it also
forms a G-invariant measure on homogeneous space W" = E n / S q n ) . c) on compact Lie groups S q n ) , SU(n), the Haar measure can be explicitly calculated in terms of suitably chosen coordinates, like Euler angles (chapters 3-4). The same could be done on non-compact (Lorentz, conformal, etc.) groups, like S q 1 ; n ) or
SU(1;n ) , by a combination of spherical and hyperbolic angles.
W is a homogeneous space of dz d y measure d p = -is G-invariant.
d) the hyperbolic (Poincare-Lobachevski) half plane
G = SL,(R). One can check (problem l), that
Y2
e) GLn(W) has a natural set of coordinates, matrix entries {zjk}'&l
of g. One can
show (problem 6) the invariant measure on GL,
dPk)=
&=
ndXjk. Jk
For more examples see problems 5-8.
Existence Theorem: i) On any compact (and more general locally compact) group
there ezists a Bore1 measure dp(x), positive on all open subsets, finite on all compact subsets and invariant under all right (or left) translations: EX) = p ( E ) , for all subsets E c G and elements x E G. ii) A right and left Haar measures: d,x; d,x are unique up to a constant factor. iii) On a compact group G the left and right Haar measures are equal, furthermore dp is invariant under the group inversion and conjugation,
dp(x-') = dp(x), dp(g-'sg) = dp(z). Let us briefly outline the proof.
Existence: To construct a right-invariant measure dp, we pick a small neighborhood U of the identity e E G to serve as a gauge. Given a (open, closed) subset E optimal (least) covers of E and G by translates of U , m n E C u U z j , G = U U yj*
j=1
j=l
C G, we choose
51.3. Regular and induced representation
26 The ratio
represents an approximate relative size of E in G in the ”U-gauge”.
m ( E ; U ) as U-.{e}, (by standard arguments such limit always exists!), one Taking limit n(C;U ) gets an honest Borel measure on G. The limiting measure, dp, thus constructed is easily seen to be right-invariant. Also p ( E ) > 0 for any open E, and p(G)
< 00 for a compact
G , since G is covered by finitely many translates of E. Uniqueness: If dji is another right-invariant measure, then for small U ,
m ( E , U ) ef-i ( E ) while n ( G ; U )e m(U)’
subsets E.
fi(U)’
whence p ( E ) = lim
U-lel .
$ = Const f i ( E ) , for
all
I
iii) T o show that the right and the left-invariant Haar measures on a compact G are equal, we take a left translate, d j i ( z ) = dp(az), of a right-invariant measure d p ( z ) . Of course, dji is also right-invariant. Hence dji = p ( o ) d p , by the uniqueness of dp. The map a+p(a) is a homomorphism of
G into the multiplicative group R+. But there are no such
nontrivial continuous homomorphisms on a compact group (continuous functions on compact sets are always bounded, whereas nontrivial homomorphisms p: G-R,
are
unbounded!). Hence, p = 1, and d p is also right-invariant. Two other invariance properties of d p easily follow now. Indeed, change of variable 2-z-l
takes a right-invariant measure into a left-invariant measure, the conjugate-
invariance immediately follows from the bi-invariance: d p ( a r b ) = d p ( z ) for all a, b. This completes the proof. R e m a r k In many cases (e.g. Lie groups) the right and left invariant measures are dlb) absolutely continuous one relative t o the other. The corresponding density -= A(g) is 4(9) a character (homomorphism into the numbers) of G with positive real values
A:G+R,; A(gh) = A(g)A(h), called the modular function. Groups with equal right and left Haar measures, A(g) = 1, are called unimodular. Large classes of groups are known t o be unimodular, for instance, all compact groups, semisimple and nilpotent Lie groups. Nonunimodular examples include: a f i n e groups and more general Borel groups of all upper/lower triangular matrices (problem 8).
The invariant (Haar) measure, topology/metric, and the differential structure on group G allow to int,roduce a variety of function-spaces. The most important among those are
C(G) - continuous (bounded) functions {f(z)}on G, with norm
II f II 00 = S U P I f(z) I ZEG
7
$1.2. Regular and induced representation and its subspaces functions; 0
e,, -
LP-spaces on
27
functions vanishing at {cm},and C, - compactly supported
G with respect
to the Haar measure dz,
in particular Hilbert space LZ(G)with the standard inner product
0 For Lie groups G (smooth manifold) one can consider a variety of differentiable function-spaces: Y ( G ) - Um-smoothfunctions”, C”(G) - infinitely smooth functions; their subspaces of functions vanishing at {m}, or compactly supported ep, with norm
er;
sum over all
llfllrn = C S U P Jaaf(41, XEG “partial derivatives” of order I a I 5 m (the meaning
: f l * f z ) ( z= )
J
fl(zy-’) f z ~ d )y = G
J
of “derivatives”) on
~ I ( Y )f z ( ~ - ’ z d5 )
(2.3)
G
integration with respect to the Haar measure d y on
G. On
finite/discrete groups
integration in (2.3) is replaced by the sum,
f*h =
c
f(zy-’)h(y).
YEG
The meaning of convolution, as an extension of the group multiplication to functions, becomes transparent here, as each group element {z} is identified with the delta-function 6,, so that
31.2. Regular and induced representation
28
6,* 6, = SZy;for all x,y E G. For continuous groups
G convolution-integral (2.3) is well defined on compactly
supported functions (ec;ep),then it extends to larger classes by the standard density arguments7. Spaces closed under convolution forms convolutzon/group algebras. One important example is space L'(G) of all integrable functions on G. Here,
Ilf*hli
(2.4)
A similar estimate holds for a convolution of L' and LP-functions,
1 f* h Il p 5 (1 f 1 IIh IIp ; for all pairs f E L', h E LP.
(2.5)
In the standard terminology L' is a Banach algebra, while other LP are modules over L'. Estimates (2.4)-(2.5) follows from the well known Minkowski inequality,
applied t o function F = f(zy-')h(y).
The inequality (2.6) has an obvious interpretation,
any function F ( z ; y ) (z E X ; y E Y) defines a map F:Y-.LP(dz), then (2.6) claims:
1 JF(y)dyll<
lIIF(y)Ildy, where
(1 11 means the LP(dz)-norm. Other examples of
convolution-algebras include spaces of continuous and differentiable functions:
em,
e, em,
as well as bounded (Borel) measures on G, A ( G ) 3 L'(G). Condition (2.5) can be
stated as boundedness of the convolution (bilinear) map, L'*LP-LP,
or L'*LP
c LP, 1 5 p 5 03.
It is also easy t o see that L2*Lz C Loo, and Ilf*hll, 5 llfl1211 hllz (Cauchy-Schwartz). Such results can be extended to many other triples: LP*Lq
c L', and other "scales of
function-spaces" by interpolation discussed in chapter 2 (J2.2).
2.4. Continuity and unitarity. The study of group representations combines two
aspects: algebraic, which reflects purely algebraic properties of groups and operators, and analytic (topological), which involves the geometric structure of manifolds A (differentiable, complex, etc.), as well as topological properties of the relevant vector (function) spaces and operators over A. In the topological context (locally compact/Lie groups G, Banach or Hilbert spaces V) one needs to impose some continuity assumptions on 'Spaces
ec and ep are dense in all Lp, 1 5 p < co!
T.One possibility would
$1.2. Regular and induced representation
29
be to require continuity of the map g+Tg in the operator (norm) topologfl on space %(V), of all bounded operators on T. This notion, however, turns out to be too restrictive, (see problem 2, and the author's paper ', JFA, 35, 1980).
A weaker notion of continuity asks vector functions {TgfiG+T}, or the matrix t E T*-dual space} to be continuous. It is called entries of T , {tcv(g)= (T,J I ~ ) E: T;Q strong continuity. Strong continuity holds in most natural examples, including regular representations (2.1), (2.2), and (2.8) below, in different function-spaces: e; em;LP(G), etc. (both on G and on homogeneous spaces X=H\G, G modulo a closed subgroup H ) . Let us remark that quotient-spaces X of topological groups often inherit basic structures of G, like metric, differential structure, and in some cases G-invariant (Haar) measure. An important class of group representations consist of unitary representations in Hilbert (inner product) spaces:
The natural example is furnished by the regular representation R on squareintegrable functions V = L 2 ( X ) ,provided space X has a G-invariant Haar measure, d ( x g ) = dx, for all g E G. The Haar measure was shown to exist on any locally compact group G, but quotient spaces X = H\G may or may not have it, as we shall demonstrate. The invariant measure always exists, if subgroup H is compact, e.g. on symmetric spaces. However, in any case a homogeneous space (manifold) X comes equipped with some measure (volume element) dx. Group elements ( g E G}, acting on X transform dx into another measure dxg, which will differ from dx by a density factor: In case of the smooth (Riemannian) manifolds and G-actions (e.g. a(x;g) symmetric space X ) , a represents the Jacobian determinant of the map 4: x - d ,
=%.
a ( x ; g )= d e t # ( x ) .
(2.7)
We can still construct a unitary representation of G on L * ( X ; d z )by combining the group (translational) action on X with multiplication by fi,
Function a ( x ; g )is easily seen to obey the so called cocycle condition, 'We remind the reader that the space of all bounded linear operators in a normed vector space 11(11< l}, or sup{(A( 7):all ( ; q } in the Hilbert space setup.
Y is equipped with an operator-norm: 11 All = sup{IIA(Il: all
'We have proved that any norm-continuous irreducible T must be finite dimensional, which automatically excludes large and important classes of m-D representations.
51.2. Regular and induced representation
30
4";9192)= +;g1)4z91;gz); all gI;gZ E G , which yields the representation property for operators T , (2.8),
(2.9)
T g l g 2= Tg1Tg2, for all gr;g2in G . Obviously operators {T,} preserve the L2-norm of
f, so we get once again a
unitarity representation of G on L 2 ( X ) !
2.5. Induced representations. Regular representations (2.2)-(2.1) on G and quotients H\G, as well as more general representation (2.8) are obtained by the process called induction. In general, induction allows to construct representations of group G , starting from representations of its subgroup H. An alternative definition is based on the notion of cocycle u(z;g)on the homogeneous space X = H\G, i.e. a function u: X
x G + {numbers} or {operators},
that satisfies the cocycle condition (2.9)
a(z;glg2)= a(z;gl)u(xgl;gz);for all gl;g2 E G , x E X . The induced representation, T = ind(a I H ; G ) , acts on the space of (Y-valued) functionslO on X,
e = C(x;Y)
= {f(z) : X + Y },
and is given by
We leave to the reader, as an easy exercise, to check that (2.10) does give a representation of G. Furthermore, if u is continuousfunitary and X has a Haar measure
d x , then T , also becomes a continuousfunitary representation'' of G on space L 2 ( X ; Y ) , of square-integrable Y-valued functions on X (problem 9). Let us remark that the cocycle condition (2.9) for elements of stabilizer H = G, = {g: xg = z}, yields a representation ~ ( h=)a ( z ; h )of H on
Y,u ( h l h 2 )= u(hl)u(hz).So one could start with a
representation u of H and construct a cocycle u(z;g)!This alternative way to define 2nd will be discussed in $3.2. Regular representations R on G and quotients X = H\G, are induced by the trivial cocycle u,
R = ind(1 I H;G). '"In the topological (Lie) setup one takes a suitable function-space on X, e.g. continuous, differentiable, L2-functions, etc. "In the absence of invariant measure on X we can modify operators T, (2.10) by a scalar (2.7) to make T unitary. Obviously, the product of two cocycles (I and makes factor another cocycle!
fi
$1.2. Regular and induced representation
31
We shall see the induction procedure to appear throughout this book in various places and contexts, in the representation theory of finite, compact, simple, nilpotent, solvable groups alike. The role of induction will be twofold: on the one hand it extends the notion of regular representations on G and X to vector functions and sections of
vector bundles12, on the other hand many irreducible representations of groups will be constructed via induction. Here we shall illustrate the induced action by two examples of homogeneous spaces H\G, without G-invariant measure (see problem 4), and construct the relevant regularlinduced unitary representations.
+
1) Affine group G = {g = (a;b):z-iaz b = zg} of $1.1 acts transitively on R z H\G, where H denotes the stabilizer of {0}, H = {(a;O)} R* (multiplicative group of reds). The “az+b” action on R has no G-invariant measure. Indeed, the translational invariance ({b}-part) requires dp to be Lebesgue, but multiplication with the {a}-factors dilate it, dp-‘ 1 a 1 d p . However, we get the cocycle a ( rr g) --@dz =lab
(2.11)
hence a unitary action of G on L2(R),
The representation theory of affine groups will be discussed in chapter 6 (56.1). 2) Group G = SL,(R) and its subgroup of upper-triangular matrices
H
=
{(R akl)}
yield the quotient-space X N R, with G acting by the fractional linear transformations, g: 5
ac+b
3 cz+d9
for
(2.12)
=
Indeed, cosets z E H\G can be parametrized by matrices
{rz = (i
coset representatives, via the Gauss factorization: (almost) any element g = factors into the product,
(: 3 = (d-l
$(b/ld
l):z E R}
-
(ii), ( d # 0 )
1) =
To find the cocycle we take the product yzg, and factor out the h-term, y z . g = ( az+b a cr+d ) = h . y ,
where
vector bundle TI consists of the base manifold X , stuffed with fibers (linear vector spaces) {YZ} associated to points of z E X . The natural example is the tangent bundle, made of all tangent spaces {Tz: z E X } of a smooth base-manifold.
A-
51.2. Regular and induced representation
32
The density factor a is computed from the transformation law (2.12) to be a(t;g) = 1 ct+d
I - 2.
(2.13)
So a unitary representation of SL, on space L2(R) takes the form,
More general representations of SL,(W) on Lz(W) are obtained by taking complex powers of the cocycle,
(2.14) These are so called principal series representations of SL,, that will be studied in detail in chapter 7.
2.6. Representations of group algebras. Any group representation T, gives rise to a representation of a suitable group-algebra L = {f}, of functions on G, by integrating T against f , (2.15) It is easy to check that map f+Tf satisfies
Tf*,, = TfTh, for any pair f , h E L . Conversely, any algebra-representation f -tTf, subject to a minor technical assumption (span{T,('V):f E L} is dense in 'V), can be shown to come from a group representation g-tT, via (2.15). As usually, in the topological context one has to give a meaning to integration (2.15) and to find a proper class of functions { f } on G. If T is continuous (which is always the case), and f - compactly supported, integral obviously exists and yields a bounded operator, )IT,II 5 Jlf(~)IIITllld..
Tf
(2.16)
Estimate (2.16) also indicates the class of functions, for which T , can be extended, as a bounded operator. The latter depends on the behavior of function w(z) = IIT,II on G. Clearly T f extends to the space (algebra) of functions, integrable
{f:/lf(z)lw(s)ds < m}. In particular, unitary (or more general bounded) representations, I(T,(I5 C, yield bounded operators {T,} for all f in the group algebra with weight w,
L = L'(G), indeed, I)T,(II Cllflll. If T is a left regular representation (2.2) on G, then operators (2.15) are nothing but left convolutions with f,
51.2. Regular and induced representation
33
R f [ h ]= f*h. The main advantage of passing from the group representation T , to its integrated form (2.15) has to do with nice (smoothing/regularizing) properties of operators { T I } . To wit the “unitary/bounded group operators” {T,} could yield a better class of “algebra operators” { Tf}:compact, Hilbert-Schmidt, or trace-class. The latter often depends on the regularity properties of
L = em(G);P‘(G), the “smoother11
{ f ’ s } one takes the “better” { T I }results. A simple illustration is furnished by convolutions with f E Cm(G) (or Coo). It is easy t o check (problem l l ) , that any such operator
R,: ZP-.Cm
(P), i.e. Cm*ZP C em,for all p, m.
In other words, convolutions with m-smooth
{f} transform ZP-singular functions { h } into
m-smooth functions. Smoothing properties of convolution operators with nice (smooth) functions are well known and frequently utilized techniques in harmonic analysis and differential equations, that go under the name of regularization methods. We shall use them extensively in subsequent sections, but here we just give a simple illustration of smoothing by convolutions in problem 10.
2.7. Algebraic constructions. We shall conclude this section with a few algebraic constructions of representations. Direct sum of {T’; Tz;...}lacting in spaces Yl;Y,...is defined in the obvious way by taking the direct sum of spaces Y,@ Y2@ ... , and setting,
(T’ @ T2@
...)g((1;(2;...)
= TL(1@Ti(, @ ...; (j E Yj.
(2.17)
There is a continuous version of (2.17) called the direct integral decomposition. Given a family of representations {TX:XE A}, in spaces {X’} (assumed to be regularly embedded in the embient space X), and a measure dp on A, we define direct integral X’dp(X), to consist of all L2-vector valued functions on A, that for each X E A, take on values in 36,: F = f(X) E X’. We then define direct-integral operators,
to map vector-functions f(X)
+ Tx[f(X)].
Tensor (Kronecker) product T’ &I T 2 is defined in the tensor-product space’
Y1@ Y,: (T’ @ T2),({&I 17) = Tb{ &I Tiq.
81.2. Regular and induced representation
34
An analytic version requires a proper topological product
Y,@ Y2.
Thus for
regular representations R3 ( j = 1,2) in spaces YJ = L 2 ( X j ) , the product space Y = Y 1 @ Y 2 L~ ' ( X , x X , ) , a n d G a c t s o n Y b y
Rgf(";Y) = f(& Y,). The contragredient (dual) representation Pg = Ti-,, in the dual space Y* to Y.Here T* denotes the dual (adjoint) operator to T. 0
The adjoint representation: Ad = AdT in the space L(Y)-all linear transformations on Y, or its subspaces, Ad,(A) = Ti'AT,, A E L(Y). Space L(Y) is naturally identified with the tensor product of Y 8 Y' (Y by its dual Y'), where tensor (@e+"rank-one operator (el ...)?. Then one can show that AdT= T@F - tensor product of T and its contragredient. A natural example of Ad is the action of GL, by conjugations on the space of all matrices, g:A+g-'Ag; g E GL,, A E Mat,,. More generally, the adjoint action appears in the context of Lie groups acting on their Lie algebras, and will be explained in 81.4. 0
-Tensor product of 2 vector spaces W = Y,@ Yz is spanned by linear combinations of {( 8 q:( E Yl;q E Y2}, subject to the natural bilinear rules: elemental products ( ( + q ) 8 ( = < @ ( + q @ { , and similarly for ( @ ( q + ( ) . The easiest way to think of W is in terms of function-spaces: Y,N e ( X ) , Y2 N e ( Y ) (for finite-D spaces Y,sets X and Y are finite!). Then W is also a function-space e ( X x Y ) , where elemental tensors {( 8 q ) correspond t o product-functions {<(z)q(y)). Such correspondence extends to some infinite function-spaces, e.g. L 2 ( X )8 P ( Y ) u Z2(X x Y ) .
35
51.2. Regular and induced representation
Problems and Exercises: 1. Show that the fractional-linear action (2.12) of group G = SL,(R) on the Poincare, and that d p is the Lobachevski half-plane H leaves measure dp = ~ - ~ d z d yinvariant only G-invariant measure of H. 2. Show that the regular representation of G = R" (1")is strongly continuous on various function spaces over R" (1"): (i) (ii)
e, = {all continuous functions vanishing at co} with
norm 11 f 11 = ma4 f(z)I;
e r = {m-times
continuously differentiable functions, vanishing a t co), with norm IIfIIm= C la1 <mmazl'of(z)I; (iii) C F = Q with a sequence of seminorms (11 f m=O; 1; ...)
er;
Ilm:
(iv) Lebesgue Lp-spaces, 1 5 p
< 00, with l l f l l p = (llf(z)rd~)'/~.
Hint: Verify continuity of the map a + R , f , for compactly supported functions f in e, or (uniform continuity!), then utilize the standard functional analysis: density of compactly supported f ' s in e,, and LP, and the uniform boundedness of the family of translations { R,: Q E R"}, R , = I!
er
1 1
The above argument applies t o other groups G and homogeneous spaces X, and their regular representations R on e ( X ) ;Zp(X), etc. as long as transformations g: z+zg, are continuous and measure preserving, i.e. X has a G-invariant Haar measure. (v) The regular representation R on G or X = H\G is not uniformly continuous, as
1 R , - 111= 2, for all g in the vicinity of {e}.
3. Check that any cocycle a ( z ; g ) defines a representation T = T f fvia formula (2.8). Show that the product, a@, and a power 19of cocycles, are also cocycles, hence T" of (2.14) is also a representation. 4. A cocycle (I is trivial, if a(z,g)= @(z8) for some function p(x).
4.)
(i) Show that cocycles (2.11) and (2.13) are nontrivial. (ii) A trivial cocycle yields a Haar measure on X. Thus in both examples (affine and SL,) the quotient H \ G has no Haar measure.
5. Show that the Haar measure on a semidirect product G = H D K, with compact group K (e.g. Euclidian motions En = R" D Sqn)), is equal t o dg = dhdk - the product of Haar measures (Hint: Haar measure is invariant under any compact group of automorphisms).
6. Show that the Haar measure on SL,(R) = {g = (5
!) detg = l},
Use the Haar measure on GL, = SL, x R', write any g E GL,
is given by
as
and compute the Jacobian a(a; b; c; d ) y; 2;A)
' ( 2 ;
=I
I
Xz'
.'7 Show that the Haar measure on SL,(R) parametrized by its entries (excluding zll!)) is equal t o
d
g
=
1
{g
= (zjk):j + t > 2
36
51.2. Regular and induced representation where M , , denotes the 1,l-minor of g. Formulae of problems 5-6 look elegant, but found little practical use in harmonic analysis on SL,. The latter exploits a different set of coordinate, based on certain factorizations of SL, (Cartan, Iwasawa), described in chapter 7.
8. a ) Find the left and right Haar measures on the atline group G = {az and compute the modular factor: A(a; b) = l/a.
+ b } over R, or C,
b) Do the same problem for the Bore1 group of all upper-triangular matrices
9. Show that formula (2.10) defines a representation T of G. The induced representation is continuous/unitary, provided u is continuous/unitary and X has a G-invariant measure. Another simple (functorial) property of ind, ind(ul @ u z )= ind(ul) @ ind(u2).
10. Consider convolutions on the unit circle G = 1, 2rr
~ , p=] f * h = J f ( t - e) q e ) de, on L2-space ( h E L*), and show
0
i) L'-functions f yield compact operators R , ii) L2-functions f yield Hilbert-Schmidt operators R , iii) Cm-smooth functions (rn 2 2) yield trace-class operators. Hint: use Fourier expansion of f = C a m e i m t ; and some basic facts of Fourier analysis on 1: (a) f E L'*{a,}-O, as k-m; ( b ) f E L 2 e { a , } E e2; and (c) f e e m 3 a, = O(k-m), as k-m. We remind the reader that compactness for diagonalizable (self-adjoint) operators means Hilberl-Schmidt means C I A, I < 00, and trace-class is that their eigenvalues A,+O, equivalent to C I A, I < 00 (see Appendix B). 11. Show that convolutions f * h with f E em on groups R", T", take any LP-space back into Cm (interchange integration J ...d y , and differentiation 8:). The same argument works on any Lie group and its homogeneous spaces, when integration and differentiation are G-invariant!
$1.3. Irreducibility and decomposition
37
51.3. Irreducibility and decoinp~tion. Section 1.3 studies irreducibility and decomposition of representations. We establish basic irreducibility tests: Schur’s Lemma and Burnside’s Theorem, discuss direct sum/integral decompositions, equivalence, intertwining spaces, and outline some basic problems of the representation theory.
3.1. Irreducibility. One of the main goals in the representation theory is to
analyze the structure of any representation T in terms of its simple constituents, called
irreducible representations. In this section we shall discuss some general features of irreducibility and decomposition.
A representations T of grouplalgebra in space Y is called irreducible, if operators { T g }have no joint nontrivial invariant s~bspaces’~:Y~ c Y; T , I Yo c Yo, for all g E G,+ Yo = {0} or Y. An
invariant subspace Y 0 c Y reduces
T , so one can talk about a
subrepresentation (restriction) T“ = T I Yo, and the factor-representations on the quotient-spaces Y/Yo.Irreducible representations play the role of elementary building blocks, in the decomposition of an arbitrary T . In this respect they resemble eigenvalues of linear operators, or characters (joint eigenvalues) of commuting families of operators. Two general results provide useful irreducibility tests: Schur ’s Lemma and Bumside (von Neumann) Theorem. Each has a purely “algebraic” (finite-dimensional) version, and a “topological” (unitary) version. Since the arguments are very similar in both cases, we shall discuss them in parallel. Given a representation T , we introduce its
commutator algebra:
Com(T)={A:AT, = T,A, for all g E G} - all operators that commute with T . Schur’s Lemma: (i) An irreducible finite-D representation T in complex space Y has scalar commutator algebra, Com(T)= {XI}. (ii) A unitary representation T is irreducible, iff Com(T) is scalar.
Proof: (i) If Com(T) has a nonscalar operator Q, then any (nontrivial) of Q, is obviously invariant under T , which means
eigensubspace E , = {(:Qt = A(} reducibility.
(ii) The unitary version requires some basic facts of the spectral theory of unitary and selfadjoint operators, described in Appendix A. Let us remark that algebra -In the topological context (Banach/Hilbert spaces subspaces {Yo}.
Y) one usually considers closed invariant
81.3. Irreducibility and decomposition
38
Com(T) is closed under conjugation Q+Q* (adjoint operator to Q ) . So a nontrivial algebra Com(T)# {XI} has a nonscalar selfadjoint operator Q. The role of the eigenspaces in part (i) will be played now by the spectral
subspaces15 {E(A)} of Q. Each subspace E ( A ) is known to be invariant under the commutator algebra of Q , since the corresponding spectral projections { P ( A ) } are approximated by “functions (polynomials) of Q . The argument is easily completed now. A nontrivial Q = Q* in Com(T) would yield a nontrivial T-invariant spectral subspace E(A). So irreducibility of T implies Com(T) = {XI}. Conversely, if T has an invariant subspace Yo, then the orthogonal projection P: T-+Y,,
gives a nonscalar
element of Com(T),QED. Let us remark that for general (nonunitary) T , Schur’s Lemma gives only a necessary condition of irreducibility. It is easy to find examples of reducible T with scalar commutator, for instance, all upper-triangular matrices in Cn (problem 3). An immediate corollary of Schur’s Lemma is Corollary: A n irreducible finite-D or unitary representation of a commutative
group/algebra is one-dimensional,
T, = x ( g ) - a character of G.
Remark: There is a version of Schur’s Lemma for representations in real spaces Y. Here irreducibility of T does not always imply Com(T) to be scalar. But the commutator algebra must contain nonsingular matrices only, hence it makes a division
algebra. There are only 3 such beasts: reals W (the corresponding T a r e called real-type); complex numbers 43 (complez-type T), and quaternions Q (puaternionic-type T). All 3 cases can be characterized in terms of complexification of T , i.e. the family of operators {TC(E+iq)= T[+i(Tq):(,qE V}. Then “real type” yields an irreducible representation TC= T @ C; “complex-type” - a pair of inequivalent (conjugate) representations: TCN T I @ T while quaternionic-type results in a pair of equivalent representations:
TCN T $ T . A simple illustration of this statement is given by 3 algebras, Iw;C;$, themselves, of a selfadjoint operator Q in Hilbert space 36, consists of { A E R: so that (AZ-Q) is not boundedly invertible). Spec(Q) is always a c l w d subset of R, and operators with 1-pointed spectrum, spec(&) = {A), are known t o be scalar, Q = XI. Furthermore, any closed subset A C R has an associated spectral subspace E ( A ) and the corresponding spectral projection, P ( A ) : 36-+E(A).Spaces { E ( A ) }are invariant under Q, and have spec(Q I E ( A ) )c A n Z . For operators with discrete spectrum { A k } , space E ( A ) = @ E(Aj),consists of all eigensubspaces with A j E A . An easy way to obtain { E ( A ) ) is to use the canonical model of a selfadjoint operator Q,namely, a multiplication: f(A)-,Af(A), on the space of square-integrable (scalar or vector) functions on C = spec(&), f E Z*(Z;dp). Then E ( A ) consists of all L2-functions vanishing outside A (see Appendix A). -Spectrum
$1.3. Irreducibility and decomposition when realized by matrices (problem ll),
R = { [ a ] ; a E W}; 43 =
{;[
] a , b E R}; and
Q=
39
{b
: ] u , b E C}.
Burnside (von Neumann) Theorem: (i) A finite-D representation T in space
(3.1)
Y is
irreducible if and only if t he algebra A = A(T), spanned by T, coincides with the full operator/matriz algebra L(Y) N Matd, where d = d ( T ) = d i m Y , called the degree of T. (ii) A unitary (infinite-D) representation T in Hilbert space 36 is irreducible, i f and only af the closure of A ( T ) in the weak operator topology16 coincides with the algebra of all bounded operators %&).
Sufficiency follows from the irreducibility of the full matrix/operator algebra Mat,/%(%). The proof of necessity: “irreducibility” ==+ ‘@ll-operator algebra”, exploits a
L‘bootstrap’’argument (see problems 7,8). In the m-D case one could not expect the purely algebraic (Burnside’s) version of the Theorem to hold, as there are many transitive operator-algebras, different from %(%), like all finite-rank operators 5,or compact operators R in 36. But the weak (operator) closure of both coincides with
%(36)! A Theorem of von Neumann extends the algebraic result to symmetric transitive subalgebras A c %(%), by combining LLbootstrap”with some standard functional analysis, to show that A must be weakly dense in ’3(36).
3.2. Decomposition. If a finite-D representation T on space 36 is reducible, one X0,so that T I Mo becomes irreducible.
can always find a minimal invariant subspace
Hence, any such representation can be brought into a ‘quasi-triangular form” with irreducible diagonal blocks. But for unitary T each invariant subspace 36, has also an Tinvariant orthogonal complement 36, = 36 8 Xo. Therefore, quasi-triangular reduction for unitary T amounts to LLquasi-digonalizationnor complete decomposition. Proposition 1: A finite-dimensional unitary representation can be decomposed into
the direct orthogonal s u m of irreducible components: T = 9 Ti, i.e. 36 = @ Mi3 3 direct s u m of invariant subspaces, and T3 = T I Xj. In other words a unitary finite-D representation is completely reducible. Thus irreducible representations provide the “building blocks” of which any T is made. In this sense Proposition 1 extends the well-known diagonalization of a unitary/self-adjoint
16Weakly dense algebra A means that any operator B E %(%) can be approximated by r) E %}, equivalently, by all traceoperators { A E A } , relative to a sequence of seminorms { ( A t I r)):t, class operators K , identified with functionals on %(%), via the natural pairing K : A-.tr(KA). In other words the distance between operators { A ; B } is measured by n-tuples of vectors {ti;qi: 1 5 j 5 m}, or trace-class operators { K } : { ( ( A- B ) t j I qi)} or {tr K ( A - B ) } .
51.3. irreducibility and decomposition
40
operator (Appendix A). There exists a continuous version of Proposition 1, called the direct integral decomposition of a unitary representation in m-D case. We briefly introduced it in the previous section and will discussed later in 56.1. Let us remark that the decomposition of Proposition 1 is not unique in general, due to possible multiplicities of irreducible components of T , the same way as multiple eigenvectors of a matrix are not unique. We shall describe a canonical (unique!) decomposition of T into the direct sum of multiples of irreducibles, ?r@m,where m rn(?r;T)denotes the multiplicity of
?r
in
T.
Theorem 2 (primary decomposition): A unitary finite-D representation T in space 36 is uniquely decomposed into the direct sum of multiples of irreducible
representations, i.e. % = @36, - direct sum of invariant subspaces, so that k T/%, N x k @ mk - a multiple of an irreducible ?rk, and
(3.2) with mk = m ( x k ; T )- multiplicity of 'a in T . Multiples of irreducibles Irk}are often called primary components of T , spaces { % k } are primary subspaces, and
(3.2) is referred to as a primary decomposition of T .
The main difference between Theorem 2 and Proposition 1 is that the primary subspaces Xj = %(d) are uniquely determined, whereas within each Nj there are many ways to break T
I Xj into the sum of irreducibles. So the primary subspaces represent
noncommutative analog of the eigenspaces of a linear operator. There are different ways to establish the primary decomposition. Some depend on specific structures of groups in question. For instance, for finite and compact groups G (3.2) could be derived via irreducible characters of G and the associated primary projections (see
chapters 3-5). Another method is based on central projections of the commutator algebra
Com(T). Here we shall exploits matrix entries of T: given a pair of vectors ( , q E 36, we def ( 9 ) = (Tg( I q ) defines a function on G. For each vector ( E 36
define the matrix entry t
€v
we introduce a cyclic subspace, Y (() = Span{Tg(: g E G}. Obviously, under T. Any vector q E 36 defines a linear map,
Y(() is invariant
9:, (-+(Tg(I q ) , that takes V(() into a
subspace '3 of functions on G (matrix-entries), invariant under right translations, f ( z g ) E GJ, for any f E '3. In fact, Qq intertwines the subrepresentation T regular representation R 1'3,
RgQq = QqTg,for all g E G.
I Y(()
and the
a
51.3. Irreducibility and decomposition
41
This shows, in particular, that any irreducible T can be embedded into the regular representation on a suitable function space. We say that vector (E 36 transforms according t o an irreducible
A,
if all matrix-entries { t
span T ( A ) of matrix-entries of
A,
(2): r)
€r)
a R-invariant subspace of L(G). An equivalent
statement: all irreducible constituents of T I Y(() are copies of claim that T I 36(()
E 36) belong to the linear
T , 80
T I Y(() u T 8 rn. We
u A has multiplicity 1. Indeed, by Schur’s Lemma one can show that
the only cyclic subrepresentations of
S = A 8 m are copies of
A.
Now we are able to
identify primary subspaces { % ( A ) } of Theorem 2: % ( A ) = {( E 36: transformed according T}.
Spaces % ( A ) are obviously T-invariant, their intersection % ( T ) ~ ) % ( A ’ ) = 0, if
# A‘,
and their sum, @ % ( A ) , spans 36. Thus we get a unique (canonical!) primary
to 7r
decomposition of T, QED.
Proposition 1 and Theorem 2 naturally lead to the following problems:
(I) Construct and classify all irreducible representations of the group/algebra G. (11) Decompose the given representation (regular, tensor-product, etc.) into the sum/integral of irreducibles. These are indeed two basic questions in representation theory, that will be addressed at length in chapters 3-7. We shall construct, classify and analyze irreducible representations for many different classes of groups: 0
finite and general compact groups (chapter 3);
0
classical compact Lie groups S q 3 ) ;Sl.42) (chapter 4), as well as their higher-D
analogs: SU(n);S q n ) ,... (chapter 5); 0
semidirect products, including the Heisenberg group:
W,
N
R D Cn, the
Euclidian motion groups: E I , = R” D S q n ) , and affine groups (chapter 6); 0
the celebrated SL, in its various incarnations: conformal SU(1; 1); Lorentz
Sql;2) and S q l ;3), and also the Poincare group: P,
N
R4D S q 1 ; 3 ) in chapter 7.
Once the list of irreducible components of G is found (usually denoted by
G,and
called the dual object of G ) , one can turn to the decomposition Problem (11) for specific classes of representations. The most interesting examples include regular and induced
representation R on G , or homogeneous spaces X = H\G. We shall see that in the compact case ( X or G), the decomposition is always “discrete”,
-l F CB
C 3 m ( T ) - direct sum of irreducibles.
For non-compact groups/spaces the decomposition is typically “continuous”
51.3.Irreducibility and decomposition
42 (direct integral),
or more generally a combination of the "discrete" and the "continuous" parts, as is the
case with SL,(W) (chapter 7). The resulting measure dp on G (whether discrete or continuous) is called the Plancherel measure of R. It will be one of principal objects of our study. The classification problem could easily become "unmanageable", if one insists on "all irreducible representations" of G, since any T can be deformed in infinitely many ways by conjugations with invertible operators,
Q : T ~ +T$ = Q - ~ T ~ Q . Such deformations, T-T', however, retain many essential features of T (e.g. spectrum), and could be regarded trivial. Thus we are lead to the notion of equivalence: a pair of representations, T and S in spaces 36 and W , are called equivalent, if they are conjugated (intertwined)by an invertible linear map Q: %+W,
The space of all intertwining operators is denoted by Int(T;S), its dimension (if finite) is called an intertwining (branching) number m(T;S).Operators that intertwine T with itself form the commutator algebra of T , I n t ( T ; T )= Com(T), and other spaces Int(S;T ) acquire the structure of {Com(T);Com(S)}- bimodule, i.e. Com(T) acts on Int(S;T )by the left multiplication while Com(S)by the right one. Problem (I) usually refers to classification of equivalence classes of irreducible representations, rather than individual {T's}. In some cases, however, equivalent representations T and S may be given in different realizations. Then one is interested in specific intertwining operators that implement the equivalence. Let us mention yet another useful application of Schur's Lemma. Proposition 3: For any pair of irreducible representations T and S, space Int(T;S ) is either 1-D,if T and S are equivalent, or (0). So the intertwining number m(T;S ) = 1 or 0. Indeed, for any
Q E Znt(T;S), its range %(Q)and kernel N(Q) are invariant subspaces
of S and 7'. So space Znt of an irreducible finite-D pair ( T; S ) consists of invertible operators Q. But QIQ;'ECom(S)
for any pair Q I Q z in Znl(T;S), so by Schur's
51.3. Irreducibility and decomposition
43
Lemma, QIQ;’ = XI, i.e. Q1 = XQ2. The unitary version requires a slight modification. We observe that two spaces Znt(T;S) and Znt(S;T)are naturally conjugate one to the other by taking the adjoint Q-Q*. X-’/*Q
By Schur’s Lemma Q*Q= XI, hence operator
is unitary and invertible, and the rest of the argument proceeds as above.
Our final result in this section gives irreducible unitary representations of the direct product group G = H x K in terms of its factors.
Theorem 4 Each irreducible representation
of G = H x K is decomposed into
?r
the tensor productl’l of two irreducible representations of H and K ,
?r
= u @ 7,
where u E 8, 7 E I?. Conversely, the product of two irreducibles u E and 7 E k , is an irreducible representation of G, T = u @ , ~G. E So the dual object of G = H x I( also factors into the product, A
1Proof:The second statement follows directly from the Burnside’s Theorem and the known facts of multilinear algebra about tensor-products of operator/matrix algebras (problem
1%
Mat,, 8 Matm N Mat,,;
(3.3)
or in terms of operator algebras
%(X)8 %(Y)
2: %(36
T o prove the first statement we take an irreducible first factor
A
8 r). A
(3.4)
E G in space 36, restrict it on the
I H, and write the primary decomposition of Theorem 2, r1HN
$Uk@mk; k
where ak E if, and mk- the multiplicity of uk in
A
IH.
so the representation space
36 N $36, - direct sum of primary subpaces of H. Since each 36, is invariant under the k
commutator Com(r I H), hence under K , it follows that 36 has only one primary component, 36 N Kk,and
A
I H N u 8 I,,,.
In other words, space 36 is a direct sum of { m }
copies of the “irreducible space” V of u,
36 N Y a3 ...$ Y N Y 8 cm.
x
Any operator A in 36 can be represented by a block-matrix A = (aij), aij E Mat(Yi;‘Tj). We are interested in operators A E Corn(* I H). By Proposition 3 all “matrix entries” {aij) of such A must be scalars: aij = yijZ; yijE C. When restricted on subgroup K,
matrices (yij(k)) clearly yield a representation of K. So space 36 is decomposed into the product
Y 8 C d , as well as operators {rg), g = (h;k) E G, rg = u,, 8 rk. Obviously, for A
17The reader should not confuse the tensor product u 8 y of Theorem 4 with the Kronecker product of 51.2. The latter is obtained, when the tensor product u@y of the product-group G x G is restricted on the diagonal subgroup ( ( 9 ;g)) N G C G x G .
51.3. Irreducibility and decomposition
44
t o be irreducible, y must also be irreducible, which completes the proof!
Problems and Exercises. GL, and u(n) on C" are irreducible.
1. Show that the natural actions of
2. Show that an eigenspace E A or a root subspace ET = {<: (T-Al)m( = 0) of a matrix T in C" are invariant under C o m ( T ) .
3. Show: the group B , of all upper/lower triangular matrices in R"; C" has trivial (scalar) commutator, though B, is clearly reducible (This exercise shows Schur's Lemma t o provide only a necessary condition for general non-unitary representations!).
X = {1,2, ...n} induces the regular representation R on the function-space: e ( X ) N C", R , f ( j ) = f( j ' ) .
4. A symmetric group W, (permutations of n objects) acting on
(i) Show that e ( X ) is the sum of two invariant subspaces: f(j ) = 0). constant function fo( j ) = 1, and 36, = 32f:
C
X0 = {Afo} - spanned by the
(ii) Show that subrepresentations ro = R 1 360-trivial, and r1 = R I M1-irreducible (Use Schur's Lemma for rl, represent any operator Q on e ( X ) by a kernel/matrix [ q j k ] ,
QfW =
qjkfk).
5*.(i) Show that the adjoint representation Ad,(A) = g-'Ag, of the group G = SL, on the space of all traceless matrices: Jb, = { A E Mat,,: t r A = 0) is irreducible (Hint: any Q E C o m ( A d ) is uniquely determined by its.values on the subspace '3 C A, of diagonal matrices, moreover Q:P+P, commutes with the natural action of W, on 9. Use problem 3!).
r''..]
(ii) Prove the same result for the special unitary group SU(n).
6. Show that a pair of matrices: A =
)t n
(diagonal), B =[1
1) (cyclic
permutation), generate the full matrix algebra in C", provided all {Aj} are different.
7. Let A be an operator-algebra in space Y, vector ( E Y. We denote by ) =I(() the annihilator of in A, ) = { A : A< = 0) - a left ideal A, ( A ) C I), and by Yo = Yo(<) - a cyclic subspace of (, Yo = {A<: A E A).
<
(i) Show that Yo is A-invariant, and the restriction of A on Yo is equivalent t o the regular representation of A on factor-space A / ) , R,(B) = A E ( m o d 8). (ii) If )(<) c )(9), then there exists a linear map Q: Yo(<)+Yo(q), that intertwines two representations of A on Yo(() and Yo(q). (Set: Q [ A < ]= A?, for all A E -4, and check that Q is well defined!).
8. Prove the Burnside's Theorem by the "bootstrap argument", as outlined below. It is convenient t o use the following terminology an operator-algebra A is transitive (irreducible!), if for any vectors ( # 0 and q there exists A E A, that takes A: <+q; algebra A is double-transitive, if for any pair (t1;t2) of linearly independent vectors and any pair (q1;q2) there exists A E A, so that At l = q l , A t Z = q2. Similarly, a k-transitive A takes any linearly independent k-tuple k-tuple (ql; ...qk), i.e. A t i = q3., j = 1; .A.
(F1;...(k) into any
81.3. Irreducibility and decomposition
45
Clearly, a n-transitive algebra in n-space coincides with the full matrix algebra! The idea of ”bootstrap” is to show that “ t r a n s i t i v i t y ” ~“2-lransilivity”* transitivity“.
...“k-
Step 1’: Show “transitivej2-transitive”. For any vector [ E Y denote by 3 = I ( [ ) the annihilalor of 5 in A (problem 6). Given an operator-algebra A and a left ideal 1, consider 1-cyclic subspaces Yo = Yo(q) = {Bv: BE 7 E Y. Each of them is A-invariant, A) c ). So for transitive (irreducible) A, each space Yo must be either 0 or the entire Y.
a},
Pick a linearly independent pair (tl;t2), and consider the ideal ) = )(tl) of t1,and the C )-cyclic subspace, Yo = Y0(t2). Use problem 6 to show: if Yo = {0}, i.e. extends to an intertwining operator Q E Com(A), impossible for then map Q: irreducible A (Schur’s Lemma)! Conclude, that Yo = Y,i.e. for any q E Y there exists A E A, Show that the latter implies double-transitivity of A.
50
that Atl = 0, A t 2 = q.
Step 2”. Demonstrate in a similar fashion that “k-transitive*(k+l)-transitive”, completes the argument of Burnside’s Theorem.
and
The same “bootstrap” applies to transitive algebras A in (oo-D) Hilbert spaces, to show that A is k-transitive for all k! Then fairly standard functional analysis implies that A is weakly closed in %(36) (von Neumann). 9.
Prove: (i) the commutator algebra of a primary representation T N ~ 8 m , Com(T) N Mat,,,, and any Q E Com(T) factors into the product I, 8 Q ,in the tensorproduct decomposition of space YT N Y, 8 Cm (rn = multiplicity of A in T!). (ii) If representation T N A’ 8 mj - is completely reducible (direct sum of primary components), then the algeura spanned by T,A ( T )N @ Matd , @ I m( .d j = d e g d ; m3. = multiplicity of A’ in T), while the commutator algebja
’
Com(T)
2:
e,3 I d j 8 Mat,
’
..
1
(iii) The intertwining number of a pair of representations m(T;S) is defined as d i m I n t ( T ; S ) .Show, if T N @ ~ @ m ( r ; Tand ) , S2: @ n 8 m ( n ; S ) then ,
10. Show that the tensor product of matdx/operator algebras is equivalent to the full matrix algebra on the tensor-product space: Mat, 8 Mat,,, u Mat,. ,,,; and %(X) 8 %(Y) N %(36 8 Y) (Use rank-one operators and vectors). 11. Check formulae (3.1) for C and Q in terms of real/complex matrices. Show that complexification of C yields the direct sum of two conjugate representations: z + ( z ; 5 ), z E C; while complexified Q yields two copies of Q itself.
$1.4. Lie groups and algebras; The infinitesimal method
46
$1.4. Lie groups and algebras; The infinitesimal method. Here we shall introduce two principal players in the Lie theory: Lie algebras and Lie groups. Although different in appearance both are intimately linked through the ezponential map, and its inverse log. Two maps allow to express the local group structure (multiplication) through the Lie bracket (Campbell-Hausdorff formula). We discuss the basic classes of Lie algebras: solvable; nilpotent; simple, semisimple, and state the decompition result (Levi-Malcev). The section is concluded with an infinitesimal method, that allows to reduce group representations to those of its Lie algebra, and thus provides a powerful tool in their analysis.
4.1. Lie groups and Lie algebras. We remind the reader that Lie groups are
smooth manifolds G with differential, (sometimes analytic, or algebraic) group structure. This means that the group operations: ( ~ , y ) + z y and z-u-’, are differentiable (analytic or algebraic). The most common examples of Lie groups are subgroups of matrices, G c Mat,,, like the general and special linear groups: GL,, and SL,; orthogonal, unitary, symplectic, and other groups described in 51.1. Let us remark, however, that not all Lie groups can be realized by finite-D matrices. One such example is a simply connected cower of Sf,, G = SL, (examples in 4.2 below). The class of Lie groups admits all natural operations, performed with groups in general, like taking closed subgroups, factor-groups, direct and semidirect products, etc. All those result in new Lie groups. N
Lie algebras can be viewed as “linearizations” of Lie groups, although formal definition does not allude to an underlying group structure. Formally, Lie algebra 0 is defined as a vector space equipped with the Lie bracket (product) [X;Y], a bilinear map 0 x 0 + 0, that satisfies: (i) skew symmetry: [X;Y] = - [Y;X] for all X,Y.
(ii)Jacobi identity: [X;[Y;Z]] + [Y;[Z;X]] + [Z;[X,Y]]= 0, for all triples X, Y,Z. We can rephrase (i-ii) is in terms of the so called adjoint action of Lie algebra on itself, adx(Y) = [X;Y]; adX:@+@. Then, (i) adX(Y) =
.-
ady(X), for all X, Y
(ii)adlX;yl = [ad,; ady] = adxady - adyadx,
(4.1)
51.4. Lie groups and algebras; The infinitesimal method
47
In particular, the Jacobi identity means that map, X+ad,, is a representation of the Lie algebra 6 by linear transformations over (5. It can also be written as ad,[Y;Z] = [adx(Y);Z] -t- [Y; adx(Z)], for all Y,Z, (4.2) which means that operators {adx} are derivations of the Lie algebra. The examples of Lie algebras are abundant. Any associative algebra (e.g. matriz/operator algebra Mat,) turns into the Lie algebra with respect to the commutator bracket: [x,y ] = xy - yx. Considered as a Lie algebra it is usually denoted A n ) . a
Any vector space A with the trivial bracket, [x;y]= 0, gives a commutative Lie
algebra.
*Important examples of Lie algebras are given by vector fields on a manifold {X=Cui(x)aj} (in local coordinates { x l ;...z,}), with the standard Lie bracket of vector fields, [X;Y] = X Y - Y X =
x ( x u j i 3 j b j - b j a j a i ) a i = C ( a x [ b i ] - d y [ a j ] ) a i ; (4.3) i
i
j
Here aX;ay denote directional (Lie) derivatives along fields X,Y. Those include the algebra 9(A)of all vector fields on A, or its subalgebras of fields, preserving certain structures on A: volume, symplectic (chapter 8), etc. Many examples arise as Lie subalgebras of Mat,, and will be examined below. There is a close relationship between Lie groups and algebras.
Theorem 1: (i) Any Lie group G has an associated Lie algebra (5, which consists of infinitesimal generators of all one-parameter subgroups of G. (ii) Conversely, given a Lie algebra with algebra (5.
(5
one can construct a ‘’local” Lie group18 G
(iii) A local Lie group can be extended to a global group in many different ways. Namely, there exists a unique simply-connected extension G, and any other Lie group G with Lie algebra (5 is isomorphic to a factor-group G/r, G modulo a discrete subgroup of the center
r c Z(G).
Theorem 1 summarizes the basic facts of the Lie theory. We shall not provide complete details of the proof, but rather demonstrate different statements in several 18By local Lie group we mean an open neighborhood of the identity {e} in a Lie group, i.e. a manifold, where the group operations: z.y, and z-*,are defined only locally.
$1.4. Lie groups and algebras; The infinitesimal method
48
important cases. The easiest case to consider are matriz groups and algebras, i.e. subgroups and subalgebras of Mat,. We shall start with infinitesimal generators. It is well known that any (continuous) one-parameter subgroup { U , } of Mat, is generated by a matrix A, in the sense that
U , = etA; where A = dt Such A is naturally to call a generator of subgroup of Mat,.
Lo.
(4.4)
{U,}.Let now G be a closed (Lie)
Proposition 2: (i) The set of infinitesimal generators { A } of G coincides with the
tangent space Te(G)at the identity.
(ii) The commutator of two generator A , B E Te(G) is also a generator, [A;B ] E Te(G),so tangent space forms a Lie subalgebra (5 c Mat,. Proof: (i) Obviously, generators {A} C T,, and the latter forms a subspace of Mat,. To show that {A} is a subspace in T, we pick two subgroups of G { U , = etA], {V, = efB}, generated by A , B E T, according to (4.4), then a subgroup generated by A
+ B, can be
approximated by an “amalgamation” of U , V,
= hence { e t ( A t B ) }
liyi(ezp(;A)ezp(;B))n,
(4.5)
(I G (problem 2). So A+B E T,, and clearly XA E T, for any scalar A.
Conversely, if matrix A is tangent to G, i.e. tangent t o a curve g ( t ) c G, A = g’(O), then we can easily construct a I-parameter subgroup tangent to A, by setting,
u, = g~&(t/4)n.
(4.6)
The generator of U , is A (problem 2), thus we show that generators {A} span the entire space T,! (ii) It suffices to exhibit [ A ; B ]as a tangent vector of some curve in G. Once again we take a pair of subgroups U t = e t A , V, = e t B , and consider the group commutator g ( t ) = cI;’V;’U,V,.
Expanding U and V in the Taylor series, multiplying and collecting
terms we get g(t) = I
+ t2[A;B ] + fl (t3).
Neglecting the cubic and higher order terms the commutator becomes the tangent vector a t { e = I},
[ A ; B ]= lim t-+O
g(,) - I ; QED. t2
We have shown that the Lie algebra of a matrix group G can be identified with the “set of infinitesimal generators’’ = “tangent space of G at { e } ” . A similar description
51.4. Lie groups and algebras; The infinitesimal method
49
could be given for all (non-matrix) Lie groups, namely 6 N T,(G). Lie algebra 6 can also be characterized, as the set (subalgebra) of left/right invariant vector fields on G. Group G acts on itself by left/right translations g: z+g-lz left/right invariant vector fields, {X =Caj(z)aj
- in
(or 2-zg),
gives rise to
local coordinates (zl;...z,) on G}.
Lefl-invariance of X means that the transformed field”:
XO(z)
= g‘[X]og-’
= X,
where g’ denotes the Jacobian map of the left translate g: z+gz. Left-invariant vector fields form a subalgebra
(s
in the algebra of all vector fields 9(G)with the Lie bracket
X can be identified with its value at { e } , a tangent for any point g E G. So 19 N T,2: {left/right invariant fields
(4.3). Any left-invariant vector field
vector
t E T,,X(g)
= g’[(],
on g}.
4.2. The exp and log maps. The correspondence between Lie groups and Lie
algebras is given by two maps: exponential and tog, exp: 6+G; log: G+@.
Map exp: 6 NT,+G, sends a tangent vector/line { t t } into a one-parameter subgroup { e x p t t } of G, tangent to
<
at {e}20. Map d(<)=exp(t)
is a local
4;
= I!), and its inverse is called log. So exp takes a neighborhood of {0} in 6 onto a neighborhood of identity { e } in G, while log does vice diffeomorphism (since its Jacobian
versa. The easiest way to introduce exp and log is for matrix groups and algebras. Here two maps are given by their Taylor-expansions,
Exponential map on the full matrix algebra Mat,, = gl,, takes it onto the full linear
group
GL, = { z E Mat,,:
det(z) # 0).
Clearly, exp on
is a local
Mat,
diffeomorphism (near X = 0), but globally it may not be 1-1. For instance, all complex matrices { X } with purely imaginary integer eigenvalues {Aj = 27rim}, are taken to the identity by exp. We shall list now a few basic and familiar properties of exp and log. W -e remind the reader that a diffeomorphism (coordinate change) y = 4(z) on a manifold transforms vector field X = C a .a. into a field Y = X4 = (#[XI)o d-’, with components J Z bi = C4:jaj, where 4’ denotes a Jacobian (matrix) of 6. 20Such notion of exp arises in differential geometry and is closely connected with geodesics and parallel transport (Appendix C). On Lie groups the parallel transport is furnished by the left/right group multiplication, z-g-lz, g E G. The latter gives rise a notion of parallel field (along a curve ~ ( t ) )the , geodesic path y (whose tangent field y‘(t) is parallel), and exp:t< E Ta-y (t) - a geodesic initiated at {a} in the direction (!
<
51.4. Lie groups and algebras; The infinitesimal method
50
Lemma 3 (i) If matrices A and B commute, AB = BA, then particular {e"A}t is a one-parameter subgroup of Mat,.
= eAeB, in
(ii) det(eA) = elrA. (iii) exp preserues complez conjugation, transposition and taking adjoint matriz: exp(2) = 3, exp(=A)= TexpA; exp(A*) = (expA)*.
The proof is fairly straightforward (problem 1). Using Lemma 3 we are able to characterize Lie algebras of many classical Lie groups (orthogonal, unitary, symplectic, etc.), listed in $1.1. For instance, the expmap takes a Lie algebra of n x n hermitian antisymmetric matrices, denoted u(n), or real antisymmetric matrices, denoted o(n), into the unitary and orthogonal groups: U(n) and q n ) . Special (traceless) Lie algebras { X :trX = 0) are taken into special { detg = 1) Lie groups, like 0
d ( n )= {A:trA = 0) (all traceless matrices) goes into SL,
0
so(.)
0
su(n) = {A* = - A; t r A = 0 ) (unitary traceless) goes into SU(n).
= {TA = - A; tr A = 0 ) (orthogonal traceless) goes into S q n )
In fact, in all cases the exp map is locally onto, since by Lemma 3 generators of groups SL,; U(n); q n ) , etc., must traceless, unitary, orthogonal. We have already shown that Lie algebras are obtained as tangent spaces at {e} of Lie groups G, or as infinitesimal generators of one-parameter subgroups of G. Going in the opposite direction one would like to construct Lie group G, starting from Lie algebra (5. As we stated earlier (Theorem 1) the correspondence C k G is not 1-1, as many Lie groups could have the same Lie algebra. Although all such groups are locally isomorphic, their global structure may be different. The local group-structure, however, is completely determined by the Lie algebra formula bracket, via the so called Campbell-Haudorff-Dynkin
Here the inner sum extends over all tuples of integers { p l , q l ;... p,,q,) with the property: pj+qi 1 1, and [AP1[Bql ...Bqm]] denotes the iterated commutator [A[A...[B...[B...]]]]. The Campbell-Hausdorff series can be shown to converge for all Lie algebra elements (matrices) of norm 11 Alk llBll< and thus defines a local group
a,
31.4. Lie groups and algebras; The infinitesimal method
51
structure in a neighborhood of {0} in Q. Formula (4.8) explains, why the exponential sends any Lie subalgebra 0 c Mat, into a Lie group. Indeed, e x p A e x p B = e x p ( A t B + i [ A ; B ] t ...), so e x p o
is closed under the matrix multiplication.
The local Lie group uniquely extends to a global simply connected Lie group G (not necessarily matrix group) with the Lie algebra (5. Any other Lie group G with algebra (5 is then isomorphic to G/I', where is a discrete subgroup of the center of G, identified with the fundamental group21 of G . So all locally isomorphic Lie groups {G} with the given Lie algebra are in 1-1 correspondence with discrete subgroups r c Z(G ).
r
N
N
We shall illustrate the relations between Lie algebras Q and groups G; G and with a few simple examples.
r
1) The easiest of the sort is the abelian Lie algebra 8 = W", which gives rise to several Lie groups: G=Rn;or G=T"=Wn/Z"orG=Bm$W"-m,for a n y m = 1 , 2 , ... corresponding to various discrete subgroups N Z", or Zm, of G.
r
2) Another example is the unitary group SU(2) (simply connected), which forms a 2-fold cover of S q 3 ) (chapter 3). All other (higher-D) orthogonal groups S q n ) also have 2-fold simply connected covers, called spinor groups Spin(n), S q n ) N Spin(n)/Z2. 3) Group G = SL,(W) is not simply connected, and its simply connected cover cannot be realized as a matrix group!
N
G
Indeed, group G is topologically equivalent to the product R2 x 1 ' (this follows from realization of the Poincare-Lobachevski half-plane H as quotient SL2/So(2), and could also be established by the Iwasawa decomposition, as explained in $7.1, chapter 7). So the
"We remind the reader that a fundamental group of manifold W consists of the hornotopy equivalence classes of closed path (loops) in W . Two loops 7 1 ; 7 2 in W are equivalent, if rl can be continuously deformed into yz inside W . Any pair {yl;y2)of closed path (loops) passing through a fixed point z,,E W can be formally multiplied by combining them into a single path ~ 1 ~ ("71'' 7 2 followed by "y2"), and any 7 can be inverted, by traversing it in the opposite direction. Such multiplication and inversion are easily verified to respect the (homotopy) equivalence on the space of loops passing through a fixed point. So equivalence classes of (7) acquire the group structure, the identity being made of all path contractible to {zo}. The resulting fundamental group of A, = al(W), is a discrete group, that carries some important topological information a b u t the manifold. It acts freely (by the so called deck Iran_sformations) on a simply connected cover W of W , so the latter is isomorphic to a quotient-space &IT. The fundamental group of a E e group G is c G , hence belongs to the center of G (problem a), identified-with a normal discrete subgroup and G rz G
/r.
r
r
51.4. Lie groups and algebras; The infinitesimal method
52
fundamental group
r of SL, is isomorphic to Z,and its simply connected cover G has
Z. If SL, were realized by matrices we would get a finite-D faithful (one-te one) representation of SL, in C”.The detailed analysis of finite-D representations of SL, in chapter 4 shows that any such T has { d e t T , = 1; for all g E C}, and the same holds for representations of G (in other words T m a p S L p S L , ! ) . Furthermore, any center Z N
representation T of G or
2:
is completely reducible ( u “direct sum of irreducibles a”).
Since center of any group is represented by scalars { X I } under an irreducible (Schur’s Lemma), and for
SL,
i~
EG
these scalars must be m-th roots of the identity { w :
wm = 1 ) (due to d e t r , = I), it follows that center Z(G) is mapped by T into a finite
group of matrices with diagonal entries { w } . But Z N Z is infinite, so some g to { I } , which means T can not be exact. In other words,
Z must go
Sf, can not be realized
as
matrix group!
Let us make a few comments concerning the explog correspondence: (5-G. It clearly, preserves all general (functorial) properties of groups and algebras. So a
closed (connected) subgroups of G u subalgebras of
(5,
normal subgroups H c G u ideals @ c (5 - subalgebras invariance under the adjoint action of (5: [@;@I c @, or adx(@) c 8, for all X E (5).
Z(G), commutant G, = [G;G] (a subgroup, generated by all group commutators {g-lh-lgh: g , h E G}),etc., have proper counterparts in the Lie algebra: central ideal 3 = 3((5), commutant a other “natural mbgroups”, like center Z =
(5,
= [(5;(5] = S p a n { [ X Y ] X : , Y E (5}, etc.
Also any decomposition of group in the direct, semidirect product, yields a similar decomposition of its Lie algebra. 4.3. The adjoint action. Lie group acts on its Lie algebra by the so called adjoint action Ad,. First let us observe that G acts on itself by the group conjugation g : h6+g - ‘hg. The lakter can be lifted to the Lie algebra via exponential map:
g
-1
(expA)g = exp ( g - ‘Ag), for A E 6.
(4.9)
We call the resulting action (4.9) of G on (5 the adjoint representation Ad,. Of course, for matrix groups G,the adjoint action is nothing but a conjugation with g, Adg(A) = g - ‘ Ag . In general, adjoint action represents G by automorphisms of its Lie algebra,
g1.4. Lie groups and algebras; The infinitesimal method
53
Ad,[A;B] = [Ad&A);Ad,(B)]. We shall see that the adjoint action of group G by automorphisms of its Lie algebra corresponds to the adjoint action (4.1) of 6 on itself by derivations, adA(B)= [A;B]. In fact derivations {adA} represent infinitesimal generators of oneparameter subgroups of automorphisms {AdexptA},as will be explained in Theorem 5 below. 4.4. Basic classification of Lie groups and algebras; Examples. On the algebraic level the realm of Lie algebras (it is easier to start with them, as one avoids many topological complication) contains two large and distinct classes:
I) solvable and nilpotent algebras (the latter make up a subclass of solvables); 11) simple and semisimple algebras. We shall briefly describe both of them. The definition of the former is given in terms of the upper derived series of repeated commutators,
8 3 8'= [0;0] 3 8" = [@';@'I Here each subsequent term X , Y E d k - l ) } of the preceding term
3 ... 3
dk) = [(5@-1); d k - 9
dU is derived dk-').
3 ...
(4.10)
via commutators, Span{[ X ;Y ] ;
Algebra 8 is solvable (of step n), if its derived series terminates at {0}, (similar notion applies to groups with group-commutators { g - ' h - ' g h } in place of Lie brackets [ X ; Y ] )A. commutator 8' of Lie algebra is the smallest ideal 8 c 8,with the commutative factor-algebra 8/@, and one can check that all are also ideals of 8. So solvable algebras/groups are obtained by subsequent abelian extensions of abelian algebras22. Nilpotent algebras form a subclass of solvable, they are defined in a similar manner with a (smaller) lower derived series,
dn+') = (0)
{dk)}
8 3 8,= [@;@I 3 8, = [O;O,] 3 ... 3
[8;(5J 3
...
(4.11)
So each subsequent term is the commutator of the entire algebra 8 with the preceding term. Once again the series is required to terminate at {0}, On+' = [@;On] = (0) (step-n nilpotent). In particular, 8, belongs to the center of 8, and the entire algebra is obtained by a sequence of central extensions. The natural examples of solvable and nilpotent algebras are made of upper22The terminology came from the Galois theory, where solvability of the Galois group of an algebraic equation implies that the equation can be solved in radicals. Radicals correspond to cyclic subgroups of quotients dk)/dk-').
$1.4. Lie groups and algebras; The infinitesimal method
54
1
triangular matrices, respectively matrices with (0) on the main diagonal (problem 5 ) . Other examples incude t e affine group Affl, matrices of the form{ and
[
realized by the upper-triangular
(a step-one solvable group), and the Heisenberg groups HI
W, (step-one nilpo ent g oups) (see $1.1and chapter 6).
Next we turn to semisimple and simple algebras. The former are characterized by the property that the commutator [0;(5] = (5 (in other they have no commutative quotients), the latter (simple) have no ideals at all (hence no nontrivial
quotient^)^^.
These properties have important implications for the structure and analysis of simple and semisimple groups and algebras (see chapters 4-6). At this point we shall mention only two basic results:
i) any semisimple algebra is a direct sum of simple ones, 0 21 @I Gj;in other 3 words the adjoint action (4.1) on (5 is completely reducible; ii) all simple Lie algebras can be completely classified; they consist of the classical
series: unimodular, orthogonal, symplectic, and a few exceptional ezamples. Here we shall list the basic classical simple Lie algebras, which correspond to
classical Lie groups of $1.l. Examples: 1) Lie algebra gqn) = Mat,, of the general linear group GL,; and its subalgebra sqn) = {A:t r A = 0 } , which corresponds to a special linear group SL,. Other classical groups (11.1) preserve certain quadratic/bilinear forms J on Wn;
Cn, so that 'gJg = J ; or g*Jg = J ; for group elements ( 9 ) .
(4.12)
The corresponding condition for the Lie algebra elements {X} becomes
'XJ + J X = O ; o r X * J + J X = O , which easily derived from (4.12) by differentiating g = exptX, at t = 0. Thus we get 2) so(n)
- algebra of all n x n real/complex antisymmetric traceless matrices
{'X = - X ; tr X = 0}, which corresponds to the orthogonal group S q n ) . In the real case "traceless" condition holds automatically for all skew-symmetric matrices.
3) u(n) and su(n) - algebras of all n x n complex (hermitian) antisymmetric matrices, { X * = - X}, and traceless antisymmetric matrices { t r X = 0}, correspond to to the unitary, or special unitary groups, U(n) and SU(n). In a similar fashion one can of ad,
23We can phrase both definitions in terms of the adjoint action: simplicity means irreduci6ility on 8 , while semisimple implies that operators {adx(@):X E 8 ) span entire space 8.
51.4. Lie groups and algebras; The infinitesimal method
55
describe Lie algebras so(p;q); su(p;q) of ( p ;q)-indefinite orthogonal and unitary groups S q p ;q) and S y p ;q), symplectic algebra sdn). For instance, matrices X E s o ( p ; q ) have block structure d E sa(q), and c = Tb E Matp
4.) (problem 6). So
{X=(i
:)}with
a€+);
and similar representations could be found for ur(p; q),
vector space 4 p ; q ) is isomorphic to 4 p ) x so(q) x Mat,
Q, which
yields the dimension of Lie group S q p ; q).
We shall see more examples in subsequent chapters 4-6. Two classes: solvable and semisimple, provide the building blocks for arbitrary Lie algebras S!. The following general result will be stated here without proof. Levi-Malcev decomposition Theorem: A n y finite-dimensional Lie algebra I? has the maximal solvable ideal '32, called radical, and a semisimple subalgebra 6,so that S! is decomposed into a semidirect product: S!='[email protected] h e radical 8 is a characteristic ideal of I? (invariant under all automorphisms of S!), and obviously unique, while a n y two semisimple factors 6;6' differ by a n automorphism of !2, 6'= a(@),a E AutI?. Let us recall that a semidirect product S! = ?R@ s6, is defined for a pair of Lie algebras {%;@}, where 6 acts by derivations on '32,
6 3 z+a = az E Der('32), where a ( [ a ; b ] = ) [ a ( a ) ; b+ ] [ a ; a ( b ) ) for ; all a,b E '32. By definition, I? consists of all pairs { ( a ; z ) }with the Lie bracket (problem 6), [(a;2);(4Y)1 = ( a -t
4)); 2 -t Y ) .
(4.13)
The natural examples of semidirect products include the Euclidian motion and the Poincare algebras: En = W" @so(.) and Sp, = M" @ 4 1 ;n - 1). In both cases the radical '32 = W" (or M") is commutative, and a (simple) factor 6 II 4.) or 41;n-1) (orthogonal/pseudo-orthogonal)acts on '32 by linear transformations. Another important example, discussed in chapter 6, is a semidirect product of the Heisenberg and symplectic Lie algebras: 8, @ 4.). Further details and references on the general Lie theory could be found in [Che]; [Jac]; [Kir]; [Ser]; [Hell. 4.5. The infinitesimal method. The correspondence between Lie groups and Lie
algebras extends to their representations. It is often convenient to replace Lie group representations by those of its Lie algebra, as the latter are much easier to analyze (study irreducibility; decomposition, etc.). We shall conclude this section with a general
56
81.4. Lie groups and algebras; The infinitesimal method
result that serves the basis of the infinitesimal method in representation theory. It will require a notion of the generator of a 1-parameter group of unitary (bounded) operators U t in an oo-D space 36. Proposition 4 (i) A one-parameter strongly continuous group {U,} of matrices, or bounded operators in space 36 has a generator A, defined on a dense subspace of “smooth vectors” {t}(i.e. vectors which yield smooth vector-functiow t(t)= U t t ) ,
Conversely, any selfadjoint (bounded and unbounded) operator A ( o r more general A, subject to some technical constraints) generates a 1-parameter group, denoted by Ut( = e t A t .
(ii) Generators of orthogonal/unitary groups {Ut} are skew-symmetIic, A* = - A . The algebraic (finite-D) version exploits the well known feature of any {U,}: differentiability t o any order (even real analyticity)! The same holds for a uniformly continuous group U , in Hilbert/Banach spaces. In both cases, U , = etA, with a bounded operator24 A (problem 4). The strongly continuous (unitary) case is more subtle, as not all vectors [ E 36 have differentiable “tails” { [ ( t ) = U t [ } , in general. However, any U has smooth vectors { < : [ ( t )E el} (or
em;Coo; real
analytic), which form dense subspaces in
36: 36 3 36, 3 ...36, 3 ...X6, 3 A. Indeed, smooth vectors can be obtained by mollifying (smoothing) operators of J1.2,
It is easy to check that operators {U,} send 36 into the m-smooth subspace 36,. The density of the embedding 36, supported functions {f E
c 36 is verified by observing that m-smooth compactly
er}approximate any L’-function
(in the weak sense!). If f,4, then images U ,
on R, as well as distribution 6
[I
are easily seen t o approach any [ E 36. d Thus we get a densely defined (closed) generator: A = &Y, on 36, c 36. For unitary { U } operator A is selfadjoint, A*=A,
90
n
its exponential eitA (as well as any other function
=f(A)”) can be defined via spectral resolution (Appendix A), eitA =
I
eiXtdE(X) = U,.
Smooth vectors form a domain 9 ( A ) = 36, of a closed, skewadjoint (in the unitary case) -This fact is a generalization of differentiability of continuous one-parameter groups of numbers, or characters (~(1): t E R) c R or C. Such x is shown to be differentiable, by smoothing i t 1 where c = J f ( s ) x ( s ) d s# 0, and the out via convolution with nice f E e:(R), i.e. ~ ( t=) i;(x*f)(t), RKS is obviously as smooth as f! Now it follows immediately, that ~ ( t=) eat, for a E R or C.
51.4. Lie groups and algebras; The infinitesimal method
57
unbounded operator A . Space 9 ( A ) is closed under a stronger (graph) norm,
11 ( ]I1 = 11 ( 11 + 11 A(
1
or norm ( I +I A
b( 1 where I A I = (A*A)’/’- the modulus of A.
In fact, any such A defines a scale of densely embedded (“Sobolev”) spaces 36 3 36,3
where 36, = ( I
+ I A I )-“36
intersection 36, =
n
36,-
... 3 36,3 ... = 36,;
=domain
of A’” = {rn-smooth vectors in 36}. The
consists of oo-smooth vectors, and is also dense in 36
m>O
(moreover, 36, contGns yet another dense subspace A of analytic vectors!). All subspaces are invariant under the group U , , and of course, U,136,
is strongly continuous in the rn-
norm.
Let us illustrate the general concepts of smooth vectors; closedness; domains; “strong” vs. “uniform” continuity of operator-groups with a few simple examples. 4.6. Examples: 1) Multiplication operator, A f ( z ) = zf(z), on L2-spaces: L2(W); ~ ~ ( bl); [ a ;or more general
L’(w; dp), generates a unitary group: U,f = e i t t f ( z ) , also
acting by multiplications. i) On finite intervals [a;b],or for measures dm of “finite support”, the group is uniformly (norm) continuous, and generator A is bounded, 11 A 11 5 C ,if supp I c [ - C;C ] . The two conditions (norm-continuity and boundedness) always come together. Furthermore, all vectors {f} in L2 are “00-smooth” (even real analytic), so
36 = 36, = A = L*.
ii) On infinite intervals (e.g. W), A is no more bounded, and U , is only strongly continuous: 1 U ,f - fIP0, as t+O, for any f E L2. Not all vectors are smooth, however, the space of “1-smooth vectors”: 361 = P(A) = {f: similarly “m-smooth vectors”:
I f 111 = I (I+ I 2 I 1f(z)llL2
36, = 9(Arn) = { f :llfllrn =Il(ltI z I
)“f(.)IIL2
< CQ}’ < 00}, etc.
Vector-function ei‘”f(z) could be differentiated at t = 0, only for 1-smooth vectors { f } , to get Af = zf!
2) The translation-group W: U,f(z) = f ( z - t ), acing on L2 (or other LP-spaces) is strongly continuous. Its generator A = d is an unbounded operator with a dense domain made of “1-smooth” vectors/functions, f E L2,so that f’ is also L2,
{
1 1
36, = f:1 f 1 1 = 1 f 1 L~ t f’
L2}
- the so called first-Sobolev space.
$1.4. Lie groups and algebras; The infinitesimal method
58
Similarly, LLm-smoothvectors” coincide with um-smooth (Sobolev) functions” (see
36, =
{f : l l f l l m
= IlfllL2
-+ ... +I1 f(m)IIL2} - m-th Sobolev space25.
The last example explains the terminology “smooth vectors”. It could be extended to more general one-parameter group actions, e.g. flows on
manifolds,
4,:
z+y(z;t) on A. Any such flow is generated by a vector field
as a solution of an ordinary differential system:
X = Cajaj,
+ = X(y); y(0) = z (so we could write:
dt = exptX). Associated to any flow dt is a 1-parameter group of linear transformation on function-spaces over A, e.g. L’(A),
U , f ( z ) = ~ ( I # J , ( z )or ) ; more general V,f(z) = a ( z ; t ) f od,(z), where
(I
satisfies the cocycle condition: a(+;t + s ) = a(+;t ) a ( d , ( z ) ; s ) The . generator of
X considered as a 1-st order differential operator C ajaj on (it plays the role of D = in example 2). Smoofh vectors consist of dz
group U,, is the vector-field functions over A
functions differentiable along the flow. In the second case, the generator is the 1-st order operator of the form: A = X + q , where q is a multiplication operator with function q(z) = crt(z;O)(at- derivative of
(I
at t = 0). If function q is sufficiently smooth, one can
check that Um-smoofh vectors” of A are the same as for X.
Once the notion of generator of one-parameter groups was made clear, we can proceed to more general Lie group representations.
Theorem 5: Any representation g+T, of Lie group G in space 36 gives rise to a representation X +Tx of its Lie algebra (li: (4.14)
In the 00-D c u e generators { T x } are unbounded operators defined a joint dense core of “smooth vectors” 36, = {.$(Tg(I q ) E C?(G);for all q E 36). Furthermore, for unitary T operators { T x } are skew-symmetric. Conversely, a representation of Lie algebra 0 by (closed, unbounded) generators { T x } can be ezponentiated (lifted) to a representation of its simply connected Lie group,
T,
= exp T x .
(4.15)
Proof: On the purely algebraic level (finite-D, uniformly continuous case) the result becomes a simple consequence of the exp and log maps (defined via Taylor expansions)
1 + I D I )“f
-One can show (52.2) that the m-th Sobolev norm is equivalent to (1 I D I = ( D * D ) 1 / 2 is the modulus of the differentiation operator D = L. dz
lL2;where
51.4. Lie groups and algebras; The infinitesimal method
59
and the Campbell-Hausdorff formula (4.2). On the functional-analytic (unitary) level some technical complications appear, as generators { TX} could be unbounded (densely defined) operators, each with its own domain !D(TX).However, all of them have a joint dense core of I-smooth uecfors 36,. As in the one-parameter case, 36, contains the range
{( = T f ( q ) :q E 36) of any mollifier T with f E e,!(G). In fact, there exists the whole
f
scale of
3 6 , 3
m-smooth
spaces: 36 3 36, 3 ...36,
3
...36,
3A
(analytic vectors), with
{ T f ( q ) :E f ey(G)}. All of them are dense and invariant under the group action
{Tg}(for analytic vectors this result is due t o L.G$rding). On each space 36, one can define the m-fold products and powers of generators {TX...T X :XjE a}, furthermore 00
1
,
on analytic vectors {( E A } the Taylor series:F$TF(E), converges. Thus we get the group action on analytic vectors { T gI d:g = e x p X } , which then extends rough the whole space 36. An alternative way t o obtain the group representation in the unitary case (skewsymmetric generators) is via spectral resolution:
T X = JXdE(X), E(X)-spectral measure, whence the corresponding unitary group: exp(t T
~= T~~~~~ ) = Je’”dE(X),
as in Proposition 4. This completes the proof.
Let us remark that many natural properties of representations, including irreducibility, decompositiona6, etc. can be transferred from Lie algebra 8 to Lie group
G. However, in the absence of simple connectivity of G local representation { T e X p x } given by (4.15) may not be extendible through the whole group. Indeed, only “half’ of irreducible representations of simply connected group SU(2) extend through the representations of Sq3)u SU(2)/{i I} (chapter 4), and a similar pattern holds for all other non-simply-connected groups. Clearly, the dual object of any factor-group G / H consists of all i~ E G, that annihilate H , i~ I H = 1.
26with some obvious qualification: for any invariant subspace of the Lie algebra generators So irreducibility of { T x } means any {Tx}-invariant subspace 36 is dense in 36.
ITx} we take its closure in the %norm.
60
51.4. Lie groups and algebras; The infinitesimal method
Problems and Exercises: 1. Prove Lemma 3 (Hint: some statements, like (ii), are easy to check first for diagonalizable mairices A = U-'DU (D - diagonal, U E GL,), and then to use density of diagonalizable {A) in Mat,!). 2. (i) Prove formula (4.5) for matrix subgroups, using the Taylor expansion of { e x p t A } . It is known as Trotter product formula (4.5), and has numerous extensions and applications to groups (and semigroups) of operators in spaces of finite and infinite dimensions. (ii) Show that formula (4.6) defines a 1-parameter group with generator A. 3. Show that subspaces d,; so(n) and
4.)of g/, = Mat,
form Lie algebras.
4. A (strongly continuous) one-parameter group of unitary Hilbert-space operators { U t } has a selfadjoint generator A = A*, i.e. U , = exp(iAt). Show U , is uniformly (norm)
continuous iff operator A is bounded. Steps: i) Use spectral decomposition of a selfadjoint (not necessarily bounded!) operator: A is unitarily equivalent to a multiplication, r : f + r f ( z ) , in LZ(R;dp) or in the direct sum
e,3 L2(R;dpj).
ii) Show: A is bounded iff s u p p t d p } or U s u p p { d p j } is compact in R. iii) Unitary group { U t : f-e""f(z)} compact.
in L2(dp) is norm-continuous iff s u p p { d p } is
5. Show that all upper triangular matrices over R or C form a solvable Lie algebra 23,. Find the derived series for B,, and the corresponding Lie group. Do the same for the the algebra U, of all upper triangular matrices with 0 on the main diagonal (show that U is nilpotent, find its lower derived series, and its Lie group).
6. Find block-structure (as explained in example 3) of Lie algebras: 4 n ) , su(n), s o ( p ; q ) , w ( p ; q ) , sp(n), and use it to compute dimensions of the corresponding Lie groups.
7. Show that (4.13) defines Lie bracket on the semidirect product % @ 8 of Lie algebras.
r
8. Show that a normal discrete subgroup of a Lie group G belongs in the center Z ( G ) (Hint: conjugate each y E r by elements g E G, y+g-'-yg, and observe that all {g} in a vicinity of {e} fix y).
Chapter 2. Commutative Harmonic Analysis. Schur's Lemma implies that irreducible representations of a commutative group G consists of I-D characters, i.e. homomorphisms { x:G+C* numbers)}, or {x:G-+T
(multiplicative complex
= {e"} - unitary characters}. We shall mostly deal with the
latter. The characters can be multiplied and inverted: (x1;xz)
+
x1x2;
x
x-l =
-+
1 Xb)' -
so they also form a commutative group, called the dual group
e. Dual group of a locally
compact G , is itself locally compact [Po,], hence carriea an invariant (Haar) measure. Three commutative groups lie a t the heart of the classical harmonic analysis: R";
Z" C R" and 1"N R"/Z". The characters of all three consist of exponentials X,(z)
= e i t . '.
In the case of R", parameter ( runs over R",
80
the group becomes isomorphic t o its dual.
For Z", identified with an integral lattice {m = (ml; ...mn)] in R", characters
x = xO(m)= e " . m ,
...
are labeled by the dual torus, 0 = (Ol; On), 0 5 O 5 2u, so
A
Z" u T". The characters of torus T" = {(el;...On) : 0 5 Oj 5 2 x 1 , or more general (0 5 Oj 5 Tj], are in turn labeled by the lattice points,
x = Xm(4
eim .O.
, m E Z",
so 1"N Z". Let remark that the mutual duality of groups 1" and
Z" exemplifies the
general Pontrjagin Duality principle for locally compact Abelian groups [Pon]: the second dual group (characters of G) is isomorphic t o G itself, via the map: u-G(x) = x(a), a E G (elements a E G define characters on G). So the classification problem (I) of
chapter 1 for commutative groups G amounts t o characterization of
6,
while the
decomposition problem (11) for the regular representation R will lead to study the Fourier transform 9 on L2-spaces and other spaces. This marks the starting point of the Fourier analysis. Chapter 2 gives the basic overview of the Commutative Fourier analysis, and some of its applications, the main emphasis being on differential equations.
$2.1. Inversion and Plancherel Formula.
62
$2.1.Fourier transform: Inversion and Plancherel Formula. In this section we shall introduce the Fourier transform/series expansion on R"; 2"; T", discuss their basic properties (with respect to translation, differentiation, dilation, rotations. Then we establish the key result of the Fourier analysis, Plancherel/Inversion formula, in 2 different ways. One exploits Gaussians and Schwartz functions, another Poisson summation.
1.1. Fourier transform 4 on group G = R";T" is defined by integrating functions {f} against characters {x = e i t e Z } ,
while on
Znintegration becomes summation,
Sometimes we shall indicate the
"I
to
transform by TZjt.
From the
definition one could easily check a few basic properties of 4. 1) Convolution-+product, 4:f*g---+?(t)G(t) 2) Translation-tmultiplication with character, 4:R, f = f ( z - a)-+eia't?(t)
3) Differentiationjmultiplication, 4:8,f +itj?(t); hence 4:aaf+(it)"Y Here
{aj} and (8%= ( a l...; a,,)}denote partial differentiations in I . The latter
relation is easily verified via integration by parts:
/(a"f)g = / f ( - a ) a g ;
with g = e T i Z . t .
We introduce an involution on functions { f ( s ) }on G,as a linear map,
-
f(-.),
*: f(.)-f*(.) = with the obvious properties: ( f * g ) * = g**f*; (f*)* = f . Clearly, 4) Involution-tcomplex conjugation, 9:
f*(z)+?(t).
5)A linear change of variable, A:z+Az
( A E GL,)
on Rn,
defines a
transformation of functions,
f + f A ( z )= f o A = ~ ( A I ) . Fourier transform takes such linear change into its inverse transpose, precisely 4 : f AdetA + ~ ~('AA-'). o
$2.1. Inversion and Plancherel Formula
63
Finally, relation (3) differentiation-+multiplication, can be reversed into
6) m ultiplication-tdifferentiation, 9:xjf(x)+$3,,f^([); As we mentioned earlier characters
+~~f-(ia,)~?
{x} describe all irreducible representations of
the commutative group G. On the other hand they can be interpreted as joint eigenfunctions of all translations { R,:a E G } , respectively, of their infinitesimal
generators {aj} (on 53"; U"), as well as convolution operators: R f [ g ]= ( f * g ) ( x ) . Formulae (1-3) can be rephrased now in the language of the representation theory by saying: map 9 diagonalizes regular representation R of G,
9R9-
N
"direct sum/integral of characters".
Precisely, any function f (in a suitable space) can be expanded into the direct sum/integral of characters
x = eix
*
€,
C
1
f(x) = bEeix* t; or f(z) = b( [) eiz ' '%(, (1.3) with "Fourier coefficients/transform" b(() = ?([), and the regular representation turns into multiplication on the space of "Fourier coefficients",
Later on (chapters 3-7) we shall establish various noncommutative versions of this result.
1.2. Plancherel and Inversion formula. The fundamental property of exponentials { e i < ' + } is their orthogonality and completeness. In case of compact G (e.g. U"), orthogonality can be understood literally',
as all
{x,}
are L2-functions on G. Completeness means that any f can be
approximated, hence uniquely expanded into the Fourier series. An elementary proof for
T"exploits the
Cezaro means (problem 3).
More general explanation has to do with compactness of convolution-operators, R f:u (z )d f*u (+) , in L2 (LP)-spaces on 1" (Appendix B). Compact self-adjoint (f = f*)
operators, are well known to have a complete orthogonal system of eigenvectors, and exponentials {xm = ei+ * m} are precisely the eigenvectors of R
f , f*eiz
*
= F(m)eic ' €.
'The same holds on an arbitrary compact group G (see [HR]; [Loo]),namely,
(X, I xa) = I&',&d+
= I G I 6km;
I G I = Ud(G);6km-
Kronecker 6.
62.1. Inversion and Plancherel Formula.
64
Similar arguments apply to all compact groups (chapter 3).
For non-compact groups W"; Z" orthogonaZity should be understood in a generalized (distributional) sense,
l(eit.z I ,ir)
.z)=
c
,i(t-r))
*
zdz
(1.5)
In other words, the role of Kronecker 6,k = S(m - k), being played by the Dirac 6-function. The integral (1.5) is strictly speaking divergent, but it makes sense as a distribution. Namely, for any nice (testing) function f(q) on R",
We shall see that in such form orthogonality is equivalent to the
Inversion/Plancherel formula for 9. Namely the transform,
4':h(q)-tL(x) = ( h I eWiV.') = Ih(q)eiV*'dq, takes
f^
(1.6)
back into Const x f. Therefore, (1.6) defines the inverse Fourier
transform of h. The precise result is usually stated for a suitable class of
functions on G. We shall treat separately compact (T";Z")-cases and noncompact Rn- case.
1.3. Compact case. A suitable class of functions on torus, called Fourier algebra, consists of absolutely convergent Fourier series, A = {f(x) = CakeiZ * k: Clearly, algebra Ace(U"),
c I ak I < m}.
is made of continuous functions on
U", and
contains sufficiently many smooth functions, A 3 em,for m > n. The latter is easy verified using differentiation formula 3). Indeed, Fourier coefficients of a partial derivative
aaf = C(ik)aakeiz'k , by (iii). So if f E em, then its
Fourier coefficients ak = o( I k I,),-
which
guarantees convergence of
c l%I. Uniqueness and Plancherel Theorem on T": (i) Any function f E A(Tn) is uniquely determined by its Fourier coefficients, {ak}*f(z)
(ii) For any function f =
=
Cakeiz' '.
CakeiZ " E L',
In other words 9:L2(T")+12(Z"), is a unitary map. Both results are fairly
52.1. Inversion and Plancherel Formula obvious corollaries of the completeness and orthogonality' (problem 3).
65 of {eiz * k: k E Z"}
1.4. Continuous case W". Here a suitable class consists of all functions { f},
integrable along with their 9-transforms,
{f E ~
T E L I } = L' nA.
1 :
As above, A denotes the Fourier algebra {y(z): f E L'}, of absolutely convergent Fourier integrals. As with T" algebra A is shown to consist of continuous functions, decaying at {oo}, a consequence of the Riemann-Lebesgue p ' - compactly supported mTheorem (problem 1 of 52.2). Furthermore, A 3 C smooth functions3, so
C'y c A c C',,
for m > n.
Inversion/Plaucherel Theorem on R": (I) A n y function f E A n L', can be uniquely
recovered from
1 b y the Inverse Fourier transform
(11) transformation 4,normalized b y factor L2(Rn) onto L"), i.e.
1becomes a unitary map from (27r)"/2 '
The Fourier transform of an Lz-function in (1.9) should be understood in the "square-mean" sense, i.e. by approximating f with nice (L'; compactly supported) functions. Let us make a few comments: 0
the general inversion formula (1.8) is equivalent to a special case z=O (and is
often stated in that form), G
multiplication of ?(() by e i z * < corresponds to a shift f(O)+f(z), under the inverse transform 9':
?+f.
'Indeed, for any complete orthogonal system (basis) { $ J ~in } Hilbert space 36, any f E 36, is uniquely decomposed into a series f = C akdlr,and 11 f 11 = C 1 ak I Gk (Parceval identity). Clearly, factor ( 2 ~ ) "in (1.9) represents L2-norms of Fourier harmonics {gk=
1) 1 '
3The inclusion
er c A
j(<= ) U( I < I -"'), Slf(f)ld<.
whe_nce
follows from the differentiation formula (3): 9[13"f] = (i<)"j for anyf E, :e which yields absolute convergence of the integral
52.1. inversion and Piancherel Formula.
66 0
Plancherel formula (1.9) follows from the Inversion (1.8) applied to a function:
g(4=
If(. + Y)f(y)dY
Indeed, from (i)-(ii) it follows that
5 via (1.8), 0
?(()
=
f*f**
(1.10)
=I ?(() f, while g(0) = Ilfll'.
So inverting
we get the Plancherel formula (1.9) for f .
The result holds for any locally compact Abelian group
4 transforms functions { f(z)} on G into defines a unitary map: L'(G)+L2(8).
{ j ( x ) } on
G ([Loo];[HR]). Here
e, takes AnL'(G)+AnL'(G), and
We shall present two arguments based on very different, but equally important ideas. One of them exploits the Gaussian and Schwartz functions, another a "discrete (lattice) approximation" of
W", and the so called Poisson summation formula.
The Gaussian method. Obviously, it suffices to prove (1.9) for a dense class of functions in AnL' or L '.
One possible choice are infinitely smooth {f},rapidly decaying a t {co}
with all their derivatives
I z I -"aaf(z)-+0,
as z - o q for all m and a,
equipped with a suitable system of seminorms, e.g.
IIf llaP = "$2 I ."f%) These are
90
1
called Schwarlr functions on R". The class of Schwartz functions and
distributions is discussed in detail in 52.2. Here we shall only remark that the Schwartz class is sufficiently rich. It contains, all C3-
compactly supported
{f}; the Gaussian
.
function: G = G,(z) = e-tz2 (z2 = z z = I z I 2), and all products: G x "polynomials".
T h e latter class, 9 = Szf = p ( z ) e - z 2 / 2 : polynomial p ( z ) } , will be used in our proof of inversion Theorem. Let us remark that space 9 is dense in C,(R"),
by the Weierstrass
"uniform approximation" principle, as well as in other spaces: Lz, ZP, A n L ' ,
Y, with
appropriate norms. The inversion for the Gaussian itself is verified directly (problem 5):
9:Gt-Gt = (2nt)"/2e-t2/4', i.e. Gaussian is Fourier-transformed into another (differently scaled) Gaussian, whence follows G(z) = T-'[C]. Then it remains to apply the differentiation and multiplication rules 3)-6), to get Fourier-inversion for any product-function, f = p , ( z ) G in 9. The proof is completed by the standard density argument: once Plancherel formula is established for a dense subspace 9 C Lz, it follows for all f E Lz.
1.5. Poisson Summation Formula: Let functions f and
1
be integrable and
52.1. Inversion and Plancherel Formula
67
continuow on Rn (e.g. f E A n L'). Then for any period T
> 0, (1.11)
Similar result holds for an arbitrary lattice: r = { m = Emitj}, spanned b y vectors { t , ;...tn} c R", and its dual lattice, r1 = (7= E kjaj}, where a;. t . = 2x6. .. Namely, 3 83 (1.12)
where J L = R"/r
21
T" denotes the fundamental domain of
r.
The proof is fairly easy and straightforward. We define a new f-periodic function:
Series obviously converges and gives a continuous function on torus R " / f
N
Tn. Function
F is expanded in the Fourier series,
with coefficients {a,) on the dual lattice f':
Next the torus inversion formula applies,
and the result follows.
Remark Formula (1.11) has many different meanings, interpretations, and S(x - rn) on Rn, "6applications. One of them is in terms of distribution: 6, = function of r".The LHS of (1.12) pairs b r to a nice test-fun%sxf(z), while the RHS defines the Fourier transform of 4 = S f . By definition, ($If^) = ($If), for any distribution 4 and a test-function f. So (1.11)-(1.12)reads,
c
in other words LL6-function of
r is Fourier-transformed into -volX1
"6-function of
rl".
$2.1. Inversion and Plancherel Formula.
68
In such s form Poisson summation turns into a special case of the general result for arbitrary Abelian locally compact groups: if
r
is a closed subgroup of G, and
r'
denotes
r in G, r'={ x : x I r = l}, fhen 6-funcfion of r (understood as (6rIf ) = Sffdy,dy-Haar measure on r), is Fourier fransformed into the 6function of Z", T:Cr-Constbrf. The key idea is that averaging over r, sends functions the annihilafor of integral
f(x) on G into functions on the factor-group G / r ,
f ( . ) + f ( z + r )= Jrf(z+V)dY = 6 r * f . The dual group l o G / r is identified with
r', and
the Fourier transform takes the A
A
projection map: f --df *f (from G to G / r ) ,into the restriction map: f -f
1 r'.
Poisson summation immediately yields the inversion/Plancherel formula on R". We let T-co in (1.11.), and remember that f(z)-+O at {m}. The LHS of (1.11) clearly goes to f(O), while the RHS converges to
&/?(O&, so f(0) =
(2x1
?(t)d<, QED.
Further applications of the Poisson summation will appear in $2.4 and chapter 7.
52.1. Inversion
69
and Plancherel Formula
Problems and Exercises: 1. Show that any continuous character on R (a homomorphism x:R-C') is of the form x ( a ) = eZa, for some z E C (prove it first for rational a E Q, then use continuity of x ! ) . Deduce the character formulae (1.0) for R"; Z"; 1". 2. i) If a sequence of functions {Fn]on R"; 1" approximate &function: (F, I h) -h(O), as n-co, for any continuous h, then the corresponding convolution kernels: F,*f(z)-f(z), as n+m, for all {f} in C and Lp, 1 5 p < cm (such sequences are called approximate unities).
ii) Any sequence of positive functions {Fn] with the properties: (a) J F , d z = 1; (b)
1
1x1
F,dz
-.)
0, as n+m, for any
6
0,
56
forms an approximate unity. 3. Completeness of characters
eim * =} on 1":any f E e, or Lz is approximated by
{x,=
trigonometric polynomials,
c
PN=
lml I N
bm
eim.I
Hint: I t suffices to check the result for continuous {f] with the standard 11 1 1 ~ . Note, that partial sums, may not converge t o f E e uniformly, in general. However, certain means of s, do, like Ces a Po, i) Show that both sums s, and cn are given byconvolutions, s,(f) = S , * f ; c , ( f ) = F,* with kernels, which in
sn(e) =
f,
1-Dcase take the form, sin'? ( l - Ln+ )e1 i k e = - , sin':
sin (n+,$)~
called the Fejer kernel.
sin^
; andF,(B)=
Ikl 5"
ii) Show that the Fejer kernel forms an approximate unity (problem 2), hence F,*f(z)+f(z), uniformly in e, as well as in any LP-norm, 1 5 p < 00. 4. Establish the decay rates of Fourier transform of m-smooth functions
1"and
emand
36,
on
R": estimates (1.11); (1.0); (1.0).
5. Fourier integrals on R are often evaluated by the techniques of complex analysis: calculus of residues, branch-cuts (if necessary), changing path of integration. Use complex analysis t o evaluate the following integrals:
-t2
i) Gaussians: G, = T(e-t22/z n- D).
sin? iii) T(-) = 2n sinh bz cos@ + ~ b
= G, (in 1-D); and (znt)-"/ze-t2/zt (in
on R 0 < a < b (calculus of residues).
0 ~ 5 '
b
Integrals (ii-iii) are evaluated by the residue calculus. They represent the Poisson kernels (j2.2) in the half-plane and a strip.
70
$2.1.Inversion and
Plancherel Formula.
The following 3 Fourier integrals appear in the harmonic analysis on group SL,(R) celebrated Selberg-trace formula ($7.6 of chapter 7):
00
vi)
xis
d 1 - 2Xcos 0 +
X2
dX = -
rsinhs(O+r) sin 0 sinhns
' -'<
- the
< O'
6 . Show that the Fourier transform o f the radial function f = f(lz I) in R" is also radial, find the convolution of two radial functions f(r), g(r) in polar coordinates. 7. Show that the Gaussian G,(z) and the Poisson kernel Pi form convolution-semigroups:
G,*G, = Gt+,, and P,*P, = Use the Yproducl,+convolutionn property of 4, and explicit form of
and
p).
52.2. Fourier transform on function-spaces
71
52.2.Fourier transform on function-spaces. In this section we shall briefly discuss the basic properties of Fourier transform on various function-spaces, including LPspaces, Sobolev spaces, distributions and analytic functions. The material will provide some background for a subsequent study of differential equations in 52.4 and other parts of the book.
2.1. L p - spaces and interpolation. Plancherel formula implies unitarity of the normalized Fourier transform '3':f+(27r)-"/23 on L2, On the other hand Riemann-Lebesgue Lemma means that 5:L'+Lm
boundedly,
Such estimates can be extended to other LP-spaces by the Riesz interpolation Theorem. Namely, S:L*-+Lq, where q = L,is the dual Holder index, furthermore P-1
1-
(Hamdorff-Young inequality).
The exact constant in the Hausdorff-Young was calculated by Beckner in ~ O ' S ,
2
and was shown to be attained on the Gaussian function f =e-" 12. Interpolation provides a powerful tool in estimating linear (bilinear, multi-linear) operators in { L p } and other scales of spaces. The simplest linear version states,
Riesz-Thorin Theorem: i f operator T maps boundedly Lpl(X)+Lql(Y), and L P 2 ( X)+LQ2(Y ) , IITf IIQ' I ClIlf then for any pair of indices:
IIP1'
1-l-s Bp1
andllTf
s +--' P2'
I Q2
1-14
I C2Il.fIIP2'
&
B -T i+ operator T is bounded from L p ( X ) to Lq(Y), and its
I T IILPLQ I c:-sc;. So function
4($f) = ZoglITllLpLqis
convez in the plane of reciprocal parameters
{++I. The Hausdorff-Young inequality arises via interpolation between pairs of Holder indices ( 1 ; ~ )and (2;2) for '3. Of course, interpolation does not yield the best constant
$2.2. Fourier transform on function-spaces
72
( 2 . 1 ) , as C , = C2= l : + C ( p ; q ) = 1 , for all p . The interpolation also applies to convolutions of Lp-spaces. This time we deal with triples of indices so that Lp*Lq c L', in other words, (f;g)--+f*g,maps
(f;$i),
boundedly Lp x Lq4.L'. The general principle for bilinear T remains the same, as for
and LP2x LQ2+L'2, with
linear operators. So if T maps boundedly LplxLql+L'l, bounds C,;C,, then for any interpolating triple:
1-L
8-1
1-
8
1 - A s-1 T-rrl+F;
8-1
q-q+%; T maps Lp x L b L ' , with the norm estimate p-pl+pz;
To apply ( 2 . 2 ) to convolutions of LP-spaces we observe that L'*Lp words operators { R : f E L'} are bounded in L p by
f
llfl/l).
c L p (in other
On the other hand
L2*L2c L". Interpolating between the resulting triples ( 1 ; h fand ) ($;;;O), we get the Young's inequality,
Young's inequality holds for all groups G = R"; T";Z". Moreover, for compact groups (like T") the range of (2.3) can be extended to { ( p , q , ~ ) : 2 $ 1 - l } , due to the
+
obvious inclusions: L"
c LQ,for all 1 I q < p I 00.
The proof of the Riesz interpolation Theorem is based on a version of the maximum principle for holomorphic functions in the strip, called 3-line Lemma (see problems 2,3,4). Let us remark, that so far we didn't use any special features of R"; 1" or Z", so all results about Lp-spaces, which involve interpolation (like Young and Hausdorff-Young
inequalities) extend to arbitrary locally compact Abelian groups [HR].
2.2. Smooth functions and Sobolev spaces. Riemann-Lebesgue Theorem (problem
1) implies that continuous functions on T" (or continuous, compactly supported
R") have
{f} on
bounded, decaying at {m} Fourier transform. Thence one could easily show
(problem 5) that m.-smooth functions f E Cm(Tn), or er(R"), have their Fourier transform/coefficients {T(k)}; {f^([)} decay at a polynomial rate:
Conversely, if Fourier transform
T([)
decays as
I [ I -p,
with p
> m+n,
then f(z)
is at least m-smooth. Both results are easily verified via the differentiation formula:
( 8 f j= (it)'?([),
and the obvious relation:
f^ = O( I ,$ I -"-')+?
E L'. Hence, by
52.2. Fourier transform on function-spaces Riemann-Lebesgue Theorem,
73
f = S-l(f^) E C,.
Thus Fourier transform relates the "degree of smoothness of f" to the "decay rate of f^ at (00)". More precise results, however, could be stated in terms of L2-norms, rather than uniform (L"). The main disadvantage of smooth functions from the standpoint of harmonic analysis, is that erndoes not allow a simple characterization in terms of Fourier transform/coefficients, which could done for Lz-functions. Thus we are naturally lead to the notion of Sobolev spaces.
Sobolev space X,, all their derivatives
{ti(@)
of order rn, consists of functions {u},which belong to L2 with E Lz: I (I I 5 rn}. It is equipped with the norm
The Plancherel Theorem and the differentiation rule of 52.1 give an explicit characterization of 36, in terms of Fourier transform/coefficients {ti(()} of u,
- sum of weighted L2-norms with polynomial weights w(()= (a. It is easy to check that the sum of monomials {ka} in (2.6) (respectively partials
{aa} in (2.5)), could be replaced by powers of (1+ I k I 2)m, respectively (l-A),,
where
A = V . V = Ca: is the Laplacian on Wn. So norm (2.6) is equivalent to IIuI?mz J ( 1 t
~E~2)m~~(()~z=~~(l-A)m~2~~~~.
In other words 4 takes Sobolev space 36, onto the weighted Lz-space, Lz(wmd() (or L2(wm)in the Tn-case), with polynomial weight turn(() = (1
+ I(I
2)m,
and
This allows to extend the notion of Sobolev n o m and derivatives (powers of
Laplacian) from integers to all real (fractional; negative) values of m. By definition,
I
(l-A)-% = T 1 ( 1 -t I [ I z)-s/24u = I(l+I ( 2)-s/2G([)e""'td[.
(2.7)
Fractional Laplacian (2.7) are given by convolution kernels, Bs*u, with the so called Bessel potential -s/2
B s = ~ ~ , ( I ~ I ) = ~ - "It1 ( 1 t) I, where K,(r)-modified Bessel function (3-rd kind) of order v = 2
Thus we get a scale of Sobolev spaces {X,:s 2 0 } ,
(problem 3).
62.2. Fourier transform on function-spaces
74
36, = (1-A)-"/2[L2] = B,*L2 = 9-'[L2(w,d()], where w, = (1+ I
t I ')'.
The above definition also extends for negative s by duality. Namely, we define space 36-, as the dual space (of distributions) to 36,, via the pairing,
( F I f) = J f ( z ) F W s =
J 3 ( 0 f i ( O d t , f E 36,;
F E 36-s.
(2.8)
We shall list a. few basic properties of Sobolev spaces. Theorem 1: i) Spaces (36,) form a continuous scale of embedded Hilbert spaces:
with 36-, = "dual to 36, via the natural pain'ng (2.8)".
ii) O n t o w 8" ( o r more general compact manifolds, domains) the embedding 36, c 36, is compact f o r all t < s. iii) Sobolev embedding: spaces (36,) are related to
{em}in the following way:
Xm c e, for m > n/2, more generally,
ern+'c 36, c em-n/2,for any m I s < m+1, Proof: (i-ii) Each 36, is mapped onto 36, by a fractional Laplacian (l-A)-m, which is Fourier-equivalent to a multiplication of ['-sequences
m = d-t 2 '
of coefficients by a
decaying sequence { a k = (1+ I k I 2)-m/2}, clearly a compact operator. (iii) Multiplying and dividing a sequence of Fourier coefficients of f E 36, by a polynomial weight (1+ I k
c
lakl
I 2)m/2,
and applying the Cauchy-Schwartz inequality, we get
=~~(1+~kI2)m/2a~J(1+I~I2)-n~2~~~{(1+I~I2)m'*~~}~~e2~(l+IkI*)-m
which shows that the Fourier series of u(z) =
C akeik
'
is absolutely convergent, whence
follows continuity.
Remark 1: The last statement of the Theorem extends to embedding of Sobolev spaces 36, into the LP-scale. The general result (Sobolev-Hardy-LittlewoodTheorem) claims: space
36,
c LP - continuously
embedded, i.e.
II f llLP I I ( 1 - 4 , h for all
;1,
(2.9)
$ = f - 3 (on R"), and $ 2 f - 3 (on 1").The argument exploits interpolation for
Bessel potentials B , E Lq, with q = y . Convolutions, u+B.*u, with such B, shifts any LP-space into Lr, where
$2.2. Fourier transform on function-spaces
Consequently, for negative s, a,,,,,
75
2 36-, 2 '3,,,-n,2. Therefore,
as claimed in (i).
Remark 2: Sobolev spaces provide a natural framework to analyze solutions of differential equations, as outlined in $2.3. This often requires Sobolev spaces of both integral and fractional order, as well as Sobolev spaces on manifolds and domains
RcW". Let us also notice that spaces 36, constitute "m-smooth vectors" of regular representation R on L2(Wn), or L2(T"), in the terminology of $1.4. 2.3. Distributions { F } are defined as continuous linear functionals on suitable spaces of test-junctions {u}, like em, em, etc. One usually writes them as formal integrals
/
( F I u ) = F(z)u(s)dz.
(2.10)
So for the class of bounded continuous functions, f E e ( X ) , all functionals are given by finite Borel measures p E A ( X ) , ( p I u ) = u(z)dp(z).Test-spaces en, em we equipped with a sequence of seminorms,
I
{Ilullm=
c
maz I a"u(z) I ;m = 1;2;...},
therefore the corresponding distribution spaces, called gm,goo,consist of functionals satisfying
m
I ( F I U ) I ICCIIuIIj.
(2.11)
j=O
Using (2.11) one can show Proposition: Any distribution F E bDm (m = 0; 1;..LO),is equal to a fznite sum of
derivatives of Borel measures, (2.12)
Integer m is called the (differential) order of F . Derivatives of measures in (2.12) are defined via pairing to smooth functions:
I f) =
/ aaf(.)dP(.).
The proof of (2.12) is fairly straightforward. Indeed, the m-jet map
f ( z ) 4 ) ( 4 :I a I I ml7 takes P ( R ) into the direct sum:
e, =
C(R") @ la1 5 m
- N-component vector functions on R";
N = (n+m m 1-
$2.2. Fourier transform on function-spaces
76
Distribution F defines a bounded linear functional (in the supnorm of
em c eN,which can be extended
through the entire space
vector-valued continuous functions, Idpa: I a
eN, is
e,)
on the image
e. But any linear functional on
given by an N-tuple of Bore1 measures
I I m}, QED.
Let remark that all natural linear operations on test-functions can be transferred to distributions by duality. For instance, differentiation is introduced via
( P F I u ) = ( F I (-a)%), by analogy with the standard "integration by parts" formula for test-functions. Clearly, daF E am, for any F E am, in fact aa:am+gm+I a I . Thus we get the following chain of test-function spaces along with their distribution-spaces
1-
00
...
ln7eEC ... C e m c ... c e c L ' c ~ = P o c ca,c
co
...c a = ua, (2.13) 0
Examples of distributions include (i) &functions and their derivatives: 6 z ( x ) = S(x-z), or (6(");u)= ( - 1) I a I u(a)(O), as well as their linear combinations, e.g. hA = cb(x-m) (A- lattice in Rn); (ii) &functions of hyper-surfaces and submanifolds: (bC I u) = J,udS, where dS is the natural surface area element on C, like the spherical distribution SS = (S = S,- sphere of radius r in Rn), that appears in the wave propagator solution of the wave equation); (iii) Certain singular integrals, like Halbert transform,
I
(HI u ) = P.V. +x
(principal-value integral) = lim
WX,
as well as its higher-D analogs, called Calderon-Zygmund integrals, can be interpreted as distributions ([Zy];[St];[SW]). (iv) Distributions often arise as limits of families of regular functions4, e.g. lim 1_ = H i~(x). ,4J z+ae
+
Another linear operation on functions, the change of variable, f +f o 4, can also 4Such approximating families {fc(z)}, called regularizations, exist for any F E 3.The easiest way to construct a regularization is to take a em- smooth approximate unity {hr+6}, e.g. h, = c-"h($), with h E ( h 2 0, and J h = l), and to convolve h,*F = f, (problem 1).
cr
$2.2. Fourier transform on function-spaces
77
be transferred to distributions (problem 2), def
( F 0 4 I 21) = ( F I
'&'4-7 jF ( l ) ( h ) 4-Y+. =
0
0
Distribution spaces on torus can be characterized by their Fourier coefficients, the same way as smooth functions. For any F E !Dm we define
bk = ( F I e-iz * k =
j
F(z)e-ik*z.
T"
Fourier coefficients of continuous functions, u(x) = x a k e i k ' 2 , are bounded and go to 0, as k-too, those of cm-functions decay at a polynomial rate, ak = o( I k I -"). Since, ( F I 21) = x a k b k (an extension of Plancherel's Theorem to pairs 'functiondistributions"), it is natural to expect {ak} to increase polynomially. Indeed, if F € 9,
I l b r l = l ( F l e i k * z ) l
(2.14)
for any F of order m. We shall illustrate (2.14) by the example of &function and its derivatives,
6=
C eik
*
I , so all bk= 1;
6(")=C(ik)a eik", so bk - (ik)a* The latter extends to all (integer and fractional) powers of the Laplacian:
(-A)sb=
C 1k12seik'z,or(1-4)'6=
~(1+Ik12)Seik*z.
Hence both belong to space gm,for m > 2s. Remark 3 On R" test-function and distribution spaces are defined similar to the compact (torus) case, but additional constraints are imposed at {co}. If no such constraints are
imposed (spaces
em and P)one gets
compactly supported C-spaces 2(';:
The R"-Sobolev spaces 36,
compactly supported distributions '3. Conversely,
Cr) give all distributions (with unlimited support).
satisfy all the properties (i-iii), except compactness of the
embedding 36, c 36,. To show for instance, the embedding: 36, c C,, for m
> n/2, we
replace Fourier sums by Fourier integrals,
If
3 E L',
then f =
the rate of decay of
fF'(3)E C,. 1 at
As in the torus case smoothness of f can be linked to
00,
f E cy *?(c) = O( I < I Conversely, sufficient decay rate O( I C I -"')
or
f E 36, *(I+ I < I m ) j E ~
2 .
off. o f f guarantees (m-g)-smmthness 2
52.2. Fourier transform on function-spaces
78
2.4. Schwartz functions and tempered distributions. The most convenient classes
of functions and distributions combine both features (smoothness and the decay rate ay {co}). These are so called Schwartz finctiom 9 and tempered distributions Y'. Space I consists of €?-functions, that decay faster than any polynomial, Y = {f:z*f (z)+O,as
(a)
z+co}, along with all their derivatives. It is equipped with a sequence of semi-norms,
I(f 1I.p
=
1 z*@f I Lm
; or equivalently
1 f llpq
1
= (1t I 5 I ')P(l-A)qf
/IL=-.
(2.15)
Let us that the choice of supnorm in (2.15) is not essential, it could be replaced with any other Lp-norm. Also the order of two operations (differentiation and multiplication) could be interchanged (problem 4). So space 9 can be characterized by semi-norms,
lp-A)p(l+ Space
Y
I I 2)Pf11L2 <
is sufficiently rich, it contains
(2.16)
{ep}; the
Gaussian { e - t z 2 } ;
{polynomials}x { e - t z 2 } , etc. The important features of the Schwartz spaces are due to the fact that T:Y+lf, represents an isomorphism of 3, i.e. semi-norms (2.15) or (2.16) of f and are equivalent. Schwartz (tempered) distributions Y' are linear functionals on Y. So distribution F E Y", if it satisfies,
7
I ( F I u)I 5 C
~ I 1lpq; I Ufor any Schwartz u.
This class includes all examples we encountered so far, like the lattice &function,
8,
6, = c 6 ( z - k ) ; its transform;
= z e i k z ; &functions of
hyper-surfaces, or
submanifolds in Rn; their derivatives, etc. We shall list a few other examples of Schwartz distributions and their transforms (see problem 7,8): i) Dirac-6 is transformed into constant 1, hence distribution 6 E KS, for s > n / C . ii) &function of subspace E
c Wn, 6E+(27r)n-k6E1;
where E' is the orthogonal
complement of E , k = dimE, iii) F = bS (&function of the r-sphere S) + P(() = cn(r I ( I )-"l2+lJ n/2-1('
I€ I)
- the Bessel function of order (;-I);
iv) fractional derivatives of Dirac 6: (-A)-",
(Riesz potential), and (1+ I ( I 2)-s/2
or (1-A)-s6
I ( I -2s
(Bessel) potentials (problem 9).
v) &function of the light cone C = { z = c
m
} in
W3 is Fourier-transformed
into 8&;0;C)
turn into
1 =r n H ( c C-
m),
52.2. Fourier transform on function-spaces
79
where H is the Heaviside function. So the Fourier transform of 6, is supported in the dual cone C ' = {(:(.z 2 0). isomorphically.
Y the Fourier transform maps Y'+Y'
As with
2.5. Analytic functions and the Paley-Wiener Theorem. After the Fourier analysis of LP-spaces and smooth functions
decaying functions LP, = { f :f (z)e"
{ern;Xm}, we turn
our attention to exponentially
I I E Lp}, and yet smaller spaces of compactly
supported functions. Clearly, in both cases the Fourier transform ?(() extends from the c Cn. In the former case the domain of
3
real space W" to a complex domain {( = (+iq}
is a tubular neighborhood of the real space R", O,= {(+if): 171
imaginary iq, furthermore,
1
I,r,z(d()
5 eprlllf
(2.17)
llL2'
The necessity of (2.17) is fairly straightforward. The proof of sufficiency is more involved. T o give the reader a flavor of the arguments we shall briefly outline the 1-D case. The proof exploits the so called LZ-Hardy spaces, made of analytic functions { F ( z ) } in the half-
>
plane H = {Sz 0}, square-integrable along lines {% = yo}, parallel t o the real axis, such that all L2-norms are uniformly bounded, m
]' I F(z+iy) I ' d z 5 C; for all y > 0.
-m
Any Hardy function F ( z ) is obtained by Fourier-transforming an Lz-function f ( i ) , supported on the positive half-line [O; m) (problem 6): 00
F ( z ) = f(t)e""' tdt,
z
E H.
(2.18)
0
A similar result holds for functions { F ( z ) } , holomorphic in the lower half-plane, the corresponding {f(i)} being supported on the negative half-line
(-00;
increasing functions { F ( z ) } (2.17) the support of {f(t)} gets shifted by
01. For exponentially p to the left/right, an
easy consequence of the Fourier-translation formula of J2.1:
eizpF(z) = 9 t - , J f ( i + ~ ) I , and the fact that {eazpF(z)}belongs to the Hardy Lz-class. Therefore, analytic function F ( z ) , exponentially bounded in both the upper and the lower half-planes, has its inverse 9-transform inside the interval [-p;p], QED.
62.2. Fourier transform on function-spaces
80
Problems and Exercises: 1. Riernann-Lebesgue Theorem: Fourier 1,ansfot-m ?(<) of any L'-function f on R" is continuous and goes lo 0, as [+m, i.e. f 6 e., Hint: Check the result directly for indicator functions {xe} of rectangular boxes: B=[al; b,] x ...x [a,;b J. Then use the standard density argument: any f E L' is approximated by linear combinations of xw 2. Three line Lemma: If function F ( z ) is holomorphic in a complex stn'p 0 < 9?ez has maximal values on both sides of the strip:
< 1, and
M , = auP(F(it)\ M1=sup I F(l+it) 1, t then F is bounded inside the strip'and /F(F(s+it) I
5 MA-";,
for all z=s+it.
1'. Truncate F ( r ) by a family of exponentials {Mb-'M;e('2-')~n}, 2
(i) product G,(z) = MA-zMfe(z -"/"F(z),
to ensure that
goes to 0, as z=s+it+m, and
(ii) maximal values of G, on both sides of the strip I1.
2 . Use the maximum principle for holomorphic functions to show IG,(z) Observe, G , ( z ) + F ( z ) , as n-m!
I 5 1, for all
z.
3. Prove the Riesz interpolation Theorem,
I s ( T f )(z)s(z) dz I IMA-":
IIf llp(s)II9 ll*'(s);
P'
=
gig
for a dense subset of stepfunctions functions {f;g}, i.e. linear combinations of the indicator functions {xi = ,yo ,} of subsets {Q C X).
1 . Represent: f =
Cajx3;
ck bkXk; with coefficients
3
g=
j
ie . a . = r .e 3; b, = pkei4,. 3
3
2 . It is convenient to use reciprocal Hijlder indices:
a =:;; p = I. 9' a 0- Po'
a,=+ 90
al, pl; a(.) = (1-z)ao+zal,
etc.
Introduce two families of functions (one-parameter deformations of f ; g) depending on complex z , - Cr+z)/aeiejxj; gz = c p j ,1-@(z)l/(1-@)ei4kxk, z -
3
where a,p in the denominators of powers of {r .}, { p , ) mean a ( s ) and p ( s ) for some real 0 < s < 1 (without loss of generality assume 11 = 1; 11 g 11 l-p = 1).
f]la
3 . Consider a holomorphic function F(z) = of the strip: 0 < 32er < 1. For
s T(f,)g,dz,
and estimate it on both sides
express F ( i t ) = $T(f)Gdz,in terms of new functions f = Crjao/aez(ej+...) x j ; G = C P k(l-@O)/(l-@)ei(~k+"')xk.
Apply the (ao;C)o)-estimate to the pair { j ; j } ,to show
Is
m
GdzII...
I~ollflll~aollGll'~@o=~o.
1-
Similarly, on the other side of the strip
IF(l+it)I
=I S T ( f , + i t ) S l + i t d ~ ( I M lIlf Ilo119111-p=M1.
Conclude the proof by the Three Line Lemma (problem 2).
52.2. Fourier transform on function -spaces 4. Outline the proof of the Interpolation Theorem (specifically the construction of holomorphic function F ( z ) ) , for a bilinear map T.
5. Compute LP-norm of the Gaussian and check the Beckner estimate of the YoungHausdorff constant (2.1). 6. Hardy classes on the half-plane [Hofl are defined by (2.18) with an L2-norm (or, more
generally, LP-norm, 1 < p < m). i) Hardy functions are known t o have the (limiting) boundary values in Lz (Lp):
1
F(z+ic)-F(z), as (-0, and F ( z )
1 5 C.
This holds for yet more general harmonic functions u(z) in {Sr> 0) (solutions of the u = 0, while F ( r ) solves only the Cauchy-Riemann Laplaces equation: Au = equation: 4 F = O!), as a consequence of the Poisson representation of u (§2.2),
I Py(z-r)u(T)dr, Y =r(22+y2)'
u(z+iy) = (Py*u)(z)=
with the Poisson kernel P&Z)
The latter forms an approximate unity {Py}, as y-0, i.e. P , 4 , and Py*u+u, for all u E LP (see problems 2,5,7 of 52.1 and problem 2 of $2.2). ii) Furthermore, analytic (Hardy) functions F ( z ) are represented through their boundary value by the Cauchy integral:
a convolution of the Cauchy kernel K , ( z ) = 2 4 2 parts are
WK = py(z) =
-+@(z) 2r(zZ+y2)
SK =Hy(z)=
2r(z2+y2)
+ iy)'
whose real and imaginary
- Poisson kernel;
+m;
where H denotes a distribution P.V.(i), called HilberZ transform (see 32.2, and [SW]). iii) The FouGer tran_sforms of both distributions, Hilbert H and Dirac 6, on R are easy t o compute: 6 = 1; H ( t ) = sgn t (52.2). iv) Use (i-iii) t o show that, any Hardy function F(z) in H is given by Fourier transforming an Lz-function f ( t ) on the half-line (2.18). The latter provides a crucial step in the proof of the Paley-Wiener Theorem.
7. Smooth regularization of distributions. Take a smooth compactly supported function h ( z ) , consider a family of dilations: h,(z) = r - " h ( z / f ) , and show that convolutions {h,*F} yield a family of regular functions, that converge t o F: ( f c 1 u)-(F I u), for any test-function u.
8. a ) Show that (6 o 4) = y
d ( z - y ) ,
E ToI 4 " V ) I
for the Dirac delta on R, sum over all zeros
ro= {y} of functiond. b) If$: R"+R, then (6 o 4) =
dS
4(z)= 0vgl' I
where d S is the Euclidian surface
element on the level set {z:d(z) = O}. 9. Bessel and Riesr potentials are formally _defined as Fourier transforms of tempered distributions: B , ( ( ) = (1 1 ( I z)-s/2, and R , ( ( ) = I ( I -'on R".
+
81
52.2. Fourier transform on function-spaces
82
i) Use polar coordinates and the Bessel-function representation of is = Jv(r ...) (on the r-sphere) to compute B,( I z I ) in terms of the modified Beasel function K,,. A Hankeltype integral relates Bessel functions J and K of the 1-st and 3-rd kind:
In our case v = n- 1; p = S- I. 2 2 ii) Check that Fourier transform takes homogeneous functions/distribuGons of degree s: f(tz) = t-'f(z), t > 0, into homogeneous functions of degree (s-n): f(t() = t""f((). Apply it to the Riesz potential to show, R,( I z I ) = C(s) I z I Compute constant
'-".
C(S).
10. Show that the Lm-norm in the definition of Schwartz functions Y can be replaced with any other Lp-norm. Show also that operations (1+ I z I 2, and (1 - A ) can be interchanged in the definition of 9. (Hint: apply a version of the Hardy-LittlewoodSobolev inequality (2.17) for LP-Sobolev spaces: 36: = (l-A)-'I2LP, to show that 36; is embedded in LQ, for = - & and
II f 11LQII (1-A)"'2f
IILP'
$2.3.Applications o f Fourier analysis
83
v.3.Some applications of Fourier anal+. Here we picked up a few selected topics and applications of the commutative Fourier analysis: Central limit Theorem of probability; the Heisenberg uncertainty principle; Finite Fourier transforms, Bochner’s Theorem and the Mellin transform. The latter has application to special functions and differential equations in R“, and also in the harmonic analysis on SL, (chapter 7).
3.1. The uncertainty principle. The Heisenberg uncertainty principle of the Quantum mechanics will be discussed in 56.2 of chapter 6. It has a simple Fourier-
for a n y function $ in
Iz I
and
I< I
Rn, such that $ and
4 are both square-integrable
with weights
respectively.
Rephrasing (3.1) in terms of the derivative (gradient) a$, we get
In this form it becomes an easy consequence of the standard integration by parts formula, /z$# =
/z(i I $ I
2)1
=
-$I 14 I ’,
(1-D); or/V$.z$ = -
I31
$ I ’(Vex) =
-51
I $ I 2, (n-D),
and the Cauchy-Schwartz inequality applied to the LHS,
I/z$.v+(/
lv$,~yz~
Izdi~)li2(l
Relation (3.2) implies that one can simultaneously localize functions $ and
4 at
I $ 11L2
(0) maintaining the Lz-norm = 1 (LHS), in other words, localizing supp($) in zspace will ‘‘stretch” its Fourier transform $, hence increase gradient, and vice versa. In quantum mechanics operators z: $+$, and &$+a$, represent the “position and momentum” of the quantum particle, while integrals
(1 I4
V d x ) and (IlW 1 2 4 ) estimate errors in their measurements. So the Heisenberg principle prevents
simultaneous precise determination of both operators to any degree of accuracy.
3.2. Finite Fourier transform. Here we shall briefly discuss the Fourier transform on finite commutative groups. The simplest of them is a cyclic group Z,. Its characters
52.3. Applications of Fourier analysis
84
coincide with the n-th roots of unity ( x Q $ j ) = w J } , where w = e z p ( F ) . So the dual group Z, N if,, and t,he Fourier transform on Z, becomes
c5
s:f(j)+?(e) =
f(j)w
3e.
(3.3) Characters { x Q= (wQJ)} form an orthogonal system of eigenvectors of the o53
n-1
translation operator R:f(j)+f( jtl),on space L*(Z,)
N
C"
, with norms
(x I x) = IIXIIZ = n. The inverse transform takes the form,
c
+i(j)= A
4-1:qe)
o
while the Plancherel formula,
a(e)w~l,
5 Q 5 n-1
llfllZ= ~lf(dlZ=AE I?(e,12=11?112, both being obvious consequences of the orthogonality of {xl}.
":I
We shall study now the czrculunt mat& A, that appears in many applications of finite groups (in linear algebra, probability, etc.),
[ :; :
... A = ... ... ... ... ...On-1 an a2
1 . -
each subsequent row of A is obtained by a cyclic permutation of the n-tuple (ul; ..a,,). Notice that matrix A is a convolution with function u(k) on Z,,
A f = a* f. So Fourier transform (3.3) applies to diagonalize A (find its eigenvalues and eigenvectors), compute inverse, determinant, etc. A specific model of such A is given by the
random walk on
Z,, where
az =a,
=f
(probabilities to j u m p from point (1) t o its neighbors (2) and {n) in a unit time), and the rest aJ = 0. More generally, we consider a stationary stochastic matriz':
C a , = 1 (aJ measures probability
aJ
2 0;
to j u m p from cite (1) t o { j ) ) .
Characters { A @ } form a complete set of eigenfunctions of A with eigenvalues,
c~,~J~,
A, = q e ) = - the l-th Fourier coefficient of function { ~ ( j ) l} ,= 0;1;...n-1. Diagonalizing A, via 4, 'Stationarity means that all rows of A are permutations of a single row, so the process is independent of the starting point: probability of "i+j"-jump depends only on the difference i - j E 2,: a . . = a(i-j). 13
92.3. Applications of Fourier analysis
85
allows to compute d e t A ; A-', all iterates A"', and yet more general "functions of A", B = f ( A ) . Indeed, any such B is itself a convolution with function b ( j ) , whose Fourier transform, b = f i?, so A
0
b(j)=T'[foi?] =
Sc e
foZ(e)wje.
As an application to random walk on Z,, we immediately find the probability distribution of the process after m steps (units of time), i.e. find entries of the m-th iterate Am. If p$:)+k denotes the probability of jump from j to j+k after m steps, then p$:)+k = s-'[xT].
Hence, pi:) =
ax
and P$:3)+k = 4
cosm (T); 2=e
e
c
cosm(@) cos(@).
e
Remark: More general finite groups G are known t o decompose into the direct product of cyclic groups, G cz
n Z,.
Hence, G =
nZ,
u G, and the Fourier transform reduces to
the Fourier transforms (1.3) on cyclic components of G. One interesting application of such finite Fourier transform is the random walk on the n-cube (problem 9).
3.3. Central Limit Theorem. Another interesting application of the Fourier Analysis is to r a n d o m walks on groups W , Z,or more generally, to sums of independent r a n d o m variables, S , = X,+X,+ ...+X,. Here we shall establish the Central Limit Theorem, that describes asymptotic distribution of S,, as n + w .
Any real random variable X defines a probability measure (distribution), d p X on R. Independence of two random variables X , Y means that their joint distribution d p x , y ( z , y ) on RZ is the product d p x ( z ) d p r ( y ) . So the sum of two independent random variables X + Y has distribution,
+x+,
= dPX*dClY (3.4) W. Another simple transformation of X ,
- convolution of two probability measures' on
scaling X + o X , results in the dilation of the measure dPx+dPx(3
(3.5)
We consider the sum of n independent identically distributed r a n d o m variables,
s,
x,,
= XI+ ...+
with equal probability distributions, d P x . = 4% 3
and ask about the asymptotic behavior of sums {S,} at large n. 'Convolution of L'-functions f*g on R" (or any group G) can be extended t o bounded measures {dm}, and other (compactly supported) distributions/generalized functions (see 92.2), via pairing to continuous test-functions
If).
$2.3.Applications of Fourier analysis
86
A typical example will be random walk on the lattice Z. Here variable X takes on values f 1 (jumps from
k t o k f 1) with equal probabilities
f, so dp = 36,+6-,).
Then the sum
of n variables, S,, measures the position of the walk after n time-steps, relative to the initial point.
It turns out that the limiting (large n ) behavior of sums {S,} carries some universal features, independent of a particular distribution dp. In what follows, we always talk about convergence of random variables {Y,} in the distributional (weak) s e m e , namely,
Yn+Y, if
I
fdpy
n-I
f d p y , for any continuous f(z)on W.
Theorem 1: L e t { X n } r be a sequence of independent identically distributed r a n d o m variables with mean: E = xdp(x), and the dispersion
I
0=
J
I(x-E)'dp = z'dp - E2 > 0.
Then n
(i) S, = EXi has n o limiting distribution, as n+w. j=1
.
(ii) $ , 4 , n
(iii) L E ( X I . - E ) + N ( E ; U ) = 0.
normal distribution with dispersion
J;;1
Rephrasing Theorem 1 we can say that the mean value of sum: E(S,) increases linearly with n, while its dispersion o(Sn-nE) u f i , grows as f i .
-
= nE,
The proof exploits the notion of the characteristic function of variable X ,
the Fourier transform of measure p. The characteristic function has the following properties:
A ) d(() is continuous and positive-definite 7, for all tuples
{t1;... tn} in R and {al; ... an] in C; also d(0) = 1; Id(€) I 51.
B) if measure dp has all moments to order k, i.e. 'The converse result is also true (see Bochner's Theorem below). is also true. Namely (problem 12): any positive-definite continuous function on group R; R", or more general commutative G , is the Fourier transform of a positive (probability) measure.
52.3.Applications of Fourier analysis then function
I
I z J I dp < co, j=O;
d(<) if k-smooth, and its derivatives:
87
1; .A,
#(O)
= iE; d"(0) = -o-E2.
C ) For any sequence of random variable { Y k } measures dp,
iff q5~n(<)+&(<),
n
uniformly on compact sets of R. The first two properties follow directly from the definition of 3 and positivity of measures dp. The last one relies on the fact that families of exponential {e'"(} with ( varying over compact sets D, approximate all continuous functions {f} on compacts in {z}. So the convergence of characteristic functions is equivalent to the weak convergence of measures. We consider characteristic functions
{dn} of S,.
By (3.4)
d"(0 = %*P*...*P) where
d(<) = b(() -
= d(O",
the characteristic function of a single variable distribution dp.
4 1, i.e. p = 6. This proves (i). Next we apply (3.4) and (3.5) to the =ergodic mean" AS,,, Property (A) shows that
d(<)" does not
converge to a continuous function, unless
ds,/,(€) = d(
d(() at
{0}, we get
{ 1 + iETi € + O(n-')}"+e'tE= '5(15,),as n+co. This implies by (C), dps /,,+bE,
(*r,
as claimed in (ii). Finally, Fourier transforming
of variable Y , = L ( S ,
distribution-measures dpy
- nE),
we get by (3.4)-(3.5),
J;; where 4 denotes thne shifted measure d p ( z + E ) , with the zero mean,
I
z d p ( z + E ) = 0.
Taking the Taylor expansion of 1 €2
(1 -lo,+O(n
-3
d(<) at 0 we get by (B)
11n+,-oE
2/2
(e-z2/2u)
=9
- the Gaussian.
Sequence of measures {dp, } converges, as above, to the normal distribution, QED. n
We shall illustrate the foregoing argument with an example of the random walk on the lattice aZ (problem 8). Then measure d p = f ( ~ 5 ~ + 6 - has ~ ) , characteristic function $6 = corn( = (1
jxdp = 0,
with mean: and dispersion: Hence, the limit:
-$[ "...},
0
= jx2dp = 2.
(1 - 2n a2t2
+ ...) +e -a2(2/2*
88
52.3.Applications of Fourier analysis 3.4. Positivedefinite functions and Bochner’s Theorem. In the previous part we
I
encountered Fourier transforms of positive (probability) measures q5(€) = e-jZ ’ (dp(z), which are positive-definite on R (or W”) in the sense of definition (6). It turns out that all positive-definite functions are obtained in this way. Bochner’s Theorem: Any continuous positive-definite function q5(()
on
W” is the
Fourier transform of a positive (probability) measure,
q5(() =
j
ei= * % p ( z ) .
Two direct arguments, outlined in problem 5, utilize spectral theory of selfadjoint operators, and regularization of distributions. There is yet another fairly general argument, based on the convezity and eztreme points. It applies to all locally compact groups (and even more positive functionals on Banach algebras). We observe that the set of all continuous positive-definite functions {q5} is a convex cone 36 in space e(W”), aq5 b$ is positive-definite, along with {q5;$}, for any
+
pair of coefficients a , b > 0. When normalized at {0}, subset 36, = {q5:q5(0) = 1 ) becomes a convex compact set in the Banach space
e.
c 36,
Any convex compact I< in a
+
Banach space has the set of eztreme points, e z ( K ) = {$:$ # adl ( l - a ) & } . By the general Krein-Milman Theorem, any point inside K can be represented by a “convex linear combinations” of extereme points. In other words, for any q5 E I<, there exists a probability measure d,u&$), supported on ez(K), so that q5 is a baricenter of dp,
(3.7) Applying this result to positive-definite functions one first shows that ex(%,) consists precisely of all characters {$,(()
= eiz‘t} (“extreme directions” in 36 must be
translation-invariant!). Hence, by the Krein-Milman, any q5(() = je-jr
*
(d,u(~),QED.
Remark The Bochner’s Theorem has a far reaching extension to noncommutative groups, known as Gelfand-Naimark-Segal Theorem. The latter characterizes positive-definite functions on G in terms of matrix-entries of unitary representations. The extreme points of % are then identified with irreducible representations, and formula (7) gives a decomposition of a unitary representation Td, associated to 4, into the direct sum/integral of irreducibles.
3.5. Mellin transform. Mellin transform of function f(t) on the half-line t E [O;m)
defined by integrating f against a family of multiplicative characters {X,(t) = t S ; sE C} with respect to the multiplicative-invariant measure d p = $ on R+;
is
$2.3.Applications of Fourier
89
analysis
t,7 E R+,
d p(t7) = d p ( t ) , for all
m
-m:j(t)+3(s) = p ( t ) t s - ' d t .
(3.8)
0
So A can be viewed as Fourier transform on the multiplicative group R+. Its inverse 31t-l is obtained by integrating T ( s ) over the imaginary line: {Res = b } i.e. { s = b iu},
+
F+f(t) = &Jmf(.)t-Sds.
(3.9)
b-im
A few important examples of the Mellin transform include 1) r-function:
r(s)=
T
e-tts-ldt
= ~ t L [ e - ~and ];
0
7
2) ~ - s r ( s= ) e-"ts-'dt
=~ [ e - ' ~ ]
0
the latter can be also viewed as the Laplace transform, L:f(t) = ts-'+F(X) = ~ ( S ) X - ~ ; W
3) r ( s ) r ( l - s) = /&t% 0
=
-+&I
4) Bessel function of the 3-rd kind (McDonald):
I%-$('
W
K,(r) =
+ l/t)"tu-ldt = At+v[e-i(t
+ l/l)z]
(3.10)
0
5) Riemann Zeta-function:
Formulae (1) and (2) are essentially the definition of r; function I{, of (4) is directly verified to satisfy the modzjied Bessel equation (problem 10); two remaining relations (3), (5), however, would require further explanation. The former (3) is based on the so called B-function,
for special values p = s, q = 1-s. The latter exploits formula (2) and the definition of Riemann zeta-function.
Let us remark that all examples (1)-(5) of Mellin transforms involve integrals of the form
$2.3. Applications of Fourier analysis
90
m
Z(s) =
F(t) tS-'dt, 0
with certain meromorphic/rational function F. Such integrals can be evaluated by the Residue Theorem, combined with the branch-cut along the half-line R+, Z(S)
=2 1-e2"'8 %
Res,j(F)z;-';
sum over all residues (poles) of F.
Fig.1: Branch cut and residues of e-z/2
F(z) = 2 sinh 212 in the Zeta-function transform. Here poles: zk = 2 r i k ; k = f 1; f 2; ...
Thus we compute the right hand side of (3) and (5), and find (3.12) ) s - l ( z b ) " z . - ](5) &[ A 2 sinh t / 2
- sin(?)
As the result we derive the functional equations for I' and Z-functions, which relate their values at {s} to those at {-s}, (3.13)
The Mellin transform, in general, and functional equations (3.13), in particular, play extremely important role in the analysis of both functions, numerous applications (see
r
and 2, and their
[Tit];[Ven];[Ter];[HST]). We shall see m in the next section,
and later in chapter 7 (representations and harmonic analysis of SL,). The standard notion of Zeta-function (11) can be extended to Zeta-functions of differential operators L (e.g. Laplacian on a manifold),
Z L ( s )= tr(L+) =
C Ale-';
{ Ak}-eigenvalues of L.
Analytic continuation (regularization) of Z L ( s ) , particularly, its zeros, poles and residues in C, contain some interesting spectral and geometric information about operator L (57.6). Among other things it allows to define a seemingly absurd (but highly useful) entity, the "determinant of L", det(L) = e-' one finds from (3.13), Z(0) = -f; Z'(0) = -$ln(27r).
I
('1. For the standard Riemann Z
52.3.Applications of Fourier analysis
91
Problems and Exercises: 1. Work out an example of the random walk on Z with uneven probabilities: p (jump to the right) and p = 1-p (jump to the left). What is the most likely position after n (large) steps? What is the probability t o find oneself in the range [ ( p - q ) n + a f i ; ( p - q)n P f i ] after n steps?
+
2. Apply the finite Fourier transform to the random walk on the n-cube
11
I, x
... x Z,
x
to
find P$:$~. 3. Verify that function K = K , ( z ) of (10) satisfies the modified Bessel's equation in variable I, a K " + $ K ' + ( l + Sz) K = 0. 4. Evaluate integrals (12) and establish functional equations (13) for
r and Z.
5. Prove Bochner's Theorem in two possible ways.
i) Using spectral theory of self-adjoint operators (Appendix A). Hint: show that the quadratic form, determined by 4,
(S,?
I 3)= J I d(t-~)3(O%dtdq > 0,
is positive for all functions f E 2'
nA, hence defines an inner product on A n L', II f Ib = (043I 3).
A
,
.
The regular representation, R,f = f(<-u), is unitary in Ailbert space Xd = 11 lidclosure of A n L ' . Furthermore, space %+ has cyclic vectors: any f, whose Fourier transform f(z)# 0, for all 2. Observe, that Bochner's Theorem is equivalent to the spectral decomposition (existence of spectral measure dp) for a family of commuting unitary operators {R,) on Xd, or for their infinitesimal generators (Appendix A). ii) Direct prpof of Bochner's Theorem on groups R"; Z", exploits a regularization of distribution d(z) by an approximate unity (problem 2): pJ2)
= €-"p(f),
p(z) 2 0; J p d z = 1.
Set Jc = pcq& and show that {$<} defines a family of positive linear functionals on continuous functions e(R") (or T"),
I(i,lf)l 5d(O)llfIlm; ( & l I ) Z O , Conclude that
forf20.
4 =€lim4 04,is a finite positive measure d p on R" of 11 p 11 = d(0).
52.4. Laplacian and related differential equations.
92
52.4. Laplacian and related differential equations. The Fourier transform provides an efficient tool in the study of differential/difference equations on R”; T”, due to its diagonalizing properties. Solving a differential equation is then reduced t o an algebraic problem, followed by the Fourier inversion. The utility and power of Fourier analysis will be shown by the basic examples of differential equations (heat, wave, Schriidinger), associated t o the Laplacian - the most symmetric of all differential operators on R”. In all case8 we shall construct explicit fundamental solutions (Green’s functions) and analyze their properties. After the freespace Green’s functions are constructed and analyzed in via the Fourier techniques in part (I), we turn t o the boundary-value problems in special regions (part (11)). Our analysis will emphasize groupsymmetries of the problem (discrete and continuous). One such techniques exploits the reductions (separation) of variables t o bring the problem t o exactly solvable ODE. Another one, known as the “method of images”, gives solution of the boundary-value problem expressed through the free-space Green’s function, subjected t o a finite/discrete group of transformations. Our third example, the ball problem, exploits yet another conformal symmetry of the Laplacian.
4.1. The role of Fourier transform in differential equations is that 9 diagonalizes all constant coefficient differential operators: L = a , < a(,Da, ( D = +a), on R”; I Tn.Namely,
TL[u]= a(<)G((),for all
{u},
or TLT-’ = a(<).
So 4 takes L into a multiplication with a polynomial function a(() = C a a t a , called symbol of L. This clearly follows from the eigenvalue property of exponentials {eic‘”} for any such L. In fact, {eiz.(} form joint eigenfunctions of all such operators8, with symbol a(() serving as “joint eigenvalue”. In many applications operator L is the Laplacian - A , or more general elliptic operato? L = - v ‘ p v q, with a positive
+
definite coefficient-matrix p = (pjk). A few basic examples of differential equations associated to operator L include:
I. Elliptic: -(L
+ X)u = f; (called Helmholtz, or “reduced wave” equation)
8 0 n torus 1” characters {erm*’} are honest L2-eigenfunctions, while on R” they represent the generalized eigenfunctions (bounded, but not L2). ’We keep a convention of writing elliptic operators with sign (-), to make L a positive operator: (Lf I f ) =
J C p i j ( a i f ) ( a j f ) d z 2 0, for all {f}.
93
$2.4. Laplacian and related differential equations
11. Parabolic (heat-difbion): a,u
+ L[u]= f; for u = u(s;t )
111. Elliptic equation in n+l variables: (a2 - L)u = f (called Poisson); IV. Hyperbolic (wave): utt + t [ u ]= f(s;t) V. Schrodinger: atu = iL[u]. Each equation must be supplemented by an appropriate boundary and initial
conditions. On regular boundaries (hyper-surfaces C c R”) one usually takes one of the 3 basic conditions: u I = ... (Dirichlet); d,u I r:= ... (Neurnann); or more general
+
I
( a ,LM,)u = .... (mized). The condition at (00) (“singular point”) typically require u to be bounded, or to have a prescribed asymptotics (incoming/outgoing “radiation
condition”). The initial conditions depend on the type of equation. One has u 1 t=O = uo (for elliptic/parabolic problems 11-111; V), and Ilt=O uo; (for hyperbolic IV).
{;
I
t=o - u1 We recall the general definition of the Green’s function, for a boundary value problem t
M[u] = F; in D . (4.14) B[u] = h; on aD’ where M could be any differentiafoperator in a space (or space-time) region D (e.g.
M = - V . pV+q; or
at2- V . pV+q),
and E a suitable boundary/initial conditions on
a D (e.g. B = a + pa,). The Green’s function (fundamental solution) of (4.14) is a n integral kernel K(z;y), or K(z,t;y,s) (possibly distributional) on D x D , that satisfies the differential equation M in z-variables with the 6-source at {y}, (also the formal adjoint equation in y-variables with the 6-source a t {z]), and the zero boundary conditions,
BJK] = 0
M,*[K] = ~ ( z - Y ) By[K] = O *
For instance, heat-problem (11) has operators M = , = 8,
+ L,,
M*,, ,=
(4.15)
- a, + L,,
and
the Green’s kernel K(z,y;l-s). Function K of either one of problems (4.15) allows t o solve a nonhomogeneous system (4.14) with an arbitrary RHS { F ; h ] ,
+
u(z) =
K(z;y)F(y)dy P(z;y)h(y)dS(y); dS-surface area element on aD. J D So the Green’s term K gives a contribution of the “continuously distributed sources”
F I D, while the so called Poisson term P(z;y) on D x a D , those of the “boundary sourcesn h
I
The Poisson kernel P(z;y) is not independent of K, in fact, K is
constructed explicitly in terms of K and the boundary operator B. For instance, the Dirichlet condition for
L = -V .pV + q, yields
P(z;y) = p(y)a, K(z;y) (normal derivative of K in y o n the boundary), Y
while the Neumann condition gives
52.4. Laplacian and related differential equations.
94
P(z;y) = p K ( z ;y), for z E D;y E aD. Similar constructions, based on Green’s identities could be given for arbitrary M and boundary operator B.
I. Formal solutions and frespace Green’s functions. We shall construct fundamental solutions (Green’s functions) for each of problems (I-V) in
Wn, by first writing its formal solution, then applying the
Fourier analysis. Formal solutions of problems (I-V) can be written in the form of certain ‘functions” of L”. Thus we get
u = (A
+
u = etL‘[uo] and similarly,
+ L)-’[f], for the elliptic problem (I);
1:
e(t-s)L[f(...;s)]ds, for the heat-problem (11),
u = eitL[uo];for the Schrodinger problem (V). Here uo represents the initial value of u,and f - its RHS, the “heat-sources”. Elliptic L‘n+l”- problem (111), is solved by
*
u=e 4 [ u 0 ] .
< 00 (unbounded range), one takes the negative “forward-stable” exponential {e-‘...}, while finite time-intervals, a 5 t 5 b,
So in the half-line case: 0 5 t
with 2-point boundary condition require a suitable combination of 4 exponentials. The wave problem (IV) is solved in either of 2 forms,
The choice depends on the type of the LLboundaryconditions” in t-variable, either “initial” at t=O, or “asymptotic” at t = f00. In all cases (I-V) fundamental solution { d ( L ) }is given by an integral kernel,
K ( z ;y), called the Green’s function of the problem. For constant-coefficient L these are convolution-integrals K = K(z-y), due to translational invariance of operator L. The Fourier transform yields explicit form of kernels K ( z ) , z = z-y. By definition of K we have to find a distribution
K ( z ) , z=z-y,
that satisfies the equation,
luuFunctions of L” for a self-adjoint operator, could be given precise meaning via diagonalization (spectral resolution) of L. Given a diagonal (multiplication) operator D:d(A)+Ad(A), on Lz(dp)-space of functions/sequencea {d(A)), function f ( D ) is also a multiplication operator, f ( D ) :d+f(A)d(A). But any self-adjoint L is unitary equivalent to a diagonal operator: L = U D U - ’ , then f(L)= U f ( D ) U - (Appendix A).
’
$2.4. Laplacian and related differential equations
L [ K ]= 6(z-y).
95
(4.16)
Applying ’3 to both sides of (4.16) we get
where aL(<) is the symbol of L. So we immediately get the transformed Green’s function 1 R(t) = -
(4.17)
ad€)’
in other words “symbol of K” is inverse to “symbol of L”. Then it remains to Fourier-invert l / a L ( ( ) to recover K. The Fourier-inversion, however, involves the techniques of complez analysis, distribution theory, and also
reduction of variables via group-symmetries, as we shall demonstrate below. 4.2. Laplacian is the most symmetric of all differential on
Wn as it
commute
all translations and rotations (Euclidian motions). In fact, A generates all such operators, in the sense that any operator M on a suitable functionthat commutes with En is a convolution with a radiallyspace (LZ;Lf’;ew), symmetric function M ( I z I ),
WuI=
pc I
5
- Y I MY)dY.
(4.18)
The Fourier-transform of (4.18) gives a multiplication-operator with another radially symmetric function
% = F( I t 1 ),
u ^ - q I t I )u^, where F ( p ) is related to M ( r ) through the reduced Fourier-Bessel (Hankel)
transform (52.1-2.3). Since, p2 is the symbol of - A , we can call M a “function of A”, M = F ( m ) .The proof of (4.18) is an easy application of the Fourier transform. It contains 2 statements. Theorem 1: i) A n y translation-invariant operator M on a suitable function-class
(e.9. Schwartz) is a convolution: MR,u = R a M u , for all a E Rn, and all u E Y,+ M [ u ]= f * u , f o r some distribution f .
ii) Rotationally-symmetric convolutions, M T , = T,M, g E Sqn), are given by radial functions, f = f( I z I ). Indeed, any such M is taken by 9 into an an operator
% = 9M9-’,that
commutes
with multiplications by characters { e i b * t},
--
f i e i b * = eib ’ €I@, for all b E R”. A-
Hence
A
fi commutes with all multiplications, M h = h M , h in the Fourier-algebra A. But
52.4. Laplacian and related differential equations.
96
algebra A contains sufficiently many functions, so any operator, commuting with A must
= F ( ( ) , so Mu = k*u. The 2-nd statement is obvious, as 5
itself be a multiplication,
takes radial functions into radial.
Our discussion will be limited to the Laplacian, L = -A, although many results and techniques extend to more general (2-nd and higher-order) operators. The corresponding Green's functions then become inverse '%transforms of certain symbols. i) resolvent kernel: RX(z)= (X+L)-
4(, ii) heat-semigroup (Gaussian): G,(z) = etL has 9[Gt] = G ( t ; t )= e-t I ( I 2.
has Fourier transform P(t;t )
iii) half-space Poisson kernel: Pl(z)= e-td, = e-,
(a semigroup, generated by
a)
I.
JL JL'
sin I
iv) cos- and sin-type 'waue propagators": U , = c o s t a ; V , =*-
U ( z ; t )= ( 2 ~ ) - ~ j c o Is [( 1t )e'c'zd(. sintI(1
.
V ( z ; t )= ( 2 ~ ) - " 1 ~ e ' ( ' ~ d ( . or the unitary group { e
generated by
fi.
v) Schrodinger propagator W, (a unitary group generated by L),
W, = (21)-n/ei(t I t I 2+t.z)dt. In some cases the Fourier-inversion is easy and straightforward like for the Gaussian and Schr6dinger propagators (problem 5 of 52.1)'
L
I
I
I
The latter is obtained by analytically continuing G,to imaginary time, t-it. 4.3. The resolvent kernel R(z;X)is expressed through the Bessel functions of the 1-st and 3-rd kind [Leb];[Erd];[WW]of order Y = - 1,
5
(4.20)
In the limiting case, eigenvalue X = 0, the Green's functions (-A)-' with the classical Newton's potentials:
coincides
$2.4. Laplacian and related differential equations
97
To derive (4.20) we notice, that all kernels (i-v) are radial functions of z , an easy consequence of the underlying rotational symmetry of A. Writing integral R(z;A) in polar coordinates
I z I = p;
I € 1 = r, we get (4.21) The inner integral gives the
80
called Poisson representation of the Bessel function J,, u =
5- 1,
Hence we get in (4.21)
The latter integral can be evaluated by complex integration in the r-plane of holomorphically extended Bessel function J,, whence comes K , ( z ) = J,(iz). We shall illustrate the foregoing argument in R3. The inner integral r r p cos
2sin rp (4.22) sinode = -. rP ' since J 1 , 2 ( z ) is an elementary function &. We substitute the RHS of (4.22) in the outer je.
0
Jnt
4.4. The half-space Poisson kernels. Next we turn to the Poisson kernel Iin the half-space W?+'. The 1-D Poisson kernel is computed directly
- e-tl t-
(problem 5 of 52.1),
The multidimensional case is, however, more subtle, and could be derived via the so called subordination principle, which allows to express the Poisson-semigroup, generated by
fi,in
terms of the heat-semigroup, generated by L, (4.23) The derivation of (4.23) is fairly straightforward, via 1-D Fourier transform (problem 1). In the case of Laplacian L = -A on Rn, we get the Poisson kernel P ( z ;t) in RT++' in terms of the Gaussian { G t } . By (4.24) and an explicit form of the Gaussian (4.19) we find,
98
$2.4. Laplacian and related differential equations.
P, = tn(1 +
Cn Cnt ' I I 2 / p ) ( n + 1)/2 - ( p + I I 2 ) ( n + 1)/2
(4.25)
(4.26) is then given by the convolution integral: u ( z ; t )= P l * f , with kernel (4.25). As an application of (4.19), (4.25) one can show
Theorem 2 Solution u ( z ; t ) of the elliptic Dirichlet problem (I.%), and the initialvalue heat-problem: ut = Au;u I t=O = f, converges to the boundary value f(z)as t+O.
The proof is outlined in problem 2. After the Dirichlet Poisson problem is solved we can proceed to the Neumann problem for the half-space Laplacian, (4.27) The formal solution of (4.27)
-f=3-t'%
21 = = &t[fl, is given by the conjugate Poisson erne1 related to P , via
co
&(...;t ) = / P (...;3) ds, and P = $. d
(4.28)
t "Let us remark that Green's functions (i-v) are related one t o the other through various transforms. For instance, the resolvent RA is obtained by Laplace-transforming the heat-semigroup G t , (A
+ L)-'
do
=
1e-"-Ltdt, 0
while (4.27) gives a similar representation for P,. Formula (4.27) exemplifies such relations. It represents a special case of the so called Sonin-Mehler integral representation of modified Bessel functions of the 3-rd kind (McDonald functions) OD
K J r ) = i J e t p [ - ( t + t ) i ] t-'-'dt. 0
Precisely, (4.28) correspond t o order u = elementary, K 1 / J r ) = &e-'.
i, the
only case, when Bessel-function becomes
99
52.4. Laplacian and related differential equations
00
as a consequence of an identity, L-le-tL = Se-sLds. Substituting (4.28) in (4.25) we t find
So the Neumann problem (4.27) is solved by the convolution kernel ti(";
t ) = &(...;t)*f.
4.5. Wavepropagatom. We shall use Poisson kernels P and Q, found in the previous part to find the wave-propagators, U ( z ; t )= c o s c t a , V =
9
= 0,
i.e. fundamental solutions of the propagation) with 2 different-type initial conditions:
( c = speed of
u I t=O = 0 ; for V . "tIt=o = 0' "1 I 1=0 = As in the Gaussian case this will be accomplished via analytic continuation of
I t=o = *. for u, or
*
the time parameter through the imaginary axis, t-it. The Fourier transforms of both propagators: n
U ( &t ) = cos ct
sin ct I < I I ( I , V((;t= ) 1c1 ' n
*-
as well as Poisson P , are easily inverted in 1-D. Thus,
turns into a Heavi.de jump-function, truncated at { fc t } . Similarly, the cos-propagator,
cos(ct I c I ) = ;(eict I c I + ,-ict I € I )
* u = ;{*(Z-ct) + *(z+ct)},
which is the sum of the right and left-traveling *-pulses, the classical d'Alambert solution. One has to compute two limiting distributions:
In dimension n = 2 both are given by certain densities,
$2.4. Laplacian and related differential equations.
100
{I
z-y I 5 c t } . So for each fixed t > 0 support of U(..;t); V( ...;t ) belongs to a disk of radius R = ct, centered at the source12 {y} (fig.2). In 3-D space we get supported in the light-cone
1
1
2ECt
I z I 2 + (c+ict)Z - I z I 2 t (r-ictlz} = lim (c2t2- I I 2 + E 2 ) 2 + qE2
2 t 244.29)
The RHS of (4.29) represents a limiting distribution of the family
{wd=*} I ’.
with a regular function f = fr(.z) = cZtZ- I z
From the basic distribution theory of
$2.2, we find,
where dS means the natural surface element on the level-set {f = 0). In other words, the limiting distribution bf is made of the “zero-level” &function of f , divided by a continuous density IVf(z) 1 . In our case (4.29) we get
Vt(Z) = k b ( c t 2 V,(z) = a,v = + ( 4*(ct)
12 1 2 )
=(6’47rct
12
I z I - C t ) t &a’(
I -Ct) I2 I -C t )
So the 3-D wave-propagator { V ; V }can be thought of either as distributions in supported on the forward light-cone C = ((2;t): I z I = c t } c R4, or as a t-parameter family of spherical &functions ( 6 * of radius R = c t } in R3. The SR* corresponding solution of the initial value problem: u I = f ; ut I = g, in 2-D and 3-D (2;t)-variables
are given by convolution kernels,
Remark: Similar expressions hold for higher-D wave-problems. Namely, the Green’s functions V ( z ;t); V ( z ;t ) are given by distributions supported either inside the light cone
C,{ 1 z I 5 ct} (even dimensions), or on the surface of the cone { I z I = ct}, as illustrated in fig.:! below (problem 4). The exacts solution of the n-D wave problem are given in terms of spherical means of function f,
1 M,f(z) = ,n-l wn-l
s
lyl =P
f(Z-Y)dS(Y),
-Green functions are often called point-sources, aa they describe solutions produced by sharp (&type) initial impulses.
$2.4. laplacian and related differential equations
101
where w , - ~ is the area of the unit sphere in R". Then the initial data u I, = 0; ut10 = f(z), yields [Hell u(2; t ) = 2 (n-2)! -0-2
1
[
~ { M 8 f ) ( x ) s ( t z -n-3 s2)~ds
(4.31)
In special cases n=2;3 one recovers solutions (4.30). Formula (4.31) demonstrates the
Huygens principles for solutions of the wave equation: any initial disturbance propagates inside the light-cone based on supp{f) with a finite speed c. Furthermore, in odd dimensions disturbance of any finite extent vanishes after a signal passes (sharp Huygens
principle), whereas in odd dimensions once a particular space-point is reached by the disturbance will remain forever, dying out exponentially fast with time. Indeed, derivatives of integral (4.31) in odd dimensions take the form
c
Pk(t)a:[M'f
1,
with some polynomial coefficients { P k ( t ) } .
Fig. 2. The forward and backward light-cones for the wave equation. The left figure (forward) shows a point-source solution (Greens function) at 3 differenl lime cross-sections. At each 1 the disturbance is localized either on the surface (sphere S R ( x ) of radius R = ct, centered at the source {t}), or inside the ball BR(z) = { yx I 5 ct}. The right figure shows the domain of dependence: any solution u(z;t) depends on the initial data inside the ball { I y-+ I 5 ct). There are different way to derive (4.31), all exploiting to some degree the underlying group symmetry of the problem. We shall mention 2 of them: [He121 (chapter 1) is baaed on the Radon transform on R" (and rotational symmetries), whole R. Howe [How] presents an elegant derivation baaed on the representation theory of the Heisenberg group (cf. $6.2).
We have summarized the main results of the section in the table at the end of the section. It. Symmetries and differential equations in regions of R".
After we obtained the free-space fundamental solutions in Wn, let us turn to boundary-value problems. The analysis below will emphasize group-symmetries (discrete
102
52.4. Laplacian and related differential equations.
and continuous), and whenever possible provide explicit formulae and constructions of Green's functions. Of course, our discussion will be limited to special classes of symmetric regions and differential equations.
(I = f(L) L; region Q), which
4.6. Symmetry reduction and the sou~cecondition. Green's function
clearly inherits all original symmetries of the problem13 (operator
allows to reduce the number of variables in the Green's function K . For instance, hermitian symmetry,
L = L*,implies symmetry of integral kernel, K ( z ,y) = It'*(y, z)
complex conjugate. Time-independence of L means that K = I'(t-s;
5,y)
-
are time-
convolutions kernels. Similarly, constant-coefficient, i.e. translational-invariance of L in R" yield translational-invariant (convolution) kernels K = K(z-y). Larger symmetry groups play the role for special operators L, like orthogonal rotations S q n ) for the Laplacian/Helmholtz operators: A , A + m 2 on Rn, and hyperbolic rotations 5 q l ; n ) for the d'Alambertian (wave operator) 0 = 6': - A , or the
Klein-Gordon (KG) operator: a: - A f m2.The corresponding Green's functions, It'(z) or K ( t ; z ) ,z = z-y, retain the symmetries of L, which allows to reduce Zt' to a single variable function I'(r), of the Euclidian or hyperbolic radius: T = I z I , or r = In both cases the radial function I' solves the reduced ODE,
d q .
which is singular at the end-points r = O;m, hence requires an appropriate boundary condition at both. The "source condition" refers to a proper choice of the boundary condition at a "singular (source) point". We shall demonstrate with 2 familiar examples. 4.7. Elliptic Caae. Laplacian/Helmholtz-type operator: L = -A
+ q.
The condition at (0)
represents the so called Upoinl-source",
,.
p - 1 ~ 1
()Ir=O==
1
(4.33)
2rnI2 - surface area of the unit sphere in R". To derive (4.33) we take any where Cn-l =---
r("/a
function u(y) on R", write L[u]= F(y), apply kernel K(z,y) to the equation, and integrate by parts (Green identity) over the complement R = R"\B,(r), of a small ball B , ( z ) centered at {z}. On one hand,
W -e shall not pursue any systematic study of symmetries in differential equations here, but refer to the recent book [Olv], which explores the subject in great depth (see also [Ovs];[Mil]). Our goals will be limited to exploiting some apparent and well know (geometric) symmetries, rather than investigate all of them.
82.4. Laplacian and related differential equations
103
but
First integral in the RHS vanishes, since L,[K] = 0 outside of the source. Remembering that
K = K ( Iz-y
I ), we get the second integral
f while the third
4
... = Cn-l F 1 K ’ ( e )(u(z)+ o ( E ) ) , Iz-yl
=€
... FT t ” K ( c )Au(z) +O,
as € 4 0 .
12-yI = c
Thus
+ {c” K(c)}Au(z),
u(z) = lim {C,-,r”-’K’(c))u(z) C+O
for any function u, which yields the source condition (4.33). Equation (4.32) for q = 0 takes the form
+V
K,,
K r = 0,
it has 2 solutions (regular and singular): { l;logr} in n
= 2, and
{1;r2-,} for n 2 3. So
K = c1 + c2rZ--n(or logr). The boundary condition for K at {m} eliminates the constant term, while the source condition
(4.33) yields c2, whence the familiar form of the Newton/Gauss potential, the Green’s function of the Laplacian,
A similar treatment applies to the case q # 0. But this time (4.32) becomes the equation of Bessel-type, y”+qyrfXy=O,
with X = q .
It is easily reduced to the standard (I-st or 3-nd kind) Bessel equation 2
Y ” + k Y ’ f ( l - E ) Y = O , oforder v=&, 2 r2
(4.34)
by the change, y(r)
= r-’Y,(Jir).
Thus solution of (4.32) with positive rn2 becomes a combination of the 1-st and 2-nd kind Bessel functions: K = r-”{clJ,(mr)
+ cZYv(mr)},of order
v=
9. The source-condition at (0) is
furnished by a non-vanishing coeficient c2. The condition at {co} should be chosen either as incoming, or outgoing “radiation” condition
The latter with a proper normalization at (0) uniquely determines K as the MacDonald function, M, = Const ( J , f iY,), so
52.4. Laplacian and related differential equations.
104
~
Similarly, for A-m’,
the reduced equation changes into the modified Bessel,
Krr + n-1 7 K r - m2K = 0, whose solutions are
I,
-
1-n r 2 emr, and K,
-
1-n
rTeWmr.
We obviously choose the exponentially decaying Kelvin function K,, which could be normalized
to satisfy the source condition a t (0). 4.8. Hyperbolic b. The wave and KGequations:
= utt - cZAu = 6;and (0f m2)u= 6, have all the symmetries of R” +
space-time, translations as well as hyperbolic rotations
So(1;n). The corresponding Green’s functions K = K(1,z) are reduced as above t o a single variable, the hyperbolic radius r =
d-.
They satisfy the Bessel-type equation
Kr,+:K,fm2K
= 0.
(4.35)
But unlike the elliptic case it is not easy to write down the proper %wee condition” a t r = 0. In the last section we used the Fourier analysis t o derive the wave-propagators, which yields for dimensions n = 1,2,3,
(4.36) In higher dimensions K becomes yet more’singular distribution, supported in the light cone
{ I z I 5 ct) or its surface { I z I = ct} (4.31), i.e. K = d ( r ) - a distribution in r = ct - I z 1, or hyperbolic r =
d m .T h e KGGreen’s functions satisfy the Bessel-type (4.35) equation
in hyperbolic r, and we can use solutions (4.35), as proper “source-conditions”. This yields
in spatial dimensions n = 1,2,3.
4.9. Discrete symmetriea and the method of images. The method of images
(reflected sources) applies to differential equations in regions of Rn, and other symmetric spaces (see 57.6 of chapter 7), obtained by “discrete symmetries” (reflections, translations, conformal mappings), like the half-space, quadrants, strips/ slabs, rectangles, spheres, etc. The Green’s function of the corresponding problem is obtained from the free-space Green’s function KO by subjecting KO to a (discrete) set of symmetry transformations, or by subtracting from
KO a suitable regular solution.
32.4. Laplacian and related differential equations
105
Namely, K Osatisfies the source (6 -function) condition, Lz[KO] = 6 ( x - y) in R, but it fails to vanish on the boundary, B[K,] # 0. So we want to find a family of regular functions {u(z;y)}, depending on the position of the source {y}, so that
Lz[u(...;y)] = 0 in 0; and Bz[u]I r = Ko(z;y).
(4.37)
4.10. Renectiona and half-upam problems. We consider the half-space R?+’ t
> 0},
and assume operator Z to be reflection-invariant: (z;t)+(z;
= { ( z , t ) : z E Rn;
-t),
for instance
L = 8: + A - the half-space Laplacian. The corresponding Green’s function depends on the type of boundary condition. For Dirichlet Problems the Green’s function KD
= K ( z ; t , s )= K,(r;t - s) - I f o ( z ; t + s),
represents the difference between the “sourcesolution” K , ( y ; s) (e.g. Newton’s potential) and the
“reflected source” K , ( y ;
- s)
(fig.3). The reflected source gives the requisite correction u in
(4.37). So the half-space Dirichlet Green’s function represents a convolution kernel in 2-variable, K ( z - y ; t , s) of the form
If = &{( I z 1 2
+ ( t - s)2)
n-1 -2
-( Iz
12
+ ( t +.)2) -F}.
This yields, in particular, the half-space Poisson kernel P ( z - y ; t ) , derived earlier in by the Fourier-transform methods
(4.38)
Let us observe, that solution of the Dirichlet problem: Au = F ; u I r: = f; in any region R with boundary C = 80, is represented by the Green’s identity, as
4.) = where
an denotes
/
nK(z;Y)F(Y)dY
+ / za,K(z;Y)f(Y)dS(Y),
the normal derivative of the Dirichlet Green’s function in y E C. So the
Green’s function K picks up the “continuously distributed sources” F over R, while its normal derivative P = 8°K - the Poisson kernel gives the contribution of the “boundary sourcesn.
Fig.%: Free-space Green’s functions at a source {y] and the reflected source {b} cancel each other on the boundary r, separating two half-spaces (Dirichlet problem). In the Neumann problem 2 “sources” add up, so the normal derivative vanishes on the boundary.
106
$2.4. Laplacian and related differential equations. For Neumann Problem the difference of two source8 is replaced with the sum
K N = Ko(z;t-s) +Ko(z;t+s), (so that normal derivative8 would cancel each other). As application we obtain the conjugate
Poisson kernel Q in the half-plane14,
found earlier by the Fourier transform methods. In a similar vein one constructs Green’s functions for other problems, e.g. the half-space Dirichlet/Neumann “heat problem” has K(z,y; t ) = G(2 - y;t)
- the difference/sum {y’
C(z - y’; t )
of two Gaussians with “sources” a t {y = (yl; ...y,))
and the reflected point:
= ( - yl; ...yn)}. Quadrants and other product-type regions are treated in a similar fashion.
Thus the quadrant {(z1;z2) > 0) Green’s function consists of 4 reflected sources: K(z,Y ...) = Ko(z - y) - K o ( z - y’)
- Ko(2 - Y”)
+ KO(Z + y);
where y’; y” represent reflections of {y) relative to the 1-st, respectively 2-nd coordinate axis. 4.11. Green’s functions in finite regions. Green’s functions of elliptic differential operators in
bounded regions S2 (compact manifolds) can always be expanded in terms of eigenfunctions {$k} of L, subject to a proper boundary condition. We leave t o the reader as as exercise t o write expansions for the inverse operator L-’, the heat semigroup e-tL; Poisson kernel exp(-t&); wave-propagators, etc. Let us remark that the Green’s function of any regular S-L problem on interval [O;T]can be represented in two different ways: via the eigenfunction expansion 15 K(z;y) = Cfdk(Z)iJ,(Y), (4.39) k or in terms of a fundamental pair of solutions: L[ul,,] = 0, one of which satisfies the zero
boundary condition on the left (ul I t=O = 0)’ and the other on the right (uz I T = 0). Given a fundamental pair {ul; uz}, the Green’s function is constructed as, (4.40)
For constant coefficient operators L =
- 8 + q,
two representations are related by another
version of the Mtthod of images. Namely, we take source y E [O;TJ,and “reflect it” infinitely 141ndeed, solution of the Neumann problem: Au = F; a,u identity, as
I = f; is represented
via Green’s
where K = K N is the Neumann Green’s function, so Q(z;y) = K ( z ;y), for z E 0 , y E E. 151f one of eigenvalues A, = 0, then kernel K (4.5) (sum over nonzeros A’s defines the so called modified Green’s ,function, i.e. inverse of L on the subspace spanned by nonzeros eigenfunctions {dk},so the product of operators L K = b(z - y) - v,bo(z)$o(y).
52.4. Laplacian and related differential equations
107
many times by the lattice Z = {mT}. So the free-space Green's function of Z on R.
is shifted and summed over the
corresponding Green's function
K on [O;T]has different representations depending on the type of boundary condition. Namely, the periodic bounda y condition (torus), gives ~ ( z , y= )
C~
+
~- y ( mT) z
= +C'ezp(i%$z - y)); (4.41) k with eigenvalues: {Ak = ( r k / T ) ,+ q: k E Z}, while for the Dirichlet/Neumann condition the m
source a t {y} is reflected to {-y},
and then the process is repeated periodically with period 2T.
So one has
The reader should recognize relations (4.41) and (4.42), as special cases of the Poisson summation for function f ( t ) = KO(I z - y I
+ t ) on the lattice Z = {mT}. Similar considerations
apply to the heat, wave, KG and other equations, and to equations on multi-dimensional tori. Two such examples are
4.12. Dirichlet Laplacians on triangles.
Rightisoseeles triangle makes up
i of the square. The (rx r)-square eigenmodes are made of
products of the even and odd Fourier modes { c k = cos(k +:)z}, and
y).
{ C k ( t ) s,(y);
So
consist
of
products
{srn= sin2mz) (in variables z
{ c k ( ~ ) c , ( ~ ) ) , { s k ( ~ ) s , o ) and
s k ( z ) c , ( y ) } . The former two (cc- and ss-types) correspond to different families
+i)2+
+
(m +$)'}, and {4(k2 m')}, so they never mix. The latter ( c s - and types) produce eigenmodes Gkrn of the form Ck(z)6rn(Y) 2 L,(')'k(Y), that vanish on
of eigenvalues {(k
sc-
{4krn}
diagonals z = f y, hence yield the triangle Dirichlet eigenfunctions. Equilateral triangle T makes lattice
r c C, spanned by
i of the diamond,
the fundamental region of the hexagonal
{ l ;a = e i S / , } (fig.4). Its Dirichlet eigenfunctions fall into 3 families,
according to the natural action of dihedral symmetry group D, = Z, D Z, on T. The latter has 3 irreducible representations: trivial
xo = 1, x1 = f 1 (trivial on rotations { 1;u;CJ:u= eizXl3}
and
equal -1 on any reflection), and a 2-D representation u, uu:((;q)+(u(;CJq) (for rotations {I;wp}),
and u,((;q) = ( q , ( ) , for reflection r about the z-axis (see 53.3). So family (I) consists
of eigenfunctions16, symmetric relative to the D,-action
, family
(11) are rotation-invariant, but
"Dirichlet eigenfunctions are uniquely determined by the values of their normal derivatives on , on all 3 sides of T. the boundary, so the description here refers to the values of a+
108
$2.4. Laplacian and related differential equations. mapped t o -(1, by reflections, finally (111) gives a pair of conjugate values { @ ; a }{-l;-l},{Sp} , on each of 3 sides of T (see fig.5~). Our goal is to link eigenvalues of A, to the well known eigenvalues of the f-periodic A on torus
c/r, A,
= I i k + a m I = k2 +m2- km.
Figure 4 demonstrates how each eigenfunction of family (I), (11) and (111) extends t o a fperiodic eigenfunction of A: for (I)-(11) is equal to the size of the triangle (the plane is cut into the union of copies of T, and 11, is taken into all images {T‘} of T with signs f , depending on the orientation of T‘. That same reflection procedure applied to a 111-type eigenfunction requires to triple the period. Since each periodic eigenfunction is a combination of Fourier modes {exp[i(kz+m%(E::y)]:k,m E Z}, we obtain the eigenvalue-spectrum of A,,
t o be made of 2
sequences:
1 Atm
= k2 + m2 - km (for 1-11) A’km = g(k2 + rn2 - km) (for 111)
I
Fig.6 below demonstrates a 11-type eigenfunction as a combination of rotated and reflected exponents, (1, = XI - x2
+ x3 - x4 + x 5 - xs. Fig.4 shows cubic roots {w,G = w z } of 1, along with their negatives { a = - w-; E = - w } . Numbers { 1;w;G } give 3 irreducible characters of subgroup Z, c Q,, while reflection (complex conjugation) defines a 2-D representation of D,. Fig.5 illustrates a periodic extension of eigenfunctions of types I, 11, and 111. A Dirichlet eigenfunction (1, in T is determined by its normal derivatives {fi =a,$, on the i-th side Si, i = 1,2,3}.
We schematically represent each I f i } by a curve along Si; the portion of the curve inside the triangle corresponds t o a positive derivative a,f, while the outside portion t o the negative derivative. For type-I all functions { f i } are equal and symmetric with respect t o the center of the side.
So given a type-I (1, in a “+”-triangle, we reflect it t o a neighboring “”triangle, changing the sign to+-$, and repeat the process. The resulting fperiodic function on C is clearly an eigenfunction of A. For type-I1 11, (b) boundary function f is odd on
109
$2.4. Laplacian and related differential equations
each side Si and repeats itself under rotational symmetries with { 1 ; q Z }. Once again reflections +-$, from T + t o T - across {S,;S,;S,} produces a r-periodic eigenfunction of A in C. The type-111 (c) is somewhat more complicated. Here we pick a pair of complex conjugate functions {+;$}. Then one ca? easily verify that the boundary functions {fi;f i} have absolute values I f l I = I f z I = I f JI = p (an even positive function), and differ one from the other by a_ factor w , f 2 = wfl; f, = 5 f l (and similarly for f .). If we choose fl.(left side) to have phase a= then f 2 (horizontal) have phase - 1 , while f, = E p. The reflection $+-4 about side S, takes triple { a ; - l ; Z } into { u = - E ; + l ; G = a } , and the process has to continued 3-times along the (horizontal) line t o get the original triple { a ; - ; E } . So type-I11 $ yields a 3r-periodic eigenfunction of the Laplacian in C.
&,
Fig.6 shows a type-I11 eigenfunction constructed a combination of 6 exponentials: 4 = x1 - XZ x3 - x4 x 5 - x6.
as
+
+
4.13. Conformal symmetry: Green's function and Poisson kernel in the ball. The group-symmetries used so far in our discussion involved space-translations (continuous and discrete); rotations (spherical and hyperbolic) and reflections. Our last example will exploit yet another transformation, the conformal symmetry of the Laplacian, to study harmonic functions in the balls B , = {
12
I 5 R } c W".
The Green's function
A'
and
Poisson kernel P of the unit disk D c R2 = 43 are well known from the complex analysis, and can be derived in many different ways. We just state the result, referring to problem 6 for details, (4.43)
where z = reit; w = pei6 are complex points in D ( p = 1 for P). Our goal is to extend
(4.43) to higher dimensions. Namely, the Poisson kernel in the n-ball is given by
(4.44)
52.4. Laplacian and related differential equations.
110
2~r 4 2 - volume of the where 12 1 = r; I y I = 1; e = angle between z and y, and w ~ =- r(n/2) unit sphere in Wn. The derivation of (4.44) will utilizes a particular conformal transformation in R", the inversion, u: z+z*
=A *
(4.45)
IZl2'
We recall that name conformal refers to diffeomorphisms {$} of R" (or any Riemannian manifold) that preserve angles between vectors, but not necessarily norms. T h e Jacobian of such m a p $, A = $', is a conformal matriz, product 'scalar x orthogonal", i.e. t A A = AZ (problem 8). Conformal m a p u of (4.45) takes interior of the unit ball in R" into the exterior, and vice versa,
I I = 1). T o construct the Green's function of A on B (Newton's potential) K O = C, I z - y I - " (for n > 2),
and acts identically on the unit sphere { we take the free-space Green's function and &ln
Iz-y I
in R2. It satisfies the differential equation: A,K = 6(z - y), but not the
boundary condition, Ko(z - y) function u&z) in
I an= h,(z) > 0, 80
n, that satisfies A,[u,]
we need t o correct KO by a regular harmonic
I
= 0; ).(,it
an= h , ( z ) ,
K ( r ; y )= Ko(z - Y) - )(., Such uy can be constructed by inversion of KO.We shall use the following identity valid for any function u, A(
I2
I 2-nu(+)) IZI
= I z I -"(Au)
(-1,
(4.46)
IZI
The proof of invariance-relation17 (4.46) is outlined in problem 8. Its immediate corollary is
Proposition: Function u is harmonic iffv(x) = Applying Proposition t o the Newton potential
12
I "-'u(z/ I z I ')
is harmonic.
KO = C I z - y I - " we get the requisite
harmonic correction: u
Cn
(2)=
l(z/Izl-
IzlY)l"-2'
Obviously, uy is harmonic and coincides with K o ( r - y) on the unit sphere. Thus the Green's function of the unit ball (4.47)
Finally, t o compute the Poisson kernel we take the normal derivative 8°K = ny.V,K,and observe that for y on the boundary ( I y I = l), ny = y. Hence,
It remains to note that both denominators in (4.48) are equal (see fig.7),
50
we get Poisson
17The R"-Laplacian is not invariant under the conformal map u, but comes 'very near" t o it. !), Indeed, (4.30) means that the unitary operator U , : f ( z ) + I z I -"f(z*); ( I z I -" = ,/intertwines operators A I z I and A, (A I z I ' ) U , = U,A.
32.4. Laplacian and related differential equations
111
kernel (4.44),
From the Poisson kernel on the unit ball we can easily derive P for any radius p,
(4.49) Fig.?
demonstrates equality of two denominators of (4.49), that measure the distances between pairs {z;y} and
I./
I2
I ; I 2 I Yl.
Formula (4.49) has many applications in the theory of harmonic functions. It follows from (4.49)that any harmonic u(x) is real analytic, since P ( q ...) is real-analytic in x for each y on the boundary. In fact, one could estimate the radius of convergence of the Taylor series ~ ~ ~ ( ~ ) ( ofz harmonic - - x ~ )function ~ u ( z ) at each interior point {xo}
C. Furthermore, (4.49) gives the minimal and maximal value of the Poisson kernel for each 0 < r < p, they correspond to in terms of the distance from {xo} to the boundary
case=
fl,
(4.50) As a consequence of (4.50)we get an important result in the theory of harmonic
functions.
Harnack inequality: The ratio between the maximal and minimal values of a harmonic positive function u(x) 2 0 an the ball of radius r is estimated by max {u(x):I z I = r } min {u(x):I z I = r ) -
(4.51)
4.13. Green’s functions and Poiason kernels in arbitrary domains D are constructed, as in
solid sphere case, from the free-space Green’s function (Newton/Bessel potential) KO,and an auxiliary (harmonic) function u(x; y)
(2,y
E D), that satisfies the homogeneous
equation: A,u = 0; and the boundary condition: u( ...;y) or a,u
I aD = -a,Ko(x-y)
I,
= -Ko(z-y) (Dirichlet);
(Neumann). Then, the Green’s function of D,
62.4. Ladacian and related differential equations.
112
K(z;y) = Ko(z-y)--(z;y); while the Poisson kernel, P ( z ;y) = -B,K(z; y)(Dirichlet); and P(z;y) = -a,K(z;y) (Neumann). Solution of any inhomogeneous problem: Au = F; u I
=I
= f ; is given by
,my ) ~ ( y ) d y+ r aDp(z;y ) f ( y ) d w .
Another construction of P(z;y) in terms of the solid-angle form is outlined in problem 12.
Additional comments and results. Our discussion of differential operators was limited to specific examples and methods. Here we give a cursory introduction to the general elliptic theory, based on symbolic calculus. For detailed exposition see [Hor];[Tal]. 1. Symbolic calculus. We write differential operators on R", as L = Cu,(z)D", where (I
= (el;...a,) means a multi-index, Do- the corresponding partial derivative ( D =
-
partial divided by i). We define a symbol u = u L ( z ;
'
(L[ei2*
13
and the Fourier transform links symbols to operators,
L [ f ] ( z )=
I
&
cr(z;<)ei(z-y)'CF(<)
(4.52)
d<.
Another way to write (4.52) is
L =T0ULOTJ-~, here uL acts on
f
<. Formula
by a multiplication in
(4.52) could be extended from
polynomial symbols of of pdo's (partial differential operators), to more general classes of functions {u(z;<)} with "nice" asymptotic behavior at large (. This gives rise to a wider class of operators; A = u(z;D), called pseudo-differential. A $do A (4.52) is given by an integral (possibly distributional) kernel,
qz; 2-y)
I
= 1,i(z-y)
(W"
*
C4 z ; € ) 4 .
(4.53)
Pseudedifferential operators ( g o ' s ) retain many basic properties of differential operators (see S2.2). For instance, one can introduce a notion of order: symbol u(z;<) is said to be
of order rn E R (positive, negative, fractional), if u(z;() = U( I differentiation in
<
lowers the order by 1: a?(z;<) = U( I < I "'-")}.
< I "),
and each
The reflects the
natural properties of the corresponding go's, A = u ( z ; D ) .Namely, Proposition: i) A $do's A of order 0 is bounded in L2 (and all Sobolev 36,) spaces. ii) Furthermore, a $do A of order m maps properly the Sobolev scale,
A: 36,+36,-,,
for all s.
(4.54)
$3.4. Laplacian and related differential equations
113
So positive-order $do’s deregularize smooth functions, while negative-order A smooth them out. The proof of (4.54) exploits the Fourier analysis of $02.1-2 (interpolation and Sobolev estimates). tDdo’s could be multiplied A . E (product rule), inverted (in some cases) and conjugated,
A-A’,
maintaining the proper order, m ( A E ) = m(A)
+ m(E),
m(A-’) = 1 (for m(A)
elliptic A). Both operations have natural characterization in terms of symbols. To write both rules we shall use the convention: a(), for partial
- for partial derivatives D”a in (, and a (0)
aa in 2. Then
-
(4.55) the expansions are taken in all multi-indices
(I
...
= ( a l ; (I”). Asymptotic sign
-
reflects
our emphasis on local (in 2-variable) properties of $do’s {A}, determined (via 41) by the
behavior of symbols at (-00. So “large a-terms” in (4.55) define “locally small (smoothing)” $do’s A, = a0( 2 ; D ) (each differentiation in ( lowers the order of a by
l!). Moreover, “smallness” of {A,} could be established not only in terms of their smoothing (order), but also operator norms of {A,} could be estimated, and shown to decay to 0 sufficiently fast. So the resulting expansions (4.55) can be interpreted as exact relations with “small remainders” in either sense. Let us remark, that both rules correspond to the standard convention of writing differential operators: derivatives {a,(z)}.
A=
Other
conventions
Caa[au(z) ...I),
{aa} (on the (multiplication
right) followed by multiplications with followed
by
differentiation,
i.e.
or Weyl-convention (where 8 and z are symmetrized, chapter 6)
give somewhat different rules. Thus $do’s {a(z;D)} form an algebra, that extends the algebra of all (variable-coefficient) differential operators. Product of $do’s is easily seen to maintain the order,
m( A * E ) = m( A) + m(E ) ; 0-order $do’s are bounded operators, and negative-order A’s (order - m ) have “nice integral kernels”, estimated by
1 K(2;y) I 5 Const 1 2-y I m-”.
(4.56)
Constant in (4.56) is bounded by certain seminorms, that involve symbol {a(z;()} and its derivatives in
(2;
(). The goal of symbolic calculus is to utilize symbols and estimates,
like (4.56), to (approximately) construct inverses {A -’},
powers {A’}, or more general
“functions of A” { f ( A ) } ; to find the kernels of {f(A)}, and to analyze their properties.
114
52.4. Laplacian and related differential equations. The crucial role here is played by the Fourier analysis. 2. Elliptic theory. Differential operators (pdo’s) or $do’s with nonvanishing (at {cm}!)
symbol a ( + ; ( ) are called ellipfic. For 2-nd order pdo’s, A = Caij(z)c3ij,this means that their symbol, quadratic form a ( + ; ( ) = C a i j ( ; t j is definite (positive or negative), a(+;()
> 0 or a(+;() < 0, for all < # 0.
Hence,
c1It I I a(+;€) 5 C, I €I *; for all (,
(4.57)
the standard notion of ellipticity. Estimate (4.57) plays the crucial role in the approximate inversion of elliptic operators. Indeed, inverse $do A - (if it exists) has the principal symbol given by the product-rule, uA-l
-1 (l.o.t), +... a(+; 0
and 1 is also an elliptic $do or order -m. As a consequence, one can construct an
4%; C)
approximate inverse B N A - ’ for any elliptic &lo A. Namely, one defines a $do B by (4.37) with symbol 1 , then shows via the product-rule (4.39) and norm estimates 4 r ;C)
(4.40), that
A - B = Z + R (remainder), where remainder R has negative order -c, hence operator R is smoothing in the Sobolev scale (4.38),
R: 36,+36,+,.
c R”, any such R becomes a compact operator, due to compactness of the Sobolev embedding: 36,(R) c X t ( Q ) , for any pair s > t (Theorem 1 of 52.2). In other words approximate inverse B = &(+;D) becomes the When restricted on a compact domain R
Fredholm inverse of A (inverse modulo compact operators). This result along with the
basic spectral theory of compact operators (Appendix A) has important consequences for the theory of elliptic equations. Before we state the general results, let us mention that the notions of symbol, &lo, Sobolev space, etc., could be extended from R“ to manifolds and domains18, i.e. boundary value problems (see [Hijr];[Tal]). Theorem: Let A be an elliptic operator on a compact manifold, o r domain (with proper boundary conditions). Then (i) spectrum of A consists of a discrete set of eigenualues
{Xk-+m};
each eigenspace Ek ( o r root subspace f o r non-self-adjoini A ) being finite-dimensional;
T -o construct &lo’s on manifolds { A } one exploits local formula (4.36), then patches together “local pieces”. Such process, however, defines a +do A only approximately to the leading order. So { u L ( z ; ( ) }on rnanifolds could be understood only as principle symbols, a consistent choice of other lower-order terms, requires additional structures. In some cases, groupstructure or geometry provide necessary tools to build a “complete symbolic calculus on manifolds” ((Be];[Un];[Ur];[Wi]).
52.4. Laplacian and related differential equations (ii)
for
any
X
4 spec(A),
operaior
(A - A)
is
inueriible
and
115 (A - A ) -':36,*36,+,;
(m = order(A)). Hence, differential equation: (A - X)u = f , with any f E
K,, has solution
In particular, f E Coo, yields oo-smooth soluiions. Also
u E 36,+,.
(iii) for all X E C operaior (A - A) is Fredholm-invertible. So equation (A - X)u = f, has solutions for all f , orthogonal l o the null-space {$}, of ihe adjoini operator: ( A * - X ) $ = 0 (Fredholm alternative); (iv) eigenfunctions: A 4 = Ak$, are oo-smooih. In case of (real) analytic A 'COO" could be replaced by (real) analyticity. Further analysis of operators {A}, via $do techniques, reveals much more. One can find for instance, asymptotic distribution of eigenvalues {Xk}, given by the celebrated Weyl (wolumecounting) principle:
5 A},
N ( X ; A )= #{Xk(A) 5 A} -+'d{(z;():a(~;<) (2*)
(4.58)
here the volume is taken in the the cotangeni bundle IT(&) = {(z;():< E T:}. The key idea in derivation of (4.58) is to compute the trace of a suitable function f ( A ) , e.g. or tr[(< - A)-'] (for sufficiently large s). On the one hand such traces represent
tr(e-"),
certain transforms (Laplace, Cauchy-Stieltjes) of the counting function N(A;A ) , e.g. tr(e-tA) =
00
C e-tAk = J e-A'dN(A). 0
On the other hand symbolic calculus yields kernels and symbols of +do's {B = f ( A ) } ,
-
u,(~)
f
ouA.
The connection between tr(B), and its symbol
uB
results from the basic
integral representation (4.37),
Hence, tr(e-tA)
-
tr(B) =
J Ib(z;()dzd(.
Ie-'"(2;0dzd(,
where the RHS gives the Laplace transform of the
volume-function V(A) = uoI{(z;():a
5 A}. Once Laplacetransformed " N ( A ) and V(A)" are
found asymptotically equivalent (at small t ) , one can go back t o the original =N and V", via Tauberian Theorems.
Remark In quantum mechanics, T*(&) gives the phase space of the classical-mechanical system (with coordinate z being the position-variable, and ( - the momentum variable); symbol a(z; () means the classical hamiltonian (e.g. energy-function), while differential/ pseudc-differential A = a(z; D)
-
the corresponding quantum hamiltonian. We shall
discuss the quantitaiion procedure in chapters 6 and 8 (see [How]).
116
52.4. Laplacian and related differential equations.
Problems and Exe.rcises: 1. Check the subordination principle by Fourier-transforming both sides of (4.47) in 1.
2. Regularity and convergence of solutions of the heat- and Laplaces equations. Show that for any f E Lp solutions of the heat and Laplaces problems: ut = Au, and utt = Au, with initial value f satisfy i) u(z;t)is ern for all t
I
> 0;
I
ii) 4 - 4 t ) LP(&) I IIf I L P iii) u(z;t)-f(z), as t-0, in LP-norm
Use the convolution representation of solutions, u = 4jt*f, with the Gaussian or Poisson kernel, 4 j t = t - W ( I z I /t),and show that both kernels form an approximate unity as defined in problem 2 of 92.1. In fact, one could show that u(z;t)+f(z), pointwise almost everywhere inside any cone: I z - zo I < c t , but this would require more powerful tools of Fourier analysis [SW]. 3. Use the Fourier transform to solve Dirichlet problem for the Laplaces equation in the strip: 0 < y < b, Au = 0; u I = = f(z); u I = b = 0.
4. Calculate wave propagators U ;V; W in higher dimensions, using the relation:
c
5. Obtain the fundamental solution of the beabproblem: Kt-AK=O - 2 )- Gaussian, * K ( z ; t ) = ( 4 ~ t ) - " / ~ e z p (41
I t = 0 = 6(z)
by symmetry reduction, using non-isotropic dilations in the
(2;
Itspace:
D,: (2, t)-(az; a%), a 2 0. (i) Show, if u ( t , t ) solves the homogeneous heat-equation, then u* = u ( a 2 t ; a z )does so. (ii) Write the fundamental solution K * = K ( a 2 t ; a z ) as a multiple of K ( t ; z ) , K * = c(a) K , compute coefficient .(a) from the "6 sourcen condition at t = 0, and prove K ( t ;z) = a" K ( a 2 t ; a z ) .Thus K is reduced to a single variable function K ( r ) ,
K = t-"/'K(rt-'/'). (iii) Verify that K ( r ) solves an ODE,
K'/ Change variable: r-$
2
+ (+ + 5)K' + ;K
= 0;
= z, and show that in the new variable K satisfies an ODE: 2r(K'+iK)'+ n(K'+:K) = 0;
(iv) Obtain the general solution of the latter,
K = e-'l2(Cl
+ C,
ez/2z-n/2dz),
show that the second (singular term) must vanish. Hence, K = e-'I2 = e-r2/4,
QED.
6. (i) Derive the Poisson kernel of the Laplacian in the unit disk D as Fourier series expansion 00 qr; 8 -t )= p I L. I - 0.
c
--m
(ii) Sum the resulting series to get
,w
52.4. Laplacian and related differential equations
117
in complex variables z = reie. (iii) Show that P- C( 0- t ) , coordinates).
as r+l,
and verify the equation A P = O (use polar
(iv) Expand the Green's function K(z;w ) ( z = reie; w = peiQ) into the Fourier series (4.59)
Km(r;p)eim(e-Q),
and demonstrate that coefficients {Km} are Green's functions of the S - L Problems:
+
+2~ = a(r - p), 2
KI' ~ K I
(4.60)
(v) Find a fundamental pair {ul(r);u2(r)} of each S-L problem (4.60) and compute functions { Km} using (4.40). (vi) Sum series (4.59) to get
(m
~ ( zw ); = k / n I *-l/tul ). Interpret this form of K in terms of reflected sources and P is equal t o the normal (radial) derivative of K on the boundary { I w I = 1).
7. Verify the Laplace equation A,P = 0, and the boundary condition, P ( z ;y) = 6(z - y), on aB - the unit sphere, for the Poiason kernel (4.44).
8. Prove the identity (4.46) for any function u on R". Steps: (i) Use a general coordinate-change formula for the Laplacian A: if map Q:Z-y = (...Qi(z)...), has Jacobian matrix A = Q', with determinant J = detA, then A,+V
. J ( = A A ) - 'V.
Written explicitly this yields,
A ( . 0 Q)=
{c ( c ajdi a k d i ) a p k u + c A Q aiu} ~ ik i
0
Q
I
(ii) Check that inversion u:z+z/ I z I ', gives a conformal map, so Jacobian = (alu; anu) satisfies tu' u' = p(z) I , with factor p = I z I - '. Hence,
ul
...
A[(*)]=
06)
II: 2).
IZI
(iii) Combine (4.61) with the product rule: A ( f g ) = A(f)g to f = I z I and g = u(z/ I z I 2), to get
A( I z I au
x
IzI -'(du-2(n-2)+.Vu
= I z I a-'{a(a
(4.61)
+ 2V f .V g + f Ag,
+ n - 2). - 2(a+n-2)+.
121
applied
V u + Au o u,
}
whence follows (4.46) for a = 2-n.
9. Mean-value property for harmonic functions: Show that any harmonic function u(z) on R" satisfies u(2) = 1 J u(z - ry)dSy - mean over the sphere of radius r; p-1 wn-l lyl = I
u(z) = 2J u(z - y ) d y - mean over the ball B ( r ) of radius r.
uolB(r) B(r)
Hint: Apply the Green's identity: JuAu - uAu = J (& - &)dS; aD an an D in region D c R" - normal derivative on OD), to the pair: { u ( z ) ; u = K o ( z - y)Newton's potential}, integrated over the ball D = {y: I z - y I _< r } .
(a an
10. Apply (4.51) to show that there are no non-constant positive harmonic functions on
118
$2.4. Laplacian and related differential equations. R". More elementary fact: there are no bounded harmonic functions on R", follows from the mean-value property (problem 9).
11. Use the mean-value property (problem 9) t o prove that a nontrivial harmonic function u(z) on R" is unbounded. Hint: w u m e I u(z) I 5 C ; pick a pair of points z,y; write u(z), u(y) by their mean values in large balls {B(r)}; estimate I u(z) - u(y) I 5 ...( r), and let r+m.
12. The Poisson kernel P(z;y), r E D , y E C = B D , in any (convex) region D, can be constructed in terms of the solid-angle kernel, T ( z , y ) = 1 - cos(I.
as
c,,-~as - ~ , - ~ ~ n - 1 '
where a is the angle between normal nu and vector (z-y),(see fig.8). Geometrically kernel T ( z ; y ) represents the density of the solid angle with vertex a t {z}, subtended by a surface area element a t {y}. It can be obtained by restricting the (n-1)-differential form (solid-angle form),
8 = 1 ( z l d z z A ...Adz, - zzdzl A d z 3 A ... A d z , r"
+ ...),
on the boundary surface C = a D . i) Show that T = 81,E (parametrize C by map @:R"-'+R"; vector surface-area element ndS = @91 A A @yn-ldn-l Y I
z = @(y)), and show that
...
where {By} are partial derivatives of @ in variables {y}, n-unit normal t o C. ii) Show u(z) = J'T(z;y)f(y)dS(y), boundary condition,
solves the Laplaces equation, Au = 0; but not the u I C = if
+ Q [ fI,
where operator Q is obtained by restricting kernel T ( z ; y ) on C. iii) Integral operator Q is compact on Lz(C) (see Appendix B), since Q has integrable So , operator ?jZ Q is Fredholm-invertible singularity, I Q(z;y) I 5 C 12-y I (modulo possible finitedimensional eigensubspace A = -?), and )the Poisson kernel,
'-".
I-
+
P = T($Z+ Q)- * = 2(T - 2TQ + 4TQ2- ...). Fig.8 Geometric view of the solid angle form, that gives a Poisson kernel
where 8 = 8(x;y) - solid angle, centered at {z}, as a function y E aD; S = S(y) - surface area element.
$2.4. Laplacian and related differential equations ~~~~
Table of basic differential equations and their Greens functions in
Equatwn
Fonnal solution
Laplace:
-Au = 6
Resolvent:
(A
Heat:
ut - A u = 0 L o =6
Schriidinger:
- A). = 6
Conj. Poisson:
RA = (A - A) - 1
Tt = e tA
ut-iAu=O UlkO
Poisson:
K=A-'6
=6
+
utt Au = 0 4t=0 =6 utt
+ Au = 0
Utlt=o
=6
p t
?t
- t o -e
=&
-
Wave (initial velocity): utt - c2Au = 0
u I = O;utlt=O= 6
I t
sin t f i
--
fi
Wave (initial disturbance): utt - czAu = 0 u 10 = 6; U t l t = O = 0
r, = cos t J E
1
6
119 ~~~
R".
~
62.5. Radon transform
120
$2.5. The Radon trandorm. The Radon transform appears in many guises and different geometric settings. The standard R2-transform takes a func_tion f in R2, and integrates it along all lines { l } , %:f-f(t). Similarly, the R"-functions can integrated along lines (X-ray transform), hyperplanes (Radon), or all intermediate p-planes (1 5 p 5 n-1). One can similarly define the Radon transform on n-spheres, integrating either over great circles (geodesics) or hyperspheres, or intermediate p-spheres. The basic problem is to invert W, i.e. recover function f from its integrals over a suitable family of lines, surfaces, etc. It turns out that the Radon transform is closely connected to the Fourier analysis in the Euclidian setup, and to harmonic analysis of relevant groups (orthogonal, Lorentz, etc.) in other cases. In this section we shall give a brief introduction to the Radon transforms on Euclidian spaces, and its connections to the Fourier transform, then mention how these results extend to other rank-one symmetric spaces.
5.1. General framework for the Radon transform is a dual pair of manifolds:
{t}c Z with a natural measure [(I = {z} c 96, with measure
where each paint z E 96 is assigned a subset [z] = d,((). Similarly, each point E Z corresponds to a subset
t
d&z). The generalized Radon transform 5 takes %-functions into Z-functions, and the dual transforms,
5:f (+3(t)
=
J "1 f (4d&);
%f(t)-.j(.) = /[,lf(Od,(t). In general, little can be said about the product and inversion of %-transforms,
5% - ? 3% ? jL-'-?
(5.1)
The situation becomes more manageable when both manifolds are homogeneous spaces of some (Lie) group G, and G-actions on 96 and Z are consistent: [tg]
= {zg:z E
[t]}, and [zg] = {(g:t
E [z]}, for all z;t.
We also assume that measures {d,([)} on leaves [z]c Z, and { d [ ( z ) } on leaves
[(I
c 96, are transformed accordingly, 9:d<(z)-.dp(z); and dz(t)-+(t)*
Group G obviously commutes with both transforms, as well as their products. Therefore, product-operators: %%kon L 2 ( 2 ) , and '%'& on L'(96) commute with the regular representations R of G on both L2-spaces. The analysis of the resulting
52.5. Radon transform
121
intertwining operators can often be reduced to special classes of functions (spherical {$}), where calculation of (5.1) can be performed explicitly. The foremost is the case of the so called rank-one symmetric space 96 (s& examples below and 55.7 of chapter 5). Here the algebra of intertwining operators is generated by a single element, the Laplacian A . So the product of two transforms and one only needs to compute the proper (1-variable!) function F(X). Formula (5.2) immediately yields the Radon inversion:
The following 3 examples will illustrate (5.2). 5.2. Euclidian space: The Radon transform on W" is defined by integrating function f ( x ) over all hyperplanes { t c W"}, that make up the dual space Z. The latter can be parametrized by all pairs { ( r ; w ) : r2 0;w-unit vector}, t = (r,w = {z:x.w = r } . Each point z E R" defines a subset [z] of all hyperplanes passing through it, which can be identified with the projective space Pn-l N S"-'/Z, = {lines in R" through 0 } , equipped with the natural (rotation-invariant) measure d x t = d t . So the entire space Z factors into the product: R+ x P"". Then the Radon transform, A
GjL:f(x)+f(r;u) =
I
fdS =
€r, W
while the dual transform & : $ ( r ; u ) + $ ( z )=
I
$dt =
[XI
1
f(x)dn-'x,
w*x=r
$(u.z;w)dS(w).
wESn-1
The Euclidean motion group En = W" b S q n ) acts naturally on both spaces,
z),
g = (a;u):x+xUta (x E Rn); (r,w+tr+a. w ; w u (€ E and transforms the corresponding function-spaces, f+Rgf. Operators ?k,%clearly intertwine the resulting regular representations, %(Rgf) = R g f . Space W" has rank 1 relative to En, in the sense that all En-invariant polynomials, respectively constantcoefficient differential operators, are functions of the Laplacian A (see Theorem 1 of 52.4 and $54.4-5 and 55.7). So (5.2) follows. Function F can be computed explicitly [Hell, and is found to be, A
n-1
9r(n/2)
F(X)= C A - 2 , with constant C = C, = (47r)
In other words the R"-Radon transforms satisfies
-.
r(:)
62.5.Radon
122
transform
To establish (5.3) we observe that set [z]of hyperplane {€} passing through obtained
from
a
single
(coordinate)
hyperplane
I,
is
to, rotated by all elements
u E K = SO(n+l). So [z]= {z+€l:u E K), and we can write (fig.9), ( T ) - ( z )=
1% +
€l)du =
I
f(z + YU)d(o(Y)dU=
K
I
(MI
I f)(z)d"-'y
5
= (5.4)
CO
= Cn-2 (Mrf)(z)r"-'dr.
K€o
0
Here du denotes the normalized Haar measure on K; d (y) =d"-'y
€
(Lebesgue)measure on S,(z)={y: I y - z l
-
the natural
to,and Mrf - the spherical mean of function f over the sphere
=r)inR",
Constant C, measures the volume of the unit k-sphere S'C Rk+'. Now it remains t o rewrite integral (5.4), as
Here we used the volume element in spherical-coordinate, dy = r"-'drdS,
and noticed
that factor F 2 d r d S , in the LHS of (5.5) turns into g y . So integral (5.5) becomes a convolution, f*$. But the convolution-kernel $ = 1 gives (up to a constant factor) a
I 2-Y I
fractional power of the Laplacian, Riesz potential: I , = ( - A ) - e , of order s = n-l 2 (see
§2.2), which completes the proof.
Fig.9 The double Radon transform: %%[f] at point z i s reduced in (5.4) to taking the Rn-spherical means {M I I f(z)} through all points {y) in the hyperplane to, and integrating them over
to.
By analogy with hyperplane transforms one can introduce the family of d-plane Radon transforms for all intermediate dimensions 1 5 d 5 n-1. Here we integrate
s2.5. Radon transform
function f on Rn over all d-planes
< c W".
123
The standard % corresponds to d = n-1,
while another extreme case d = 1 gives the famous X-ray transform". The space of all d-planes is parametrized by the points ( 7 ) of the Grassmanian Gr,(n-d), and all points E 7 . The Grassmanian Gr,(d) can be identified with the
<
double quotient space SO(d)\SO(n)/SO(n-d), of dimension, dimGr,(d) = d(n-d). The Radon space Ed can be thought of as a natural vector (n-d)-bundle over Gr,(n-d),
so its dimension = d(n-d)
+ (n-d) = (n-d)(d+l).
Once again we have a correspondence between points {xER"} and subsets
<
[x] c Zd (all d-planes through x) on the one hand, and all points (d-planes) E s d and subsets [ = {x E [} in Rn. So can define a pair of transforms %, 3, and hose turn out to be related by an analog of (5.5)
(5.6) 5.3. The spherical and hyperbolic spaces. The standard Radon transform on Sn is
given by integrating f over either totally geodesic (equatorial) hyperspheres 0N
Sn-l c S" (intersections of Sn with hyperplanes), or geodesics/great circles
(spherical X-ray transform), or intermediate totally geodesic d-spheres. This time space
Z is identified with the Grassmanian of 2-planes (X-ray transform), 3-planes, etc., in UP+'. The symmetry group here is still K = SO(n+l), and space Sn has rank one. As a consequence of Frobenius reciprocity (§5.7), the regular representation R on 5'" has multiplicity-free spectrum. Hence formula (5.2) holds with a function F(A) depending on
d. It turns out that for all even d , Fd is polynomial of degree ~131,
.+
P d ( X ) = [A
So we get
(d--I)(n-d)] ...[A +
$ (like in R").
+ 1(n-2)].,
Precisely
(5.7)
d 12- factors
-The energy (intensity) of an X-ray passing through a medium of variable density p ( z ) is attenuated at a rate proportional to the integral along its (straight-line) path I: E(incoming) In E(outgoing) - I pds. So X-ray measurements produce density integrals of p along all lines, passing through the body (of a patient), and one would like to recover the density variations (e.g. tumor).
I
52.5. Radon transform
124
Similar result holds for the rank-one hyperbolic spaces: H” = SO(1;n)/SO(n).The role of circles and planes is played here by the closed geodesics or totally geodesic dsubmanifolds. Once again relation (5.8) holds with the same constant C = C(d), but polynomial F d (5.7) ‘has all “$” signs in factor [A ...I changed to “-”,so the hyperbolic
+
Remark: Another version of the Radon transform is defined by integrating function u on R” over all spheres, passing through the origin. Any such sphere is parametrized by the radius r , and the direction-vector w, Sr,w= {z: I z -rwI = r } , %:u(z)-.G(r;w) =
J
u(z)dS(z),
(I-rwl = r
where d S is the natural surface area element on Sr,y. We refer t o [CQ] for further details. Let us only remark that geometrically the space of “spheres through the origin” is identical t o the space of “hyperplanes”, via the inversion map of $2.4, y-y’ difference is in the family of measures {dC(z)} on surfaces {(,,, Euclidean dPr(y) =
dy
{(I
-
= %. The IYI
Sr,,,}.The standard
on
hyperplanes
$y
(see fig.10). The Radon transform on spheres finds an interesting
is
repalced
by
the
spherical
measure
applications to the Darboux problem for the wave equation, utt = Au; with characteristic boundary conditions,
U(Z, 11:
I ) = f(z),on the light-cone { t = I z I } (see(CQ1).
For complete proofs and further details of Radon transforms we refer to a superb exposition of the subject in the Helgason’s books (Hell (see also [Ter]; [Wall; [Karl). Fig.10. The inversion y-y/ I y I takes a hyperplane tr= {y w = r } into the sphere S,,r,w (of radius 2/r), passing through the origin, and transforms the Euclidean measure dy on into the measure,
<
$0 = e
2
y = by. r2+y2
Chapter 3. Representations of compact and finite groups. In the first section of chapter 3 we develop the general representation theory of compact groups G , known as Peter- Weyl theory. We show among other results the orthogonality and completeness of irreducible matrix entries and characters of G, and complete reducibility for any representation T of G. Then (83.2) we shall study irreducibility and decomposition of regular and induced representations of compact and finite groups, based on Mackey's imprimitivity systems and Frobenius reciprocity. Several examples illustrates the general concepts and methods of $3.2: symmetric and alternating groups, symmetry-groups of regular polyhedra. We apply these results to "Laplacians on Platonic solids". In the last section 53.3 we carry out a detailed study of semidirect products. Our presentation here will be motivated and illustrated by specific examples of finite group: symmetric and alternating groups; unimodular group
SL, over finite fields, finite
Heisenberg groups H,. The latter are closely t o Clifford groups and algebras, and find an interesting application t o a topological problem of determining the number of linearly independent vector fields over spheres.
33.1. The Peter-Weyl Theory.
1.1. Matrix entries and characters. For any representation finite-dimensional space
Y N C", we choose a basis
{el;...;en} in
K
of group G in a
Y, and
define matrix
entries
I ek)}jk'
{"jk(")=("uej
(1.1)
More generally, we pick any pair of vectors <;qE Y,and set y7(")=(AuJ
I v).
(1.2)
The character of a representation K is obtained by taking the trace of ru,
Definition (1.3) strictly speaking applies only to finite-D representations However, we can define character
xT
distribution on G, (XT
for
a
suitable
class of
{T}.
of infinite-D (unitary) representations, as a
I f)=wJ);
test-functions
e={f(u)} on
G, provided operators
{Tr= If(u)T,du: f E e} belong to the trace-class. The latter holds for regular representations RG;RX on G, or its quotients X = H\G, for
$3.1. The Peter-Weyl Theory
126
L2; Lp. Indeed, operator {Rf:f E C!} are given by continuous integral kernels, e.g. K(z;y) = f(zg-'), on the compact space G or X. Any all reasonable function-spaces C!;
such K is well known t o be of traceclass (see Appendix B), with IrK =
I
K(z;z)dz.
In particular, character of the regular representation R on G ,
i.e. (problem 3),
XR = I I 6*,
(1.4)
We already know (1.4) for commutative groups R"; T", where it can be interpreted as orthogonality relations for exponentials/characters
{x<(2) = e i z
'
(}, and was shown t o be
equivalent t o the Inversion/Plancherel formula. In this section we shall establish a (noncommutative) compact version of these results.
Matrix entries (1.1)-(1.2) and characters are continuous functions on G, hence belong to L*-space with respect to the Haar measure' du. The characters are easily seen to be conjugate-invaiiant functions on G,
~ ( u z u - ' )= ~ ( z )for, all z , u E G, so they make up the center of the group algebra Z(G) (problem 1).
The following Theorem due to Peter-Weyl summarizes the main results of the representation theory of compact groups.
1.2. Peter-Weyl Theorem: (i) Any irreducible representation T of a compact group G is finite dimensional. (ii) Matriz entries and characters of irreducible representations following orthogonality relations
{T}
satisfy the
(iii) The orthogonality relations for characters
(iv) Completeness: matriz entries { T j k : 1 5 j , k 5 d(T);T E G} form a complete orthogonal system in L2(G); characters { x ~ : ET G } are complete and orthogonal
in the subspace of conjugate-invariant functions 'We have shown in $1.2 that the right/left invariant Haar measure exists on any compact (even locally compact) group. It defines the scale of Lp-spaces (1 5 p 5 m), and the convolution structure on G.
$3.1. The Peter-Weyl Theory
127
Z(G) = { f E C2:f ( u - ' z u ) = f ( ~ )Z;, E G } , center of the group algebra2 L2(G).
Proof involves two steps: first we show that any irreducible r can be embedded into the regular representation on LZ(G),then we apply the averaging procedure to certain rank-1 operators in L2(G),and utilize Schur's Lemma. Step 1: Embedding.W e pick a vector 'I in the representation space V = V ( r ) and define a map
I r)) = f ( Z ) l
W = W'I:€+z€ from V into L2(G). Map W is bounded:
I1f I1 = so
11 W 11 5 11 1) 11,
JI
I
(Xu€ 11)) 2du
I II 'II1 II € II 2>
and it intertwines representations "r I V" and ' R I range W C L2(G)".
From the unitary version of Schur's Lemma we easily deduce that W is an isometry, modulo scalar factor. Indeed, W*W commutes with r, hence W*W = cZ - scalar. Thus an irreducible r can be identified with a subrepresentation of R on a subspace
Y c L2(C).
Step 2 Averaging. If a compact group G acts on a vector space Y by linear transformations { T z } , all associated objects (vectors, operators, tensors) could be
averaged over G t o yield G-invariant objects (problem 7). In particular, average of a linear operator Q on Y is defined via Q d= ef
T,'QT,dz, G and is easily seen t o commute with all { T z } . Given an irreducible representation r on
Y C L2(C),we pick a pair of functions 'p, 4 in Y
and define a rank-one operator
W) = Q,@
= (f I
Q:L * ( G ) + S P ~ ~ { ~ ) .
Its average Q has the form
Q(f)=
1
R,QR;'(f)du =
1(K'fI
4 R U W = (f I
Notice that Q maps L2(G)onto the cyclic subspace of
V($) = Span{R,$:u
and
R , d R,4 du.
4,
E G},
a is 0 on the orthogonal complement of V(4) = Span{R,$;u G } . In the case of Y(4) = V($) = Y,hence a = XI is scalar on Y. E
irreducible r, both cyclic spaces coincide
Constant X can be computed from the bilinear form
'We
remind the reader that L2-functions on compact G form a convolution algebra, as
IIf * h 112 I I1f 112 11 h 112'
83.1. The Peter-Weyl Theory
128 whence
Here {r4+} denote matrix entries of the representation R. This shows that the average operator Q is indeed a A-multiple of the orthogonal projection that both Q and
P:L2+V. Let us observe
0 are given by integral kernels: Q(f)= JK(z;I)f(y)dy;
Q(f) =
J
where K ( ~ , Y =),to(zG(~),
where ~ ( zy), =
~ ( 2y)f(y)dy, ,
J $(zs)P(v)ds.
Obviously, kernel R(z;y) E L2(G x G), in fact R(z,y) is continuous on G x G. Therefore, integral operator Q: L2(C)-L2(G) is compact (even Bilbert-Schmidt). But a compact projection has always finite rank, hence dimY
< 00. This proves the first statement of the
Theorem. Step 3 Orthogonality. Given an irreducible representation r we once again pick a rankone operator Q = (... 10'1,and average it over G,
Q=
/
~;~Qr,,du. G Obviously, Q commutes with T, and irreducibility of T, implies by Schur's Lemma, that
Q = XI is scalar. But trQ = tr Q = Ad(*), whence we find trQ ('I I€) x -=Now we evaluate the bilinear form
4 x 1 4*)' (Qt' I9'), for a pair of vectors (';q'.
On the one hand
On the other hand
which proves orthogonality relation (1.5) for matrix entries of the same representation r. If entries r . and unmcome from two different (nonequivalent) representations, we take a rank-one operat,or Q from Y ( r )into
Y(u),and average it with respect to the pair
Q=
(rp),
J v;lQr,,du.
Such Q is easily seen to intertwine r and u, hence by Schur's Lemma Q = 0, whence follows the orthogonality of r j kand unm.The orthogonality of characters (1.6) is a direct consequence of (1.5)
Step 4 Campletensti of matrix entries and characters in Lz(C). Careful analysis of the argument in step 1, shows that any cyclic representation T (a representation with a cyclic vector
to),can be embedded in L*(G), also any cyclic subrepresentation T of R contains
129
83.1. The Peter- Weyl Theory a finite-dimensional (hence an irreducible) subrepresentation r. So if the span of all A
matrix entries { r j k : jk,; r E C ) were incomplete (had an orthogonal complement 'T c L2), then the restriction T = R I 'T would contain an irreducible r, a contradiction!
Completeness of characters results from completeness of matrix entries
{rjk)
via the
orthogonal projection (averaging), P:f'f(Z)
= /Gf(U-%u)d",
that maps Lz(C) onto the subspace of conjugate-invariant functions Z(G)= "center
L2(C)., QED.
Corollaries: (i) Matrix entries of irreducible representations { K j k : 1 5 j ; k 5 d(a);aE &} form a complete, orthogonal system in L2(G), a noncommutative analog of characters {~(u) = earn' "} on t o m . So any f E L2 can be decomposed into a generalized "Fourier series" in matriz entries,
f ).( = ' K , j k K j k ( " ) , with coefficients a K ,j k = (f I K j k ) d ( a ) . The series converges in the L2-norm, as in the torus-case, and one can also show convergence in all LP-norms ( 1 < p < m), via interpolation. (ii) Convolution formulae for matrix entries and characters, 1 O ; .i f k # i d ( r ) ajm,if k=i
Kjk*Kim=-{
and
I
xT; if K
X**Xa
{ 0;
if
K
I
=u
#u
Those follow by expanding each entry into the product, ajk("Y) =
Kji(z)K&(Y)? 2
using unitarity, K I.k (x-') = r I r k j (and z ) , orthogonality relations (1.5)-(1.6).
1.3. Decomposition. Formula (1.9) implies that characters { x A } form a system of mutually disjoint central projections in the convolution group algebra C(G). This property of characters will be applied now to construct the canonical decomposition of representations. In section 1.3 (proposition 3.2) we have shown that any unitary finite-D representation T is uniquely decomposed into the direct sum of primary components (multiples of irreducibles). For compact groups this result can be extended to all (m-D)
$3.1. The Peter-Weyl Theory
130 representations.
Decomposition Theorem 2 Any unitary representation T of a compact group G is uniquely decomposed into the direct sum of primary components, i.e. Jb = CBJb, - direct sum of invariant subspaces, and the restrictions T I Jb, N a 8 m - are knltiples of a E 8, with finite or infinite multiplicities m = u(a;T). The result is established by passing from the group representation to the convolution algebra e ( G ) ,or L'(G), Irreducible characters
f'Tf
{ x } yield
I
= Gf(U)Tudu.
a family of mutually disjoint (orthogonal) projections
{ P , = TX,:x E &} in 36. The fact that images, (36, = P,(36)} are primary subspaces follows from the characterization of primary subrepresentations:S N r @ m is primary, iff all motriz entries of S { s . (u) = ( S u e j l e k ) } are linear combinations of the matriz 3k entries of T.
Remarks: 1) Irreducible characters, as well as characters of more general finitemultiplicity representations, T N @ a @ m(a); m ( a ) < 00, provide a convenient way to label G, also to analyze irreducrbility and decomposition of T . Indeed, by general (functorial) properties of characters (problem 6),
+
1XT @ s = XT xs; XT gl s = XTXSli it follows that the correspondence T+xT is 1-1, and X T = Cm(a)xr.
(1.10)
As a consequence, one can show i) equivalence: T
-
S iff
xT = xs;
ii) d e g ( T ) = (xT I xT);hence T is irreducible iff 1lxT 112 = 1
(1.11)
iii) intertwining spaces: dim Znt(T;S) = (xT I xs). Formulae
(1.10) along with
orthogonality relations
for
{x,}
yield
a
decomposition of tensor (Kronecker) products of irreducible representations,
a @0 =
@-T
@C(n;a) .1
(1.12)
The multiplicities {C(a;uI T ) } called Clebsch-Gordan coefficients can computed from (1.12)-(1.10) (1.13)
2) The decomposition Theorem remains valid for Banach-space representations, in
83.1. The Peter-Weyl Theory
131
particular, the regular representation R in LP-spaces or e(G). In each case we get a complete system of mutually disjoint central projections { P % } .So any vector ( in V has a generalized Fourier series expansion, ( =
C (%; (* = P , ( ( ) .
However, convergence of
%
such series is usually more subtle issue, it holds in all spaces {LP(G): 1 < p < co}, but not in e(G).
1.4. Noncommutative Fourier transform. Given a function f on a compact group
G we define its operator-Fourier coefficients, ?(A)
= jcf(s)a,dz; so S:f-{3(n)}.
Map 9 takes convolution of functions into the product of Fourier coefficients,
( f * g ) - ( n ) = f ^ ( ~ ) g ^ ( ~ )So . any convolution algebra on G (C, L', etc.) is taken into the direct sum of matrix algebras: $ M a t Map %f share many properties of its 4x1' commutative counterpart, Inversion and Plancherel Theorem: for any function f e C(G'), or L'nL2(G), (1.14)
f o r any f E L2(G), (1.15) I
Here IIAllrrS denotes the Halbert-Schmidt norm of the matrix/operator A
I(A Ilks=t r ( AA*). Inversion
7,
easily follows from the orthogonality relations for matrix
entries and characters, while Plancherel (1.15) is obtained from (1.14), specified to functions g = f*f*. Sequence of degrees {d(a):?rE
c} defines the so called Plancherel
measure on G. Later (chapters 4-5)we shall compute them explicitly for the classical compact Lie groups. Finally,
we
shall specify the
decomposition Theorem
to
the
regular
representation R,f = f ( m ) , and describe the structure of the group algebra LZ(G). Theorem 3: i) regular representation R I L2(G)is decomposed into the direct sum of irreducibles,
RN each
~ - A B ~ ( A ) ,
nEG
appearing with multiplicity equal to its degree d(.rr). Furthermore, LZ(G)= $ L(A)- direct orthogonal sum of subspaces, A
63.1. The Peter-Weyl Theory
132
L(x) = Span{ait:l obtain by central projections, x,: L z - d ( x ) ,
5 j,k 5 d(x)},
1
x r ( f = Xr*f = f *Xr-
ii) Each subspace L(x) is closed under the convolution: L(x)*L(x) c L(x), so L ( r )
f o m a subalgebra of LZ(G),and L(x)*L(u) = 0 for x # u . iii) Each subalgebra L(x) z Matd(,) - the full mat& algebra of degree d ( x ) . Indeed, convolution formulae (1.8) identify basic matrix entries { x j k } in L ( T ) with the Kronecker (bjk}-basis in Matd. So the group algebra LZ(G)on group compact G is decomposed into the @-direct sum of matrix algebras,
1.5. Finite groups. We shall illustrate the foregoing by a few examples of finite
groups. All the above results apply here, but some can be further refined. Namely,
Theorem 4 (i:) There are finitely many irreducible representations number 1 GI = # { conjugacy classes of G}.
9 Matd(,),
v l -
(ii) The group algebra L(G) N representations satisfy:
{x}
hence degrees of
of G , their
irreducible
the order of G.
(1.16)
(iii) Degrees of irreducible representations { d ( x ) } divide the order of G . The first two statements readily follow by labeling irreducible representations by their characters
{xr), then
identifying
{x,}
with a basis of the center
Z(G).The last
statement is less obvious, it exploits some properties of algebraic integers (see problem 5 and ). Example 5: Symmetric group W, has conjugacy classes labeled by all ordered ptuples of integers: m, 2 mz 2 ... 2 mp; E m 3. = n. A representative of conjugacy class LY = (ml;...m k ) , can be chosen as a product of cyclic permutations: ol...ok; o1 = (12...m,), u, = ([m,+l] ...[ml+mz]),...
(i.e. cyclic reshuffling o1 of the first m,
numbers, then m, subsequent numbers, etc.). Two obvious examples of irreducible representations of VV, are the trivial representation x i = 1; and T: = sgng. Another one arises from the natural action of W,
63.1. The Peter-Wevl Theory
133
by permutations of C = {1,2, ..A},( R , f ) ( k ) = f(kg), for k 6C.The regular representation
R on L(E)N 43" has an invariant subspace 'V {f: C f ( k )= 0) N C"-', and one can show that a"-' = R I 'V is irreducible (problem 4, of §1.3), so R is decomposed into the sum, R T O e a"-'. The characters of all three are easy to compute,
-
x,o(s) = 1; xR(g) = #{lC E (1,2,***n), fixed by 9 ) XTn-l(g) = XR - x 0 = #{fixed k ' ~ } 1. Specifically, group W3 of order 6 has 3 conjugacy classes (by the number of
+ m,+ ... = 3) labeled by
partitions a:m, (a) =
{el;
(ab) - made of transpositions ((12); (13); (23)) (2-cycles); (abc) - 3-cycles {(123);(213)}.
So W, has 3 irreducible representations, whose degrees can be found from (1.16)
IG I
= 6 = 12+12+22 (the only way to decompose 6 into the sum of 3 squares!). Thus
we get a complete table of the representations and characters of W3.
I
I
The relative sizes of conjugacy classes {wj= C j / I G I } for VV, are
{i;;;;} (for
classes (a), (ab), (abc) respectively), and one can check directly the orthogonality of characters {xo;x1;x2}with weights {wj}.
Example 6: An ultemuting gToup G = A,, consists of all even permutations of 4 objects {1;2;3;4}. It has order 12 and 4 conjugacy classes:
ICl I = 1; (ab)(cd)- products of 2-cycles, I C, I = 3 (abc) - the class of a 3-cycle (123), I C3I = 4, - the class of (123)2= class of ( M ) , I C4I = 4. (a)={e),
Group A, is split into a semidirect product of an abelian normal group
H Z, xZ,={e; (12)(34); (13)(24); (14)(23)}, and a subgroup Z, generated by a 3-cycle, A, = H Q Z., In the next section we shall develop the representation theory of semidirect products, but here we shall use more elementary tools. We already know two of
A,-
representations: the trivial no, with character xo=l, and a 3-D representation a3 on
53.1. The Peter-Weyl Theory
134
Yoc 43,. Two other are easy to find from the decomposition (1.16): 12 = 1z+1z+1z+32. Thus ?yl = x l and ? y 2 = x z are both one dimensional. In fact both are nontrivial characters of the Abelian factor-group A,/Z, x Zz 1~ Z,. So the character table of A, takes the form: (acb) 1
1
1
1
As above one can easily check the orthogonality relations for
{xj} with weights:
(1.1.1.1) 12’4’3’3
’
Example 7 The symmetric group W, has order 24 and 5 conjugacy classes, (according to partitions a: m, + m, + ... = 4), labeled by ( a ) = {e}-trivial; (ab) - a 2cycle; (ab)(cd)- a pair of 2-cycles; (abc)- a 3-cycle; (abcd) - a 4-cycle. Hence there are 5 irreducible representations. We already know 3 of them: x+ = 1; x- = sgn; and a 3-D representation 7r3, realized in functions on a 4-element set (a homogeneous space of VV,!). Counting their degrees l Z +1’ 3’ = 11, we find that the remaining two representations have degrees: a’ bz = 13, so a = 2; b = 3 (the only solution!). We shall construct them explicitly in $3.3 (see also chapter 5, 55.5).
+
+
A similar analysis applies to W, of order 5! = 120. It has 7 conjugacy classes, labeled by {5}; {4;1}; {3;2}; {3;1;1}; {2;2;1}; {2;1;1;1} and { l ; l ; l ; l ; l }= { e } (by the number of partitions). We already know 3 of its representations: x+ = 1 (trivial); x- = sgn; and 7r4 (of degree 4). The remaining 4 could be shown (problem 8) to have degrees: d = 6;5;5;4. So the entire dual object of VV, consists of 3 pairs: { ~ * } ; { 7 r ~ * } ; { 7 r ~ * } , and {T‘}. In the next sections we shall explain the “breaking into pairs”, and the relation between representations of W, and its subgroup A,. Let us also mention that group A, has 5 conjugacy classes, hence 5 irreducible representations of degrees: d = 1;3; 3; 4; 5 (problem 9).
$3.1. The Peter-Weyl Theory
135
Problems and Exercises: 1. Show that center 1; of any group algebra L(G) consists of class-functions f(uzu-') = f(z) (Elements of 1; commute with the regular representation!). 2. Show that an irreducible character 2; into C.
xr
defines a homomorphism, f
m'
(x*f)(e), from
3. Find the character of the regular representation R on G , and show that XR = 6 ( U ) = 4r)xr. 4. Calculate the Clebsch-Gordan coefficients for K, @ r, and r, @ T , in Examples 5 and 6: groups G = S, and G = A,.
c
5. Theorem [Ser3]: For any fin_ife group order1 G I divides %he degree d(r) of any irreducible representation I 6 G. Follow steps: i) An algebraic integer in the center 2; of the group algebra, is an element u that solves an algebraic equation:
c' aju"-J= n
u"+
with integer coefficients {aj} E Z.
0,
ii) All integers in 1; form a ring, closed under multiplication with complex algebraic integers (numbers), i.e. any combination b j u j of Zintegers {uj} with algebraic integer coefficients {bj} in C, is itself a Zinteger.
c
iii) Take the natural basis in the space Z, which consists of class-functions
1. on C . the j-th conjugacy class e j = (0; elsewiere rnfjek; with integer mfj). Show that functions {ej} are Zintegers (eicej=
C
So any class-function whose values {u(g):g E G} are algebraic integers is itself an integer in Z, therefore any character x is an integer. iv) Any irreducible character
x defines an algebra homomorphism
(problem 2),
qi * Z + C,qi (u) = ' c u ( g ) f ( g ) ; d = degree of r, x that maps algebraickintegers into alghraic complex integers.
v) Apply qix to
x
to show:
qix(x) = + IG I I x) =
I G I - algebraic integer.
But any rational algebraic integer in C is an integer, QED.
6. Characters: (i) check the direct sum, tensor product and conjugation formulae for characters:
xT$s=xT+xs; xT@s=xTxs; XT = R T . (ii) Use these formulae and orthogonality relations for characters to establish (1.10)(1.13). 7 . Averaging: Show that any representation T of a compact group G in an inner product space 36 is equivalent to a unitary representation (Take the product (( q ) in 36 and average it over G, (( I q),, = (Tu( I T,q)du!).
I
8. Analyze 4 remaining irreducible representations of W, (example 7), and show their degrees to be 6; 5; 5; 4. This is the only solution of the equation a2+bz+cz+dz = 102, subject to constraint of problem 5: a,b,c,d divide 120, and another constraint: all four are greater than 1 (as any W, has only two 1-D representations, those of the quotient
136
$3.1. The Peter-Weyl Theory Wn/An!). 9. i) Show that alternating group A, has 5 conjugacy classes, represented by the following elements (cycles and products of cycles): (12345); (21345); (123); (12)(34); and {e}. Hint: take 'even" conjugacy classes in W,, and show that all of them but one, (12345), coincide with the "alternating conjugacy classes"; the W-class of (12345) though splits into two A-classes. Splitting or unsplitting of the W-conjugacy class of g E A, depends on whether the commutant (centralizer) of g in W, Z(g) = {h E W: h - ' g h = g}, 'has" or "has no" an odd permutation (check)! ii) Use Theorem 4 (problem 5) to derive the degrees of irreducible representations of As: d = 1;3;3;4;5.
53.2.Induced representations and
Frobenius reciprocity
137
53.2.Induced representations and Frobenius maprocity. Induction of group representations was introduced in seetion 1.2 (chapter 1). This procedures allows one to construct in a canonical way a representation (2’) of group G , starting from a representation of its subgroup H , and provides one of the basic constructions in the representation theory. Here we shall study induced representations of compact groups, and addreas both problems: irreducibility and decomposition. The former is given by Mackey’s test, the latter by the Frobenius Reciprocity Theorem. Several examples will illustrate the general results, but many more will come in subsequent sections and chapters.
2.1. Induction. The induction procedure arises in the context of group
G acting
on a homogeneous space X cz H\G, g:x-+xg,where H denotes the stabilizer of a fixed point x,,E X . We pick a system of coset representatives { 7 z } z E X in each class
{x = Hg}. Then any element g E G is uniquely decomposed into the product g = hy,, where x = H g is the class of g (mod H ) , and h E H depends on g , and on the choice of coset representatives. Given a pair x E X , g E G we take the product y2g and decompose it as
729 = h(x,9 ) 7,s. The resulting function h:X x G -+H satisfies the cocycle condition
h(x;gu) = h(x;g)h(xg;u),for all x E X and g , u E G.
(2.1)
The induced representation T = ind(S I H ; G ) of group G is constructed from a representation S of H , acting in space ‘V, and could be realized in many different ways.
Construction 1: We take a suitable space of ‘V-valued functions on X ,
e = e ( X ;Y),or L2;etc.,
and define operators
It is an easy exercise to check, using (2.1), that T is a group-representation, independent of a particular choice of coset representatives { r2}. Furthermore, equivalent representations S’; S2of H induce equivalent T’, T 2 of G. The natural examples of induction are furnished by regular representations on G and X , both induced by the trivial representation 1,
RG = ind( 1 I { e } ;G);RX = ind(1 I H ;G). Construction 2. We take a subspace e(G;S) c C(G;‘V) of all ‘V-valued functions on G that transform according to S under the left translates with H ,
83.2. Induced representations and Frobenius reciprocity.
138
representation
I T g f ( z )= f(zg); all z,g E GI. Here T is realized as a subrepresentation of the d(S)-multiple of the regular representation on G, T c R B I
4s)'
One can easily check that both definitions (2.2) and (2.3) of T are equivalent, by writing an intertwining operator W : e(G;S) * e(X;Y). Indeed, each function F ( g ) on G, that transforms according to S, is uniquely determined by its values at coset representatives {yz:z E X } . So we get
W:F(g)-+f(z) = F(y,). The second construction is clearly
independent of a particular choice of {7,}!
Construction 3. A subgroup H c G and a representation S I H in the standard way to an associated vector bundle W = U
Y, N Y.
Y,, over X
Y give
rise in
= H\G, with fibers
ZEX
Group G acts on the bundle W and the base-space X , and two actions commute with the natural projection T:W+X. Moreover on each fiber-space map g:Yz-+Yzs is linear. We call it u ( z , g ) .Function g(z,g) is easily seen to satisfy a cocycle condition
(2.1), with group multiplication turning into the product of operators. The G-action on the vector-bundle results in the induced action by linear transformations on vector-spaces of cross-sections T ( X ;W )= {f(z)E Y,}, equipped with usual (L', L", etc.) norms',
As above one can easily construct an intertwining map W :T ( X ;W)+e(X;Y);or
to C(G;S), and to show that all three definitions (2.2); (2.3); (2.4) are equivalent. 2.2. Commutator algebra and the irreducibility test for Ind. Let X = H\G and u be a representation of H in Y.We choose a particular set of coset representatives {y,}, and construct T = Int(o I H ; G ) in space e ( X ; Y ) .Let H, = y;'Hy, denote a stabilizer (for each fixed z) defines a representation of of Z E X . Notice that the cocycle u h ( z ;9 ) H , in Y.We call it u2,the "pull back o f u I H" by the adjoint (inner) automorphism 4,:
H-+H, = y,Hy,'; W -e assume that all fiber-spaces are furnished with certain norms (inner products), and X has a suitable (G-invariant) ,volume element.
53.2. Induced representations and Frobenius reciprocity
139
4 s ) = 4”; 9)=47A7Yz).
(2.5)
Our goal is to characterize the commutator algebra Com(T), in particular to find conditions for Com(T) to be scalar, i.e. T - irreducible. To this end we use a simple analytical result: any linear operator W on the function-space e ( X ; V ) is given by an operator-valued kernel (in the “finite cue’’ a matrix, with entries labeled by z , y E X ) ;
A(z, y): X x X-’3(V) - the algebra of linear operators on T ‘
Wf(.) = Ix4z;Y)f(Y)al; or
c
A(z;Y)f(Y)* ,EX If W commutes with the induced representation, W T , = TgW,it follows that
A(z; Ybh(y; g)[f(Y’)] = u h ( q g ) A ( z g ; y)[f(y)l; for any
(2.6)
f in e(X;‘V).Changing the variable y+yq in (2.6) we get:
= ah(z,g)A(zg;Yg); and the latter can be rewritten as A(z;Y)ah(y;g)
for
z,y E x; E
The G-action on the product-space X x X = {(z, y)} splits it into the union of Gorbits: 6 w j ; each w . = {(z,y), = (zg;yg): g EG}. One of the orbits is the diagonal 3 1 w,= {(qz)}. Obviously, the value of kernel A at each diagonal point (z,z) is determined by its value at a single point (zo;zo) E w,. Moreover, if H denotes the
in G, then the value A(z,;z,) = A, E le(V), commutes with the representation a of H, A, E Corn(a I H). Conversely, any A, E Com(u I H) can be extended to a matrix/operator-valued kernel A(z; z) on w,, satisfying (2.7). stabilizer of {z,}
Next we shall analyze orbits spanned by non-diagonal pairs { p = (z;y)}, and
H, = H, nH,. Let us observe that any cocycle o(z, g) defines a family of stabilizers {H,} in space ‘V. The RHS of (2.7) can be written in terms of the family of representations {u”}of stabilizers {Hz}, their stabilizers
of representations
(Q-’A(z;y)(u$) = A(z,y), for all g E H(z.y) = H,flHy. This implies in particular that each operator A, = A(z;y) on space ‘V intertwines
n
representations: d‘ I H, H, and a, space of two representations u’ and
I H,nH,.
Conversely, any A, in the intertwining can be extended to an
d‘, restricted on H,nH,,
operator-valued kernel A(z; y), satisfying (2.7) on the entire orbit w of (z; y). Combining both results of our analysis (diagonal and non-diagonal parts of
X X X ) , we get a complete characterization of the commutator algebra for induced
$3.2. Induced representations and Frobenius reciprocity.
140 representations.
Mackey’s Theorem 1: The commutator algebra of t h e induced representation T = ind(a I H;G) consists of all operator-valued kernels A(z,y), satisfying (2.7). Hence,
ComT IICorn(a I H).g Int(aYjI HzjnHyj;azjI HzjnHyj) 3=1
- sum over all orbits { w j }
in
(2.8)
X x X , where (xj;yj) is a point in w j .
Since any G-orbit in X X X has a point of the form (zo;y),where zo is a fixed (distinguished) point in X , we can state (2.8) as
sum over all orbits ( J c X x X . Here we picked a point y = y(w) in each orbit w , and denoted by H , its stabilizer, and by uw the pull-back u y (2.5) of u on H y ( w ) .As a corollary we obtain the folowing Irreducibility Test for induced representations.
Theorem 2: T = ind(u I H;G) is irreducible iff intertwining spaces: .Tnt(az I H ,
Remarks: 1) For
u
is irreducible, and all
nIfy;# I H , n HY)= 0,for any pair z # y.
(2.10)
the sake of presentation we stated the above results for finite/discrete
spaces X. Both of them extend with some technical modifications t o infinite/continuous spaces, the direct sum (2.9) in Mackey’s Theorem being replaced by the direct integral over the orbit-sp;ace R = (X x X ) / G , while the intertwining condition of Irreducibility Test required for almost any pair ( q y ) , or almost any orbit w E R.
2) In many cases, like semidirect products to be discussed in subsequent chapters, the “off-diagonal” part of A(z,y) vanishes for obvious reasons, so irreducibility of T reduces t o irreducibility of a!
A special case of (2.9) arises when H is a normal subgroup of G. Then space coincide with H . X = H \ G is the factor-group, and all conjugate stabilizers The action of G on H,h-tg-lhg, induces the “dual” action4 on “representations of H”, g:u-u[ = o(g-’hg).
-
Note that elernents h E H transform u into an equivalent representation: oh u.
So d depends only on the class of g in H\G, i.e. d= uz (z = H g E X ) . 4The same holds for any automorphism a E Aut(H), a:u-uE = u(ha)!
$3.2.Induced
representations and Frobenius reciprocity
141
Now part relation (2.10) of Theorem 2 takes the form
ICom(T)= Corn(a I H ) V C o n ( o " j I H ; a y j I H)I If o is irreducible and all {#:r # q,}belong to different equivalence-classes in p, then T is irreducible! This observation will be essential in the study of semidirect products in the next section and in chapter 6. But the importance of induction extends far beyond the class of semidirect products and group extensions. Here we shall briefly describe two such examples.
Example 1. Let G = SL(2;IF) be the group of 2 x 2 matrices of determinant 1 over finite (or infinite) field F, and H denote a subgroup of upper triangular matrices , called the Bore1 subgroup of G.
[t :4}
We shall pick a character (1-D representation) x ( h ) of H , and study
T =ind(G;xI H ) . To compute operators { T g }explicitly we choose coset representatives of the form
rZ= (i9,
and write the product:
This factorization shows that the homogeneous space X = H\G
can be identified
with the projective space: P,(F) N IFU{m}, group G acts on X by fractional-linear transformations g:z cz+d' and the representation T transforms functions f(x) by
T,xf(4= x @ . + d ) f ( S ) ; 9 = Using Mackey's test one can check that
3
(2.11)
T X is irreducible iff xz # 1 (Problem 3).
Formula (2.1 1) defines the so called principal series irreducible representations of SL,.
Example 2: Alternating group A,. We already know the degrees of irreducible representations of A, (problem 9 of 53.1): 1; 7r3*;7r4; and 7r5 (superscript indicates the degree of T ) . Two of the list are known to us: 1, and a 4-D representation in "functions
on a 5-point homogeneous space",
C, = {1,2,3,4,5}N A5/W4. In other words the regular representation R = ind(1 I A4;A5) of A, on C, is decomposed into the direct sum: R N 1@ a4. It turns out T' can also be realized, as an induced representation on X,. But this time the inducing character is nontrivial: x = x * , either one of (nontrivial) 1-D representations of A, (rather A,/[commutator]
N
Z3),described in example 6 of 53.1. SO
142
$3.2.Induced representations and Frobenius reciprocity.
7r
= k d ( x 1 A,;A,).
Mackey’s test applies to show that
7r
(2.12)
is irreducible, and 2 representations
T*,
induced by X * are equivalent (problem 6). To complete the analysis of A5 we need to construct a pair of 3-D irreducible representations {7r3*}. Let us remember that A, acts naturally by orthogonal matrices (dodecahedral symmetries) on I$ ($1.1). This action is easily seen to be irreducible (leaves no invariant lines or planes!), and to be of real type
($3.1), i.e. scalar commutator. Another 3-D representation of A, is obtain by conjugating with an outer automorphism 0 = (12) E W,, so 7r3-(g) = 7r3(uga).
7r3
2.3. Characters of induced representations. Since the equivalence class of
T = ind(G;aI H ) is uniquely determined by xT in terms of xg. Indeed, from (2.2),
u we should be able to express its character
Tgf (z) = u(h(z;d ) [(f4 1 , it follows that the only nonzeros contribution to t r ( T g )comes from fixed points {z} of transformation g , i.e. {z: T z g T i l E H } . Then summation over all { Y ~ } ,that conjugate g into H . We can rewrite it in the form,
where summation extends over all fixed points {z}of element g. In subsequent chapters we shall derive similar expressions for induced representations of Lie groups. After the natu.re of irreducible induced representations was analyzed we turn to the decomposition problem for induced T .
Frobenius Reciprocity Theorem: Given an induced representation T = ind(u I H ; G ) , and a representation S of G , the intertwining spaces I n t ( S ; T ) and Int(S I H ; aI H ) are naturally isomorphic. In particular multiplicity of an irreducible 7r E G in T is equal to the intertwining number m(u;alH)= “multiplicity of u in 7r I H”. Proof: We denote by Y(S); Y(u); Y(T)the representation spaces of S, u and T. Each linear map W :Y(S)+Y(T)
is given by an operator-valued function F ( z ) :Y(S)+Y(u),
defined via F(z)u = (Wu)(c),for all u E Y(S).
For operator W to intertwine S and T function F ( z ) must obey the relation:
33.2.Induced representations and Frobenius reciprocity F(z)Sgu= uoh(z,g)F(zg)u, for all z E
143
X,and g E G.
So F ( z ) is uniquely determined by its value at a single point, zo = {H} E H\G,
and
F ( z o ) = u-'(h)F(zo)Sh, for any h E H, so F(zo) intertwines representations S and u of
H. Conversely, each intertwining linear
map Fo = F(zo)E Znt(S;u), extends to an 'intertwining function" f Y z ) = u - l o h(zo;gz)FoS(gz),
which proves the requisite isomorphism of the Znt-spaces, QED.
As a corollary we recover the multiplicities of irreducible { T } in the canonical decomposition of the regular representation RG = ind(l1 {e};G), established in the previous section, RG = @ _T @ d ( T ) , TEG
so the multiplicity of each irreducible
T
is equal to its degree. We also obtain a
decomposition of the regular representation
RX = ind(1 I H;G) on
By Frobenius Reciprocity RX is made up of
{T
E
quotients X = H\G.
G}, whose restriction on H contains a
trivial subrepresentation 1 of H , and the multiplicities,
m(n;Rx)= #{H-invariant
vectors in V ( T ) } . A specific example in case is the regular representation of G = Sq3) on the 2-sphere, R,f(z) = f(z"), f E L2(S2),u E G, induced by a subgroup H = S q 2 ) . In the next chapter 4, we shall obtain a complete decomposition of Lz in the sum of irreducible 00
components @36, 0
(dim%, = 2m+l), each appearing with multiplicity 1, and each
subspace containing a unique H spherical
harmonics on
-
invariant vector (function), Ym(0), called zonal
Sz. The same holds for higher dimensional
spheres
S" N Sqn+l)/SO(n). Once again the regular representation R of group S q n + l ) on L2(S") has simple spectrum (direct sum of multiplicity-free irreducible components), as each irreducible subspace contains a unique Sqn)-invariant (see 55.7).
2.4. Laplacians on Platonic solids. We retell a horror story from a popular thriller
[Kirl]: A student of mathematics having once spent a long but disappointing night playing dice, decided to make the dice "fair". He relabeled its faces from the standard (integer) set {1;2;3;4;5;6},to another sextet, by averaging each face over its 4
neighbors. The new fractional dice played somewhat better, but not quite to his satisfaction, so the student kept on relabeling the faces again and again, maintaining the utmost accuracy he could. How close did he come to the fair mean 3.5 after 30 laborious steps
§3..2. Induced representations and
144
Frobenius reciprocity.
The reader has probably recognized lurking behind this not “very serious” problem, the cubical Laplacian,
L f ( a )= f E f ( @ ) , sum over 4 neighboring faces, acting on “faces of the cuben 4, better to say functionspace ‘Y = e(4). Clearly, Laplacian L commutes with the action (regular representation) of symmetry group G = Gcubeon Y. To diagonalize L we need to decompose ‘Y into the sum of irreducible components @ Ti,and compute eigenvalues { X i = L I Yj}. The number of components equal to the number of G-orbits: w c 4 x 9, and the latter has 3 terms: diagonal wo = {(aa)};neighboring pairs w1 = {(a@)};and opposite pairs, w2 = {(aa’)}(check that
G
acts
transitively on all 3). So there are 3 irreducible
components in ‘Y, and these are easily identified:
Yo = {constants} (deg = 1); ‘TI = {even functions of total sum 0: C f ( a )= 0 (deg = 2); ‘Y2 = {odd functions} (deg = 3).
(rE9
The value of ,C on ‘Yo is Xo = 1. To find A, we pick a particular
fl
= 1 on top and
bottom, and -f on walls of the cube, hence A, = -f. Finally, subspace Yz is represented by f z = 1 (top); - l(bottom); 0 (walls). Hence, A, = 0. Triple {l;-$;O} completely determines L and all its iterates. So we find Lm[f] (f = (1;2;3;4;5;6)) to be within E
= 2-m11
f-f
1 proximity of the mean f = (3.5; ...;3.5).
A similar argument applies to other regular dice, tetrahedron: X = 1;-1 (deg = 3); 3 octahedron: X = 1;-1; 0; dodecahedron: X = 1; -1 (deg = 5); l/& (deg = 3); - l / & (deg = 3). 5 We shall briefly sketch the last case. The dodecahedron has 12 faces and 4 orbits in 4 x 4: diagonal ( c ~ a )adjacent ; pairs
(ap); pairs
one-face-apart (a7);and opposite
pairs (aa’). To identify irreducible components of T ( 4 )N
CI2, we
break it into the even
and add parts, write
‘Yet,= ‘Yo{const}CB ‘Y, (all even f of sum
cf(a) = 0), a
while T o d d = ‘Y3+ CB ‘ Y 3 2
The former (‘V3+) consists of linear functions (vectors) ( E Ff‘ restricted on faces {a E 4 ) . We associate unit ort ea with each face a,and set
t+f&)
= ( . ea.
53.2.Induced representations and
Frobenius reciprocity
145
The latter (Y3-) is made of bi-vectors (tensors) t A 'I by a similar rule
t A 'I+f ( A ,(a) = t /l'I ea = det[t;'I;ea]* *
-8.
The eigenvalues of the even part: A,-, = 1; A, = For T, we pick f = 1 on top face, and f = -15 on 5 adjacent. For T3+ we pick vertical ort ( and compute the (components of 5 adjacent {ea} = l/&, which yields, ,A = l/&. A similar argument applies to A-, = - l/&. But we can also observe that total trace of L = 0, hence A-, = --A3+. Based on our results one can easily analyze more general symmetric random walks of regular polyhedra of (p;q)-type, i.e. probability p to stay at a given vertex/face and probability q to jump to an adjacent one, where p 3q = 1 (or p +4q = 1; p 5q = l), depending on polyhedron. The corresponding operators (stochastic matrices) are combinations: qL p1.
+
+
-+
146
53.2.Induced representations and Frobenius reciprocity. Problems and ExeTcises: 1. Show that the equivalence class of T = ind(S I H ; G ) is unequally determined by the class of S, i.e. given W , E Znt(S,;S,) construct W E Int(T1;Tz). 2. Calculate F(y,g) and show that W intertwines both representations (2.2) and (2.3). 3. Show that the representation T of SL(2;F) given by (2.7) is irreducible iff 'x Mackey's test). 4. Calculate the character of
# 1. (Use
T in problem 3.
5. Find the character of the regular representation and expand it in the sum of irreducible characters.
RX on the quotient-space X = H\G
'
6. Use Mackey's test to show that induced representation 12 of A, is irreducible. Show that representations r (2.12), induced by characters x f!c A,, are equivalent.
*
7. i) Compute irreducible characters of A, on all 5 conjugacy classes, represented by elements: a = (12345); a' = (13524); b = (12)(34); c = (123), and {e}. ii) decompose the regular representation R of A, on space e ( X , ) 2: C" of dodecahedra1 faces (X = A,&), into the sum of 4 irreducible representations: 1 @ 17' (even part of e); r3+@ r3- (odd part).
$3.3. Semidirect products
147
93.3. Semidirect products. In this section we shall examine the structure of irreducible representations of semidirect products: G = R D K . Although the answer becomes more complicated, compared to the directproduct case, there is a systematic procedure t o describe and construct them completely, in the language of Mackey's group extension theory and induced representations. Our presentation here will emphasize the finite/compact examples, but later (chapter 6) many of the results will be extended to continuous (Lie) groups.
3.1. Definition and examples. Let us remind the definition of a semidirect product. Given a pair of groups H and K , K acting by automorphisms of H, u : z +xu, (uE K ; z E H ) , a semidirect product H DK consists of all pairs g = ( 2 , ~ with ) the multiplication rule (z,u).(y,v)=(z.yU
-1
;uv).
It can be easily seen that H = { ( q e ) } forms a normal subgroup of G, while I( = {(e;u)} a subgroup, so that H nI( = {e}, and the whole group G = H . I(. Natural examples of semidirect products include:
i) Dihedral group: D, = Z,
D Z,;
K = Z, acting on Z, by inversion, u: z-+(-z).
ii) Alternating group: A, = (Z, x Z,)
D
Z,. Two subgroups Z, are generated by
double cyclic elements: (12)(34) and (13)(24), while Z, is generated by a triple cyclic permutation (123). iii) Symmetric group:
W, = A,,
D Z,.
Any transposition can be chosen to represent
I( = Z, in VV,,. iv) Affine transformations: (a,b):z+az
+ b, over a finite (or infinite) field IF. Here
translations, b:z+z+b, form a normal subgroup, isomorphic to the additive group IF, while multiplications, z-+az, make up a multiplicative subgroup F* of F. So Aff(F) ? D F*. The affine group can also be realized by the upper-triangular matrices
{P PI},
There are many examples of (infinite) Lie-group semidirect products. Here we mention just two of them (the details of both will appear in chapter 6). v) Euclidian motion group: En = W"
all translations, while subgroup
D
S q n ) . Here normal subgroup Iw" consists of
Sqn)acts on W" by rotations.
vi) Another well-known example is the Poincare group of Special relativity:
$3.3.Semidirect products
148
P4= M4D Sq1,3), here M4 denotes the Minkowski 4-space with indefinite metric (+---),
and S q l ,3) the Lorentz group of all linear transformations that preserve it. Let 8 denot8e the dual object of H, the set of all equivalence classes of irreducible representations of H. Group K (respectively G) acts on H by automorphisms and this action is transferred to 8,
g: x'xg( h) = x(hg Thus the dual object
x
).
a splits into the union of K-orbits
w = {K:K =
For each stabilizer in G.
-1
xg;g E K } E a\K
= R-orbit space.
in H we denote by K , its stabilizer in K , and G, = H D K, its
3.2. Semidirect products with commutative normal subgroups. We shall first study this special case of semidirect products, as it makes the result somewhat easier to formulate and to prove. Theorem 1: Irreducible representations T of a semidirect product, G = HDK, with
commutative normal subgroup H, are parametrized by pairs { ( w ; ~ ) }w, E R - a n orbit of K in H, and U E K, - a n irreducible representation of stabilizer K , of point x E w . Furthermore, T is equivalent to induced representation, T i n d ( x 8 a I H DK,;G). N
Proof: Take a representation T of G, and decompose its restriction, T I H, into a sum of primary components:
Let us observe that the set of characters
{x E fi}
in decomposition (3.1), splits into the
union of G-orbits { w } , under the natural action of G on H . Note also that any two x's in the Same orbit,
xl=xir have
equal multiplicities, the corresponding subspaces being
transformed one into the other by
Tg:Vx1)--'T(x2). Furthermore, direct sum of primary subspaces over w,
is invariant under the entire group G. We denote by S =Sx a subrepresentation,
T g I T(x), of stabilizer G X= H D K X on the X-th primary subspace. Then irreducibility of T I G implies that the primary decomposition of TI H is made of a single orbit w. Picking a point
xo in
w with stabilizer,
Go = H
D K O , one
can easily show that subspace
$3.3.Sem idirect products
149
‘+‘(,yo) is irreducible under representation
S I Go, and T = ind(G;S I Go). Indeed, G acts
on the homogeneous space w =Go\G,
and the associated family of subspaces
{‘+‘(x)}~
forms a G-invariant vector bundle over w. The resulting G-action on
“sections” of the bundle is obviously ind(G;SIG,). Hence, the Mackey test applies to
T is irreducible, iff G
show that
xo
a c t s irreducibly on Y(xo). The verification of the
Mackey’s criteria for
T = ind(G,X@ u I H b KO), is fairly straightforward. All pair-intersections: G ,
nGXo2 H - normal subgroup of G,
and the fiber-actions of H at two different points, y,
# xo are clearly disjoint,
sxo I a 2: xo 8 z # sx 1 a 2: Hence, SxO I G ,
,I. 8
nGo and Sx I G , nGo are also disjoint!
So the irreducibility problem is reduced from representation T of G to a subrepresentation
S = Sx of a stabilizer G X = H b K X on a single fiber Y(x). But the normal subgroup H c G , acts on Y(x) by scalars, {S, = x(h)Z), so irreducibility of Sx I G , is reduced to a representation u(u) = Sx(u) of stabilizer subgroup
K X C K. This completes the proof.
Remark Theorem is a special (finite) case of Mackey’s imprimitiuily system [Mac]. In general such systems are made of a group G, acting on manifold X, and a pair of representations: {T)of G , and {Lf]of the function-algebra e ( X ) in space 36, that obey the natural consistency condition,
Tg-’L,Tg = L f g , for all f , g , where
fg(z) = f(zg).
G. Mackey shows that any irreducible imprimitivity system must be realized in L2 vector-valued functions {$(z)} on the orbit w C X (or a closure of the orbit), by operators, Tg$= $(zg); and Lf$ = f(z)$(z).
We shall illustrate Theorem 1 with several examples:
Example 2: The dihedral group, “9, = Zn b Z,. Its dual, 2, = Z,, each character
x I.(a)
= txp(2ri%); j=o; 1; ...n-1.
Group Z, = (1;~)and automorphism c : x ~ - + x , - ~ ,and point orbits
{w
. = (x3.;xn -3.)}, 3
2,
9)
splits into
(or 2(depending on the odd\even parity of n), and a 1-point
wo = {0}, (or a pair of 1-point orbits: (0); {;}, for even n). So the dual object b, consists of 2-D representations, corresponding to 2-point { w . } , and a pair (or two
orbit:
(9)
3
corresponding to pairs) of 1-dimensional representations of the factor-group Z, = D,/Z, (0) (and/or (9)). The reader could check directly all the general results (Theorem 4 of $3.1) for the dihedral group: (i) the correspondence between irreducible representations
and
#
of
conjugacy
classes; (ii)
orthogonality
relations,
and
the
identity
63.3. Semidirect products
150
Ed(r)’ = order(&); (iii) divisibility of the order of D, b y irreducible degrees {d(r)}. Example 3: The tetrahedral (alternating) group A, = (Z, e Z,) D Z,. We studied this example in the previous section, and now revisit it in the light of Theorem 1. Here normal subgroup
H consists of elements { e ; a = (12)(34); b = (13)(24); c = (14)(23)},
while K = Z, = {e;(123);(132)} permutes {a,b,c} cyclically. Thus
k has
two orbits:
{xo} with 3 chosacters (irreducible representations) of Z, “sitting above it”; and a 3-point orbit w1 with a 3-dimensional representation 2. One can verify straightforward that r1 is equivalent to a representation of problem 4 (51.3 of chapter 1).
wo =
3.3. General yemidirect products. Now we turn to a general case of G = H D K , with an arbitrary (noncommutative) normal subgroup H, and establish the analog of Theorem 1. This discussion will bring in a few important concepts in representation theory: projective representations; cocycles and central extensions. The first step in the derivation is similar to the commutative case. Namely, the representation space breaks into the direct sum of fibers over a single G-orbit w
Here
c c?:
x E H denote equivalence classes of irreducible representations of H. Notice, that
any automorphisrn a E Aut(H) can be lifted t o space
c?,
a:T-TE = Tha;h E H. Thus we get a natural action of G, respectively factor-group K on H . As in the commutative case the restriction Sx = T I T(x) of stabilizer GX = H D K X , is irreducible on the fiber-space T(x). However, the ensuing analysis of S x becomes more involved, since now we have to deal with equivalence classes of H-representations, rather than characters
{ x E E?}.
Cocyclea and central extensions. Given a representation
x E g , and a group K of
automorphisms of H, that leave “class X” invariant, each element u of K defines an
x.
intertwining operator W , in the representation space Yo of Of course, such W , is not unique, but is determined only modulo scalars (Schur’s Lemma). However, once operators { W,} are chosen, they form a so called projective representation of K ,
W,W, = a(u,v)W,,; for all u,vE K , with scalar factor a(u,v). The associative (group) law immediately yields a cocycle condition5 on function
53.3. Semidirect products
151
a(u;v)a(uv;w ) = a(";vw)a(v;w ) , for all triples u, v, w.
(3.3)
4 u ;v), Cocycle a is called trivial (coboundary), if
a(u;v) =
m, for a function (cochain) p(u). P(,.)
A trivial cocycle implies that W could be transformed into an honest representation of K by multiplying each operator W , with P(u). Thus a projective representation is determined up to equivalence by the 2-nd cohomology class of
0,i.e.
element of the quotient group " 2 - ~ 0 ~ y c l/eU2-coboundaries". s~ All notions of representation theory: irreducibility, equivalence, etc., extend to projective representations. In fact, any projective representations of group K can be made an honest representation of a certain central extension of I<, by group T, C*, or more general Abelian group. For a given group K and a cocycle a satisfying (3.3) on K with values in
G,
=K
T = { z =e":O<8<2s},
we define an a-central eztension6 of K ,
g T,as the set of pairs { ( z ; u): z E T;u E K } with multiplication law, ( z ; u )* (21;v) = ( z z ~ a ( u ; v ) ; u v ) .
It is easy to see that any a-projective representation of K gives rise to a
G,. Conversely, any representation of G,, restricted on a subset ((1;~)) =I<, yields an a-projective representation of K . Thus there is 1-1 correspondence between dual objects: a-k (a-projective irreducible representations of K ) and A natural example of central extension, is given by the celebrated Heisenberg group, that will be discussed at the end of the section and in chapter 6.
representation of
ea.
5The terminology came from algebraic topology. The cocycle condition can be written in a more conventional form, via c o b o u n d a y operator B on spaces e p ( G ) of cochaines, i.e. functions a(zl; "'2,): G x x G +"numbers".
...
yzz Operator B takes a p c h a i n a into a p+l-chain, 8:a-(8a)(z0; ...z p )= c (-1)3a(zo; ... zjzjtl; P
"1;
.
... z p ) .
3=0
Here we write the group operation on numbers in the additive form. Linear operator 8 sends and has the property a'= 8.8 = 0. Hence, space of null-vectors vector space ep+Cptl, Zp = {a E eP:8(a)= 0}, called cocycles, contains the image of 8, W' = { p = 8(a):aE €Yt'}, called coboundaries (or trivial cocycles). The quotient ZP/%P forms the p t h cohomology group, X P ( C ) . Cocycles arise naturally in group extensions, and in off-diagonal matrix-entries of group representations (see ch.6). 6For finite K one can show that, modulo coboundary corrections, each cocycle a takes values in a finite subgroup Z, of T (n-th roots of unity). So we can always use a finite group 2, instead of T in the definition of the a-central extension. More general notion of the central extension arises, when circle T is replaced by an Abelian group A, and a represents an A-valued cocycle.
83.3. Semidirect products
152
We proceed to analyze semidirect products. With each point
x E E (a clam of irreducible
representations of H, acting in space To),we associate its stabilizer K,, a cocycle a = aX on K , and an a-projective representation W of K X ,that results from its action on To:
~ ( h "= ) W , x(h)W,'; h E H ; u E K X . On the other hand representation T of the product E DK also yields a ,-primary subspace T(x) of TI H, on which stabilizer G X acts irreducibly by S = Sx. Space Tx
I
factors into the tensor product To8 T,, and each T h Y(x) = ~ ( h81. ) Conjugating the latter with operators {S,:u E K,} we get
S,'(x(h) 8 Z)S, = ,(hU) 8 z = (W,18 Z)(X(h) 8 Z)(W, 8 I ) . Therefore, operators { Q , = S,(W,'
8 I):, E K , } commute with all {X(h) @ I: h E H}.
By Schur's Lemma (see Theorem 3.5 of §1.3), operators {Qu} must be of the form I 8 u,, which yields a representation u of K X in Tl. Since W was a-projective it follows that u must be &projectfive in order for the product S, = W , 8 uu, u E K,, to make an honest representation of K , . It is also obvious that irreducibility of S I G , is equivalent to irreducibility of u.
The results of our analysis can now be summarized in the following
Theorem 4: Let G = H DK be a semidirect product. We take a representation x E H, and denote b y w a K-orbit through point { x } , by K , and G, - stabilizers of x in groups K and G respectively. The action of group K , on "space of X" determines a 2-cocycle (rather 2-nd cohomology class) a E X 2 ( K x ) .We also pick a pair of representations: a-projective W in space to= t ( ~and ; )irreducible, , - projective u in t,.Then (i) irreducible representations T of G are parametrized by all pairs { ( w ; ~ ) } , of K-orbit w c H,and &-projective representations u of K,, at some x E w . (ii) the dual object of G can be viewed as a fibered space over the set of orbits f2 = {w c E?}, with fibers labeled by $-projective irreducible representations of K,, the $-dual object (&K,) , A
6 = uJ w ; ( & q
A
1.
WCH
Furthermore, T is induced by a representation S of G,, whose restrictions on subgroups H and K , are:
S I H c z x 8 1 , SIK,cz
W ~ U .
Remark Theorem 4 obviously includes all previous results for direct products
$3.3. Semidirect products
153
($1.3) as well as semidirect products with Abelian normal subgroups. In both cases representation W becomes trivial, which simplifies the final result.
The following examples illustrate Theorems 1 and 4, as well as the notion of central extension.
Example 5: The affine group, G = { g : I-+ az+b} is a semidirect product F D F*, with the action a:b+b” = ab. The dual group i of a field F is identified with F by means = Xl(ab), and the action of F* of a nontrivial character xl:F+C*, so that each a-Xa(b) def on libecomes a multiplication, a:Xb+X,b. Clearly, $ splits into the union of two orbits,
{O}uw, w = F*. The stabilizer of w is trivial, so there is only one irreducible representation, “sitting above” w, rW=ind(xbI F;G), where b E B*. Thus the dual object G consists of an irreducible representation rwof degree = #F*,and all l-D characters of the Abelian factor-group G/F N F*;G = { a WUF*. }
Example 6 Symmetric group W, is a semidirect product of W,DZ,; ?!,-factor could be represented by any odd permutation of order 2, e.g. Q = (12). We already know all irreducible representations of A, (example 6, $3.1), which is itself a semidirect product (problem 7). These are (i) { x o = 1 ; x I ; x 2 } A,/[commutator] N Z3;
l-D
characters
of
the
commutative
quotient
(ii) x3 - a 3-D irreducible representation. Automorphism Q acting on the dual object A, leaves (1) and { x 3 } invariant (one-point orbits), and permutes {x1;x2}(a two-point orbit). Theorem 4 applies now, and we get a pair of representations of W, over each one-point orbit (labeled by the dual object of Z,):
{ 1}+{ 1; sgn} - l-D characters of the quotient W,/A, { ~ ~ } + ( 7 r ~ ~- +a ;pair a ” of } 3-D representations,
N
Z,;
x3 I A,, extended by g+I,
or -I.
A 2-point orbit w = { x I ; x 2 }yields a 2-Dirreducible (induced) representation r2 of VV,, whose elements are represented by matrices, for h E A,; and a: = Of course, the count of degrees: 12+12+22+32+32= 24, agrees with the general results of $3.1. Group W4 makes up the cubo-octahedral symmetries ($l.l),so it has a number of other interesting “geometric” representations, discussed in problem 8.
53.3. Semidirect products
154
Example 7: Symmetric group W, = A, D Z,. Once again we know the dual object (here we call irreducibles of A, x's, rather than 7r's). of A,: (1); {~3+;x3-~};{x4};{xs} The dual space clearly breaks into 4 orbits: 3 of them one-pointed, and 1 a two-pointed orbit. Each one-point orbit yields a pair of W,-representations of the same degree (with Z,-generator CT being represented by & I ) , while a 2-orbit yields a degree-6 irreducible representation:
(11; {w};
1+
x4 x5
--$
{7r4+}; {*4-};
+
{7r5+}; {*5-};
{x3+;x3-}+
{T".
Thus we recover all 7 irreducibles of W,,and their degrees (problem 8, $3.1).
Example 8: The Hewenberg group G = W, consists of all 3 x 3 upper-triangular matrices
Group G is nilpotent of step 2, i.e. the commutator subgroup the center,
[G;G] is equal to
Group G can be viewed as a semidirect product, F'DF, where IF2 is identified with the normal subgroup
and subgroup
acts on F2by linear transformations a:(b;c)+(b;ctab).
The dual group g2is isomorphic to IF2, where character and the dual action of IF on
x&b) i2becomes,
= Xl(XCtPb),
53.3. Semidirect products a: (A;
F-orbits in each nonzeros A,
@)-(A;
iz are identified with
155
@+.A).
lines parallel to the @-axisthrough (X;O), for
wA= {(A;b):b€IF}
and points {(O;@): PEF} for A=O. Stabilizers of the first family are trivial, while for w = { (O;@)} they coincide with the center 2.The resulting irreducible representations, become
T ~ ind(xX10 = I IF~;G),for A
+ 0,
(3.4)
and all characters of the factor-group G/Z=F2. The reader could write down explicit formula for T X(problem 3 and chapter 6), T X g f ( x )= xO(Xc t x)f(x t a ) , (3.5) where x o ( a )is a nontrivial character of the additive group F, a finite-field analog of eza on R. So the dual object G consists of two families of representations: ITX}, F* of degree #F, and 1-D characters {x E i '}. On the other hand group G can be viewed as a central extension of an Abelian group
by a cocycle, a(h; h') = ab'.
Indeed, multiplying matrices
So irreducible representations (3.4) of
G can be viewed as a-projective
representations of the commutative group F2,with cocycle a (3.6)! More general (finite) Heisenberg group H, represents a central extension of F2"by 2 = F, i.e. elements {g} are given by triples {(a; b;c): a, b E F"; c E F}, with multiplication a ( g ; 9')
= (I .6'.
(3.7)
H, is also a semidirect product, F"+*iF",and has a family of irreducible representations
{T':x F*} (3.5) in spaces of functions on F", T l g f ( z= ) xo(~c+~b.+)f(z+a). Those along with characters
{ x ~ D }of the commutative factor-group
H,/Z
N F2"
$3.3. Semidirect products
156
comprise all irreducible representations of H,
(8ee
chapter 6 for more details). We shall
describe now an interesting topological applications of the finite Heisenberg group.
Example 9: Vector fields on spheres. The problem is to find the maximal number of tangent vector fields on spheres 9 - l C R", linearly independent at each point x E s"-1. Let D(n) denotes their number. If fields {El;...ZD} form such a system, then the standard Gram-Schmidt orthogonalization yields m orthonormal basis of fields
;<&)I,
{ & ( x ) ; ...
tj(z). z = 0; (I.(z) . ( k ( i ) = 6I.k ; for all j , k; at each x .
Taking linearization of fields
(3.8)
{ t j }we get D matrices { A l ;...AD}, so that t j ( z )N Ajx.
(3.9)
We shall apply the representation theory to solve a more simple linear problem
(3.9), although the final answer, number D(n), turned out to be the same for both (3.9) and (3.8). The orthogonality conditions (3.8) imply that matrices { A j } are antisymmetric and satisfy A; = -1; A j A k = -AkAj, for all j # k; (3.10) (problem 3). We want to find the maximal number D of such matrices in R". In the language of representation theory we deal with a group G = GD in D+1 generators {a,; ...a D ; b } , that obey the following relations, b2=l; a~k .a = a a b; (3.11) k j and ask for a minimal degree representation of G , that takes generator b into - I . We claim that G is a finite Heisenberg group 04, over the field7 IF = Z,. Precisely, a'. = b; 3
(i) G contains center
{
Z,; for even D 2 = Z ( G ) = Z, x Z,; for odd D; and the quotient GJZ N Zf is commutative. Center Z is spanned by { b } in the former (even) case, and by ( b } and a = al...aD in the latter case, so (problem 2),
BID,,; even D
'Its official name is the Clifford group, as it generates the Clifford algebra, that plays an important role in representation theory, geometry, topology and Physics. It serves the basis for so the called spinor-groups (2-fold simply connected covers of orthogonal groups), gives rise to spinor structures on manifolds and the Dirac operator.
53.3. Semidirect products
157
We shall state a few other simple properties of G, which the reader is asked to verify (problem 4):
...a . , or
(ii) any element of G is uniquely represented either as ai (il
1
< ... < ip);
2P
as bai l-aiP
(iii) square of any element g in G belongs to the center, moreover, P(P+l)
9 - b 2 , for elements of length p .
Form the previous example 8 (see problem 3) we already know all irreducible representations of the Heisenberg group. They are either 1-D characters of the commutative quotient G / Z (trivial on Z), or an irreducible T = T A ,which takes central generator b to -I (the only possible nontrivial character on Z2!). Representation T Acan D-1 be realized on functions on Z;l2 (even D), or 2 7 (odd D), (problem 3). Hence it has degree: D-1 n = 2’f2; or 2 2 . It might seem we have essentially completed the argument. But there is a small catch, not to be overlooked! Construction (3.4) gives complex irreducible representations of Wn, whereas our goal was a representation (3.10) of real type’. Of course, any complex T can be thought of a real representation of the double-degree, but our goal was to find the esact minimal D to represent (3.10).
So the argument would be completed after we determine the type of representation 2’ (real, complex or quaterni~nic~). The latter could be accomplished by a type-criterion due to Schur. Schur’s Criterion: An irreducible representation T of a finite/compact group G belongs to a real, complex or quaternionic type, depending on the sign of the integral, Schur(T) =
I
G
XT(g2)dg=
+; real-type
- ;quoternionic
(3.12)
The proof of Schur’s criterion is outlined in problem 6. We apply it in our setup. P(P+l) Remembering that all squares { g 2 E G } are &I, depending on parity of 7 (for elements of length p , (iii)), we find the Schur’s number (3.12) to be ‘ha1 type T means that the representation space Y has a real invariant subspace Yo, so T I Y becomes a cornplexification of T I Yo.
-
’We remind the reader that an irreducible unitary representation T in complex space Y belongs to the quaternionic type, if T T (complex conjugate, or contragredient of T ) , i.e. T has real but there is no real invariant subspace in Y. character xT =
xT,
$3.3.Semidirect products
158
...}
x(e)(l- 4-(f)t):( - 7 (3.13) the “double-alternating” sum of binomial coefficients, the tnth term corresponds to
elements of length rn in G. The latter can be written as R(l+i)D - S(l+i)D.Thence, one finds that the sign of Schur(T) depends on the remainder of D modulo 8, and is given by the following table 1
2
3
4
-
-
-
5
0
6
7
+
+
Dt2
22;
I
if D=8m+2, or 8m+4 D-1
22;
if D=8m+7
As an off-shot; of our analysis we see that the number D(n) depends only on the maximal factor 2P in n ! Our exposition closely followed the Kirillov’s book [Kirl].
53.3. Semidirect products
159
Problems and Exercises. 1. (i) Find all irreducible characters of the affine group and show their orthogonality.
(ii) Do the same for the finite Heisenberg groups H, and H,.
2. Show that the Clifford group G of example 9 in D generators {ui:l 5 i 5 D;u:= -1; u i u j = -ujui} is isomorphic to a finite Heisenberg group over Z,: HDIz (for even D)and H(D-l I z x Z , (for odd D). Hint: for even D = 2n, elements { u ~ u ~ ; u ~ u ~ ; . . . ; u , ~ ~ ~ u ~ ~ generake a maximal commutative subgroup, A N find another subgroup E N 2, to get a Z2-cocycle cy(u;b). For odd D = 2n+l, we still have subgroups A , E 2: Z;, but the center is now spanned by b=-Z and element c = ul...uD Show that c commutes with all { u j } . 3. i) Show that irreducible representations { T A }of H, can be realized in space of functions {f(z)} on F, by operators,
T A f ( z ) = xo(Ac+Ab.z)f(z+u), where g = (u,b;c); A E F*, anc! xo(u) - a nontrivial characte! from the additive group F to complex numbers, an analog of exponential function {e"} on R. It is well known (and could be easily verified) that any character x = XA on F is obtained from xo by a "A-factor" : X A ( 4 = xo(Aa); A E F. an analog of xA(u) = ei'a, on R. ii) establish a similar formula for the n-D Heisenberg group H, (the characters of F" have the form: x P ( u ) = xo(p.a);p E F"). 4. Verify the relations (3.10) for matrices { A . } (Use a characterization of antisymmetric 3 matrices, as { A : A z .z = 0, for all E E R"}).
5. Verify properties (i)-(iii) of group G,. 6. Sehur's "type-Criterion". i) We denote by S'(T), A '(2') symmetric and antisymmetric tensor-ezlensions of an operator/representation T in space Y. Show xT(g2) = tr s'(T,) - t r A '(T,), for any g.
ii) Observe that tensor-product space Y @ Y is identified with the matrix-space Mat(Y), whereas subspaces of symmetric and antisymmetric tensors turn into symmetric and antisymmetric matrices { A = ( u j k ) : u k j = & a j k } , and group G acts on Mat(Y) by conjugation, 9: A+=T,AT,. (3.14) iii) Representation (3.14) on Mat(Y) is also equivalent to a G-action, T i ' A T , , on space Mat(Y;Y*). Integrating the latter over G,
-
A+A = $, T i ' A T , , on yields G-invariants l:Y+Y*,intertwining representations T in Y and T in Y*. Note that Mat(V) N S'(Y) @ A ,(Y) - direct sun of symmetric and antisymmetric spaces, and show that Schur integral, IxT(g2)dg, gives the difference of two intertwining numbers: Im(?
T ) on symmetric space PI- Im(T; T
on antisymmetric A
'1
iv) Conclude that for complex-type T, Schur(T) = 0 (space Int(T;T) = O!). For realtype T , the only intertwining operator { A } lies in the symmetric subspace, S'(Y), so Schur(TJ = 1,while quaternionic T has Znt(T;T)1: Com(T) inside antisymmetric space A (Y),so Schur(T)= -l! In the former case, A = I , and representation operators IT,} belong to the orthogonal
53.3. Semidirect products
160 group
qn), in the latter case A =
'} and {Tg}belong to symplectic group
7. Find all irreducible representations of alternating group A, method of Theorem 1.
N
(2, x 2,)
D 2,
Sp(n).
by the
8. Find irreducible characters of W,, and make its character table. 9. Group W, acts naturally by symmetries of the cube (each h E A, rotates 2 antipodal tetrahedral into themselves, while u $ A, f l i p both). There 3 different function-space on
the cube: C (functions on faces); a) eF N ' b) eE N C" (functions on edges); c) e, N c8 (functions on vertices).
So we get 3 associated regular representations of G = W,: RF;R E ; R". i) Write each of them as the induced one, ind(l1 B;G)(find subgroups H); ii) Decompose each R into the sum of irreducible by two different methods, using characters (problem 8), or applying Frobenius reciprocity.
Chapter 4. Lie groups SU(2) and SO(3). Groups of unitary and orthogonal matrices: SY2) and S q 3 ) provide the simplest examples of the classical compact Lie groups, yet they exhibit many essential features of the general theory. Both appear in many different contexts in Mathematics and Physics, not surprising due to the fundamental role of 3-D Euclidian space and its translational and rotational symmetries. After a brief survey of groups SY2), S q 3 ) , their Lie algebras and Haar measures in H.1, we construct and analyze (54.2) their irreducible representations in different realizations: polynomials in 1 and 2 variables, infinitesimal construction. In the next seetion ($4.3) we compute matrix entries and characters of irreducible representations { P }of 4 2 ) , and establish their connections to the classical orthogonal polynomials: Legendre and Jacobi. Then (94.4) we turn to representations of SO(3) realized in spaces of spherical harmonics {Jb,}
on the 2-sphere. Here we shall see
the connection between spectral theory of the spherical Laplacian and the regular representation of SO(3). The last section 94.5 will extend the SqS)-theory to higherdimensional spheres and orthogonal groups.
54.1. Lie groups SU(2) and Sq3) and their Lie algebras.
1.1. Lie group SU(2) consists of all 2 x 2 complex unitary matrices of det = 1, { u = [ -p a- h p ]detu=Ia~+l~~=1
while S q 3 ) is made of 3 x 3 real orthogonal matrices u of det = 1. Both are three dimensional Lie groups, which means they have the structure of smooth (real analytic) %manifolds with differentiable group operations of the product and inverse. The two groups are closely related, SU(2) forms a two-fold cover of SO(3): S q 3 ) u SY2)/{center f I}.
(1.1)
One way to show (1.1) is to realize SU(2) as a multiplicative subgroup of quaternions,
<
Q = {z t iy t j z + h; 2,y , t ,u-real$, j , k - imaginary units} = { = (Y +t jp: a,,&complex}. In the (noncommutative) field of quaternions there exists the norm,
d
m
, with the property: l
Hence the set of unit quaternions (unit sphere)
U = {(:
/
isomorphic to SU(2). Conjugation, <-+u-~(u, with u E U leaves invariant a 3-D real subspace of pure imaginary quaternions {( = yi+ t j t uk}. So it defines a map from
U - SU(2) onto S q 3 ) (problems 1;2).
84.1. Lie groups and Lie algebras: SU(2) and
162
1.2. Local coordinates
Sq3).
(Ed= angles). There are many possible choices
of local
coordinates on SU(2) or SO(3), for instance, a pair ( a ; R e p ) or (a;Smp), on SU(2), or any three independent entries of u E Sq3). Another (better) choice of coordinates is given by so called Euler angles (4,B,$). They arise in a factorization formula: u = u4veu$; 0 5 B
Indeed,
'
writing
p = smZei(4-4)/2, we get '
matrix-entries
of
u E SU(2),
a = cos-e e i(4++)/2.
as
2
(1.2). For Sq3) the corresponding one-parameter subgroups
can be identified with rotations by angles 4,$ about the z-axis, and by angle 6 about the x-axis. The Haar measure (invariant volume element) on both groups is expressed in the Euler-angles as
I
du = -&1inBdBdq5d$
I
(1.3)
Formula (1.3) is a special case of a general invariant volume-element formula on compact Lie groups to be derived in the next chapter. But translational invariance of du can be verified directly. We observe that the quotient-space of G = Sq3), modulo oneparameter subgroup
H = {u4:0
5 4<2r},
G on S2.Given
,B
S2,with the natural action of E S2,we identify z with an element uz = ueull of
is isomorphic to the 2-sphere, H\G
point z=z(e;$)
N
SU(2), via the Euler decomposition. Then the product dS(z)=sinOdOd$, represents the natural (rotation-invariant) area element on S2N H\G. We take u E S y 2 ) factor it as u4uz, then multiply on the right by u E G. The product (u uz)u is factored once again
according to (1.2) into uu = u4,u
,; where
4
the new Euler coordinates are: z'= zu
(zE S2,rotated by u E C ) , and 4 shifted by an angle depending on
(2;
u),
4' = 4+4(z; u).
Since dS(z) is invariant under all rotations z-rz";u E Sq3),and d4 is invariant under all translations: +#~+q5,-,
it follows that du = dSd4 is the right-invariant Haar measure.
1.3. Lie algebras 4 2 ) and 4 3 ) . Lie algebra 4 2 ) consists of all 2 x 2 hermitian
antisymmetric traceless matrices:
while 4 3 ) is made of all 3 x 3 real antisymmetric matrices. Both have a basis of three elements:
$4.1. Lie groups and Lie algebras: SU(2) and
So(3).
163
The basic elements satisfy the commutation relations:
[H;V]= 2 w ; [v;W]= 2H; [W;H]= 2v.
(1.5)
So both Lie algebras are isomorphic, but Lie groups are not! In physics {H; V ;W } are commonly called PauJi (spin) matrices, and denoted by ao;a';a2.The corresponding one-parameter subgroups of Sy2) generated by { H;V ;W } are
[
e z p ( t H ) = ,it L
,-it
1
[
J
L
ezp(tV)=
cost
sint
-sint
cost
1
1
J
L
;ezp(tV)=
cost
isint
isint
cost
1; J
while for S q 3 ) they consist of rotations about 3 coordinate axes.
Problems and Exercises: 1. Quaternions Q can b e
i) 2 x 2 complex matrices ii) all pairs {u = ( a ; z ) : aE R;zE R3}, made of real scalars and 3-vectors, with multiplication given by the dot and cross-products: 21.
u = (ap - t .y;ay
+ p t + t x y}.
Check both statements.
2. Calculate explicitly the homomorphism Su(2)-.So(3). Hint: consider the adjoint action: u+Ad,(X)=u-'Xu of Su(2) on its Lie algebra 4 2 ) 2 : R 3 . Show that (X;Y) = -trXY, is a positive definite inner product on 4 2 ) , and {Adu:u E 342'3' preserve this product. Thus So(2) act by orthogonal transformations on 4 2 ) N R Compute matrix of Ad, in the basis { H; V; W } .
.
Another way to establish the correspondence S m Q N 4 2 ) is via the cross-product in R3 (problem 1): (u . N t x y.
$4.2. Irreducible representations of SY2).
164
54.2. Irreducible representations of SU(2). 2.1. Irreducible repreeentations of group SU(2) can be constructed starting from
its natural action on the complex 2-space C2= {(z'w)}. We shall call this representation a = a19' The action of S y 2 ) on C2 extends to a representation 7rm on the space of all homogeneous polynomials in (z,w) of degree m, !Pm = {f(z, w) = C a j zJw"-J}, as1
(*?f)(z, w) = f(G z-pw; p z
+ow)
O<j<m
(2.1)
It will be convenient to write polynomials f E Tm, in the form
Let us observe that space Tm can be identified with the rnth s y m m e t r i c tensor power of C2,
Tm N P ( C 2 ) c
-
c2El c2€3 ...El c2 = P ( C 2 )
by assigning monomials zjwm-j to symmetrized tensors 2.j = ~
-
y m el ( 8
1 ...8 e, 8 e, ~l ...8 e, ) = xc (el8 ... ~l e 0
~ ) ~ ;
(2.2)
s - (
sum over all permutations c r W,~. Here {el;%} form the natural basis of C2. All tensor-product spaces (C') m, consequently symmetric tensors Ym and polynomials Tm, inherit the natural inner product of C2: (t111) = 6.9; ( t 1 ~ 8t m 1..
I 71 Q ...Q vm) = 0 ( t j I v j ) . 3
So for symmetrized tensors (2.2) we find,
which in the TP,-space takes the form, 0;
Since the original
T
j#k
was unitary the resulting representation am (2.1) also
'We remind the reader that the groupaction on manifold g:z-+g(z) is transformed into the ) left action (representation) on functions {f(z)}, as R ~ ~ (=zf(g-'(z)).
34.2. Irreducible representations o f SU(2).
165
becomes unitary with respect to the product (2.3). Space 9, can be identified with polynomials in one-variable 2 = & by writing
It will be convenient to write polynomials F as
and relabel spaces ,T , so that 3 , will consists of (2m+l)-degree polynomials (dimy, = 2m -k l), with m taking on integer or half-integer values: m = 0;;; I; ... We define an inner product on space 3 , = { F = C a j z m + j } , according to (2.3) m+j)!(m-j)!
-m_<j_<m
and write representation
?rm
as
Theorem 1: Representations { T " } are unitary and irreducible. Any unitary irreducible representation of SU(2) is equivalent to {zm:m = 0;3;1;...}. Proof: We shall use the infinitesimal method of section 1.4 and study the associated representation of Lie algebra 4 2 ) = Span{H;V ;W } . The one-parameter subgroups of the Lie algebra generators (1.4)
after substitution in (2.5) yield ( z ~ ~ ~ = ~ e-i2mt ~ ) ( F(eZitx) z )
(.lrzptvF)(z) = (X sin +- C O S ) ' ~ F (z zcos sin i- cos1
(2.6)
(7r;ptwF)(z) = (z isin i- cos)2mF(S?;;;';;;) Differentiating the RHS of (2.6) in t at t = 0, we find infinitesimal generators of
{ H ;v;W I ,
166
84.2. Irreducible representations of SU(2).
$.
where d denotes the derivative Formula (2.7) could be directly verified to realize su(2)- commutation relations (1.5) by differential operators (problem 1). Next we shall represent generators {HIV;W}by matrices in the natural basis of monomials {zm+j}$ -m in .3 , The direct calculation with (2.7) yields
So operator 7rE is represented by the diagonal matrix, diag(2m;2m-2; .,.-2m), while 7rT and 7 rg are given by tridiagonal matrices with 0's on the main diagonal, {1;2; ...;2m) above the diagonal and {2m;2m-l; ...;I} below the diagonal. Notice, that matrices T;;" and 7rE are not skew symmetric in Czm+' 2: Tm with its natural hermitian product. But if we change it to the product (2.4), i.e. (ej I ek) = (2m-j)!j! bjk1 by conjugating {nV;rW}with diagonal matrix Q = diag (..[(2m-j)!j!]-'/'...}, bring both operators to a skew symmetric form (problem 6).
this will
Irreducibility. By Schur's Lemma we need to show that Com(7rm) is scalar. Matrix A = T; was shown to be diagonal with distinct eigenvalues. Commutator of any such A consists of diagonal matrices. But any diagonal matrix B = diag(b,;...;b,,+,) that commutes with {~;;7rE},also commutes with their linear combination. We take X = T;;" f i7rE - an upper/lower triangular matrix with a single sub/super-diagonal row of non-vanishing pairwise disjoint entries. It is easily to show that any diagonal B that commutes with such .Y must be scalar (problem 2). This proves the first statement. To show that { P : m= 0;f;l; ...} comprise the entire "irreducible list" of SU(2), we shall apply once again the infinitesimal method. Let us note that any representation of Lie algebra s\=su(2) can be lifted (exponentiated) to a global representation of SU(2), since SU(2) 2: S, is simply connected. Lie algebra si =su(2) is contained in the complex Lie algebra (5 =s(2), and forms a so called real (compact) form. This means that (5 = R e is\, as any matrix A E 0 can be decomposed into its symmetric (is\)and antisymmetric (R)parts. So any representation of s\ extends to a complex representation of (5,
54.2. Irreducible representations of SU(2).
I
167
I
In fact there exists a natural 1-1correspondence between representations of s 4 2 ) , respectively SU(2), and complez representatiod of 4, which respects irreducibility and decomposition. This correspondence, known as Weyl "unitary trick", will be examined closely in chapter 5 for more general simple and semisimple groups. Representations of 4 are easier to deal with. We introduce the so called Cartan basis in 4: h=-iH=[l-l)
X=#+W)=
[O 03 Y = - % V + W ) = k o}
and verify the commutation relations [ h ; X ]= 2X; [ h ; Y ]= -2Y; [ X ; Y ]= h.
(2.10)
-
Let r be an irreducible representation of sl, in a complex vector space 9T. We denote by h = r h ; X = rx; Y = r y its generators, respectively H = r H ; V = rv; W = rw - the generators of 4 2 ) . Operator K is diagonalizable with real eigenvalues, since H = ih is equivalent to a skew-symmetric operator (problem 3). Let X, > X, > ... denote the eigenvalues of K, and { E ( X j ) }the corresponding eigenspaces. We shall use the following A
A
-
Lemma 2 i) Ail eigenspaces of r,, are one dimensional, E(Xj) = span{vj}; vj- the
the j-th eigenvector. ii) the highest eigenualue A, = 2m is an integer, and the j - f h eigenvalue X j = 2(m-j); for 0 5 j 5 2m. So dim 'V = 2m 1. iii) operator X is raising eigenvalues: X ( v j )= ajvj-,; while Y ( v j )= bjvj+l
+
A
A
lowering them. iw) sequences of coeffzcienta { a j } , { b j } , obtained by applying eigenvectors satisfy the relations 3 = 2(" - j ) = U i + , b j - bj-l"jl.
p
2 and p
to the j-th (2.11)
'
To prove the Lemma we pick an eigenvector vi of an eigenvalue X j and apply the lowering operator rY to u j . It will move ui to another eigenvector of
7rh,
due to the
'd2(C) can be considered as either real or complex Lie algebra, so one can talk about its real or complex (holomorphic) representations. On the Lie-group level this corresponds to an additional holomorphic (algebraic) structure on G = SL,, which turns it into a complex (algebraic) manifold. In this context one can talk about the usual class of continuous representations T, or impose stronger assumptions, like holomorphic (algebraic) T, i.e. representations with holomorphic (algebraic) matrix entries: i(z) = (T%
54.2. Irreducible representations of Su(2).
168
commutation rehtion (2.10), iP(Vj)
So
P
= (Ph^+[h^;P])(uj) = (Aj-2)9(uj).
mapa E(A,j) into E(Aj-2),
and in a similar fashion, ?:E(Aj)+E(Aj+2).
shows that the eigenvalue spectrum of
h^
This
is invariant under the shift A+A f 2. To get
relation (2.11) we apply the commutator
[?;?I
to
up Next we pick the highest
A .
eigenvector
uo
and
form
a
sequence
{uj=YJ(uo)EE(&,-2j).
The
span
Yo = Sp{uo;...uj...} is invariant under all three generators {h; X ; Y}, as -
8(uj)
.
A
-
= j[&,-2(j-1)]uj.
Hence (irreducibility) To = Y. So spectrum of
h^
(2.12)
is {A,;
&,--2;...},
eigenspaces are 1-dimensional. In order for a sequence of coefficients
as claimed, and all aj
= j[Ao-2(j-1)] in
(2.12) to vanish after finitely many steps A, must be an integer. The symmetric form of sequence {A,;
(A,
- 2); ...; - A,}
results from an involution on
(5,
that interchanges X
and Y.
2.2. Matrix realization. We shall apply Lemma 2 to show that any irreducible representation x of 4 2 ) (respectively 4)is equivalent to xm of (2.1). The goal is to construct a basis in the representation space V ( x ) ,where the generators {H,V;W}, or { h ; X ; Y }could be brought to the form (2.8). We start with the highest eigenvector vo of operator g, according to Lemma 2, and iterate it by lowering operator 9, to get a sequence of eigenvectors v j = YJ(v,). Here all coefficients b j = 1, and { a j } can be calculated by (2.11), A
U1. = X , + . . . + X ~ - ~
.
=j(pm-j+l).
Thus, in the basis {vj = pj(vo)}, operators h ; X; Y are represented by matrices -
r
A
-
.,-
A
Renormalizing the basis: vj+ J M u j , we can bring x into the symmetric form: P = 2*, i.e. 0
1
In this form one easily recognizes the symmetrized form of generators p = ?d - 9 and @ = i(2 P) of 4 2 ) derived in (2.8) (problem 6). So representation x am of
+
-
34.2. Irreducible representations of SU(2).
169
(2.5), and we complete the proof of the Theorem. Remark 1) The construction of irreducible 42)-representations {r"} in (2.5) gave a t
once all finite-dimensional representations of &@).
Both series were linked via
complex/analytic continuation through their joint complex hull 4 ( C ) . Next we would like to determine all irreducible finite-D representations of 4(C), considered as real Lie algebra, respectively representations of the real Lie group G = SL,(C). Both (group and
algebra)
have an involution
g-7
(complex conjugation),
and
two series of
representations, arising from 4 2 ) : holomorphic { r r } and anliholomorphic { a
= r;}.
It turns out that all other representations are obtained by tensor/Kronecker products of two series. Namely, a;m=7r;c33g";
9EG;
and r;.m
= r;@ I + I c3 a:
for Lie algebra elements z (problem 7), a consequence of general properties
of
complexifications and representations of direct products (Theorem 4 of $1.3). Representations
T"""
of SL,(C) could be realized in polynomials of holomorphic and
antiholomorphic variables: F ( z , w ) = z a k j z k 6 j j ,by an extension of (2.5),
>;
-
( m y ?" F ) ( z ;w) = (cz+d)Zm(cG +d)Z"F(%
where matrix g =
E SL,(C). All representations
{P; 3";
2) The infinitesimal construction of irreducible representations eigendata of
(2.14) of SL, are non-
7rm
in terms of the
i,for a specially chosen (Cartan) basis ( h ; X ; Y ) of 4, gives the simplest
example of the highest weight construction of irreducible representations of simple and semisimple Lie algebras, to be studied in the next chapter.
84.2. Irreducible representations of SU(2).
170
Problems and Exercises: 1. Verify the commutation relation of 4 2 ) for differential operators { r H ;rv;
rw} in
(2.7). 2. Check: if a diagonal matrix A = diag(al; ...;a,) commutes with an upper/lower diagonal matrix B with nonzero8 entries bl;...;bn-l above/below the diagonal, then A is scalar.
4
3. Show that for any representation T of the operator T is diagonalizable (Hint: a representation T~ of a Lie group G is unitary iff infinitesimal generators { T X : X € 8 ) are skew symmetric. Any representation of a compact group is unitarizable!).
31
4. If a pair of matrices ( A ; X )satisfies the commutation relation [ A ; X ]= OX,for o # 0, then X is nilpotent, i.e. Xm = 0 for some m (Study the eigenspaces and root-subspaces
of X ) .
5. Apply problem 4 to show that SL, has no nontrivial unitary representations in finite dimension. 6. Conjugate matrices (2.8) with Q = diag [(2,,,- j)!j!]-'/'...}, to bring them to a skewsymmetric form obtained from (2.13).
(..
7.
i)
Given
a
complex
Lie
algebra
8, show
that
its
complexification
6 = 8 @ C = { X + i Y ; X , Y E 8 )is isomorphic to 0 @ 8 (direct sum of Lie algebras). ii) Complex conjugation g+
in
(s
induces a conjugation of representations:
T-rT, = T7 ; which takes holomorphic representations into antiholomorphic ii) Conclude (via Theorem 4 of §1.3), that any irreducible representation T of 8, as a real Lie algebra, is equivalent to a tensor/Kronecker product of holomorphic and antiholomorphic irreducible representations: T 1:r @F
.
54.3. Matrix entries: Legendre and lacobi polynomials.
171
$4.3. Matrix entries and characten of irreducible representations: Legendre and J a d pdynombk In this section we shall compute matrix entries and characters of irreducible representations { P }of 4 2 ) , constructed in the previous section and establish their connection to classical orthogonal polynomials: Legendre and Jacobi.
We shall find matrix entries: rrn = ( T r e kI en) in the natural basis of 9,
Notice that the inner product in 3 , can be expressed by means of the differential operations: (rn+k)!am-', associated to each monomial {P-'}, where a = -.d dz
In other words each polynomial f = ~ u k z m tisk assigned a differential operator
D f = f(a)=
c
(rn--L)!akamSk.
The inner product (4.7) in 9, is then given by rn
rn
-rn
-rn
(f I g ) = Df[g]L=o;where f = Curnzmtk, g = Cbpmtk Using this new form of the inner product and formula (3.1) we can write the knth matrix entry of am as
- b)m-k(-cz
rrn(u)= /,?am-rn-n)!( '(dz rn-k)!(m+k)!
[
+ a),+'
for any u = :]E SL,(C), or 9 4 2 ) . Changing variable, z - e = c(dz - b), and derivative az+d, = we can rewrite (3.2) as
$az,
Next we shall recast (3.3) in terms of Eder angles on SU(2), introduced in 54.1,
(3.4) = i(4-4)12. Remembering that all where the entries a = cosZe e i(4++)/2; are eigenvectors of diagonal elements { e z p t H } , that appear in the right and {ek = J ... left factors of (3.4), rz"zptH(ek)
we get
=
ikt
e&;
84.3. Matrix entries: Legendre and Jacobi polynomials.
172
the RHS to be evaluated at z = sin2B/2. Making another substitution: 2-2 = c o d (the new z bears no relation to the original variable of Tm),and the substitution 1-2 := l+z 2 = cos2e/2;
we bring
7rpn to the form
= 1-Z 2 = sin26/2 ; and
a, = -28,;
Tg(d,e,$) = e i ( t $ + n 4 ) kn(COS ~ m e), with a single variable function
We shall express functions P c in terms of the classical Jacobi polynomials [Erd], [Leb]. There are many different ways to introduce Jacobi polynomials, one of them is through the Rodriguez formula:
The Jacobi polynomials are orthogonal on [-1;1], relative to the weighted L2product:
(3.7) with weight w = (l.-z)O(l+z)P. In fact, orthogonalization of the natural basis {l;z;z2;...zm;...} vie product (3.7) provides an alternative definition of { P ( a ' p ) } . Furthermore, Jacobi polynomials are known to satisfy a second order differential equation: L[y] Amy = 0, y = P k j P ) ( z ) ,where
+
L = a(].- zz)a +
t (a+p)z~a,and ,A = m(m+a+p+i).
(3.8)
In other words, { Pk,P)}m represent eigenfunctions of the Jacobi differential operator L (3.8) with eigenvalues {Am}. This gives yet another characterization of Jacobi polynomials. Returning to functions {Pz}that appear in formulae (3.5)-(3.6) for matrix entries rpn we find them to be n-k
with coefficients
C = C(n;rn;k)= 2-
( n + k ) $ F
m+t)!(m-t)!'
Thus we get a representation of matrix entries of {P} in Euler angles as products of exponentials, trigonometric terms and Jacobi polynomials,
$4.3. Matrix entries: Legendre and Jacobi polynomials.
173
A special class of Jacobi polynomials called Legendre are given by
They correspond to special values
a!
= /3 = 0 in formula (3.6). So Legendre
polynomials are orthogonal on [-1; 11 relative to constant weight ~ ( z =) 1, 1
(f I 9 ) = and satisfy the differential equation:
jf(s)g(z)dz; -l
L[P,] t m ( m t l ) P , = 0, where L = a ( 1 - z2)a, in other words {Pm}are eigenfunctions of the Legendre differential operator L. Let us remark that many other classes of orthogonal polynomials on 1-1; 11 ( Tchebyahev; Gegenbauer; etc.) are also special cases of Jacobi, corresponding to certain values of a;@. Legendre polynomials give a special set of matrix entries,
rG($,O,$)= Const P,(cosO), called spherical functions on SU(2). More general entries of the form TO", are given by the so called associated Legendre functions:
- 2)-n/2{(1 - Z2)m}(m-n). P&(,) = -(I zmm! Namely, = ,/(m+n)!(n-m)! 1
Gn($,',+)
Family
~nm(cosO)eind.
{PL}, also forms an orthogonal system of eigenfunctions in L2[-1; 11 for
the so called u-th associated Legendre operator: 2
L, = a( 1 - z2)a- 1-22' with eigenvalues {Am= m(m+l): m 2 u}.
(3.9)
We shall examine them closely in the next section.
Problems and Exercises: relations for Jacobi polynomials and find the norming constants 2. Show that the Jacobi differential operator L is symmetric w.1. to the product (3.7):
(LfI d = (f I Lg). 3. Use orthogonality of Jacobi polynomials and the fact that the Haar measure on SU(2), du = sinf?d6d4d& in the Euler-angle coordinates, to show that matrix entries are orthogonal. Find their norms 11 rkm, 11
{rrn}
L (G)'
$4.4. Angular momentum and Spherical harmonics
174
$4.4. Representations of SO(3): Angukr momentum and spherical harmonics.
Irreducible representations of the orthogonal group Sq32 will be realized in spaces of spherical harmonics 36, C L (S'). Subspaces (36,) can be characterized in different ways: harmonic polynomials of degree rn on @, eigenspaces of spherical Laplacian A, on S2, irreducible subspaces of Sq3). We Nhall give several proofs of the main decomposition result. One of them exploits important concepts of Weyl algebra and dual pairs (due to R. Howe). It finds an interesting link between irreducible representations of Sq3) and thoee of SL,. Another (computational) argument provides a grouptheoretic base for some well-known recurrence relations of associated Legendre functions. In conclusion we apply the analysis of the regular representation on Sz to the spherical Radon transform.
In the preceding sections we constructed irreducible representations { am} of SU(2), labeled by the spin parameter m = 0;;;l;p; ... (integers and half-integers), which at the same time comprised all complex (holomorphic) irreducible representations of SL,. Only half of {a""},those of integer-valued m, factors through the representation of the orthogonal group S q 3 ) = SY2)/{ 1;-1). Indeed, operators representing the diagonal (Cartan) subgroup { m p t H } of SU(2),
become trivial at t = a (i.e. at u = -I), iff m is integer. Here we shall give another construction of representations {a"} of integral spin realized in spherical harmonics on S2.We denote by 9, the space of all homogeneous polynomials in 3 variables of degree m,
dirn9" = C "A2K = (m+2)(m+l) 2 ' and consider its subspace 36, of harmonic polynomials, i.e. solutions of the Laplace's equation,
Af = (@+ai+a;)f= 0. Group G = 5 q 3 ) acts on W3 by rotations and this action extends to polynomials,
T u f ( 4 = f ( z U ) ,u E 5q31, z E p, and x - d -the action of 5 q 3 ) on W3. There is a natural inner product on polynomials o;, 9 = 9,, which makes T unitary. It can be introduced via differentiation on UP, D= a,; Namely, to each polynomial p = t a , z q , a = (a1;a2;a3), we assign a
(al; a3).
175
i4.4.Angular momentum and Spherical harmonics constant-coefficient differential operator p(D) = x a , D a ; polynomial r' = I z 1 yields the Laplacian A. Then we set
for
instance
radial
for any pair of multi-indices a$. This product coincides with the product induced on symmetric tensor powers of p,Y"(@) E 9, by the natural p-product. Indeed, given a pair of m-tensors t = @ ...8 &,,, q = q1 8 ...8 q,; with the usual product (tI q ) = (tiI qi), and the operation of symmetrization,
n
Sym,:t-+Ym(t) =
$E€,(l)~ ...@tn(,)- s u m over all permutations
0,
the product of symmetrized tensors,
(f"(t)I Yrn(77))
=
f (tI Yrn(rl))*
--
Applying the latter formula to decomposable tensors of the form tCr=e l @... @el a q@...@q a,... whose symmetrizers
Yrn(ta) are identified with monomials zcI E,T,
we get
Theorem 4.1: (i) Space 3, is decomposed into the direct orthogonal sum Tm= 36, Q r236,-2 Q ... (44 where 36, denotes the space of harmonic polynomials of degree k. Hence the entire space 3 splits into the tensor product %@36 of the algebra of radial polynomials 3 = {f = c a k r 2 k } ,and the subspace of harmonic polynomials 36 = %Xi,. (ii) Each subspace 36,
c 3,
is S0(3)-invariant and irreducible, furthermore the
restriction of the regular representation R I 36, u T'".
From the orthogonal expansion (4.2) we easily compute the dimension of 36, dim 36,
= dim 3 , - d i m 3m-2 = 2m+l.
Theorem 4.1 gives the spectral (primary) decomposition of the regular representation R on L2(S2),and at once the spectral resolution of the Laplacian A S2'
Its first statement follows from product formula (4.1). Indeed, given any pair of functions g E 9,; r2f with f E 3m-2, the orthogonality relation (r'f I g ) = 0, for all f E 9,,,-2;
is equivalent to
176
54.4. Angular momentum and Spherical harmonics
[f(D)Ag](O)=: (f I Ag) = 0, all f E Tm-2($ Ag = 0 , i.e. g - harmonic. Therefore, space X,, an orthogonal complement of G-invariant subspaces {r2k9m-,k}k=-,;1;..., is also G-invariant. It remains to prove irreducibility of R I J6, and equivalence to 7rm. We shall give 3 arguments, based on very different but equally important ideas. One of them will exploit Weyl algebra W = W,, and so called Dual (commuting) pairs in W . As a byproduct we shall find an interesting connection to (oo-D) representations of SL,. The second argument is fairly straightforward (though somewhat tedious), it will link representations { 7r") to eigenfunctions of the 2-sphere Laplacian, spherical harmonics and Legendre functions. The third and shortest utilizes irreducible characters of 50(3),and uniqueness of the map: T*xT on compact groups, explained in chapter 3. Proof 1: The Weyl algebra W on Rn consists of all differential operators with polynomial coefficients W = {A = ca,(x)D"}. Algebra W acts naturally on space 9 of polynomial functions, as well as other function-spaces, e.g. L2(R"), and this action is irreducible (problem l), since W contains generators of all translations and multiplications on Rn, and anything commuting with translations and multiplications must be scalar. The Weyl algebra in d contains a Lie subalgebra 4 3 ) of 1-st order differential operators (vector fields), generators of rotations about z, y , z-axis,
afI
xaz;a,
xa, Yaz,
= yaz- %ay;a, = Za, = (4.3) which correspond to the Cartan basis { H ; V ; W )(1.4). Using vector (cross-product) notation we can write triple (3), as (a,;a,;a,) = 2 x v.
Vector derivative J = Z X V is called in Quantum mechanics the angular momentum operator, by analogy with the classical angular momentum x x p (the
meaning and the role of angular momentum will be explored in depth in chapter 7 (see
[BLI1. Another interesting subalgebra of W is generated by 2-nd order operators:
X = A ; h = z . V ; Y =1
~1'.
(4.4)
One can show { h ; X Y }to obey the commutation relations (2.10) of d, (problem q3.We denote by A, c W the associative algebra (hull), generated by 4 (4.4), and -A, will denote the associative hull of 4 3 ) in 9.Obviously, algebras A, and A, commute
54.4. Angular momentum and Spherical harmonics
177
in W , since all three generators (4.4) are rotation invariant. Furthermore, using Weyl's "orthogonal polynomial invariants", one can show (problem 3) that 4, A, form a mazimal commuting pair in W , called dual pair, according to R. Howe [How]. We shall use the following general result. Proposition: If an irreducible algebra of operators W in space V has a mazimal (dual) pair (&;A,), then space Y is decomposed into the direct sum of temorproducts V N gTmB%m,
(4.5) Moreover, all are pair-wise different.
so that A, acts irreducibly on each T,.,,, while A, on each 36.,
representations {AoI V,},
respectively {A, I % ,},
The argument was essentially outlined in Theorem 4 of 51.3 (irreducible representations of product-groups G = H XK ) . As a consequence, we get a 1-1 correspondence between irreducible representations of A, and A, that appear in the decomposition of W . In our case (V = 9 or L'), the decomposition takes the form:
where % denotes the radial functions { f ( r ) } (polynomial or L2), and X, - harmonic polynomials of degree m. Both algebras B=so(3) and Q = s l , respect this decomposition, B:36,-.Xm, and $:%-+%. Hence, due to the uniqueness of components (4.5), the resulting pairs of representations: a = R 1 J6, of So(3); and Tm of SL,(R) on k must be irreducible, QED. Remark: The "dual pair" approach yields more than was claimed in Theorem 4.1, not only we are able to realize irreducibles of So(3) in spaces of harmonic polynomials 36, but as a byproduct we obtained a series of irreducible representations of SL(Z;R), realized in various radial components of the decomposition 9 = @ GJb 8 36,. m
These are
80
called
discrete s e r i e s representations, studied in chapter 7.
Proof 2. Harmonic polynomials {f(x)} c J6, can be restricted on the unit S2 c W3, due to homogeneity. The resulting functions {Y(x)= f I Sz: f E X,}, are called spherical harmonics of degree m . It turns out that spaces 36,(Sz) form a complete orthogonal system in L2(S2)with the natural (invariant) L2-product, 00 LZ = 63 36,. 0
'The same clearly holds in any space R", n 2 2. In fact, lurking behind the scene there sits a much larger symplectic algebra d n ) , and its natural 'oscillator" representation in Lz(R") (see chapter 6).
178
$4.4. Angular momentum and Sphericdl harmonics
In fact, each 36, will be shown to coincide with an eigenspace of the spherical Laplacian As. We shall find a basis in spaces of spherical harmonics X, and then compare two actions of Lie algebra generators { H , V , W }in R 136, and in am. The W3-Laplacian A in polar coordinates ( r ;4; 6 ) takes the form
A = a; + gr+ j d S ; where As denotes its spherical part,
As =
ap + cotes* t #in 6 +-a2
4' Given a homogeneous function of degree rn, restricted on the sphere, f(z) = 1 z 1 mY(z'), z1 = ./ 15 1 , the Laplacian of f , A(f ) = ~"'-~[m(m+l) As]Y, is reduced to the spherical Laplacian of Y . This shows that %,-spherical harmonics on S2 are eigenfunctions of the A,: ASY m(m+l)Y = 0, (4.7) of eigenvalue A = rn(rn+l). The standard separation of variables 6,4 in (4.6)-(4.7), Y = F(b)G(B), yields a pair of ODE'S for functions G(6) and F ( 4 )
+
+
F"
+ k2F = 0 (periodic in 4) => F = e G"+-&G'+
{m(m+l)-*}G
srn 0
ik4
= 0.
(44
Changing variable: z = c o d , equation (4.8) is transformed into the k-th associated Legendre equation (3.9)
Lk[G]= - a(1-z2)aG ++G (1-2 1 = m(m+l)G.
(4.9)
Solutions of (9) is the k-th associated Legendre function (of degree m) G = Pk; the m-th eigenfunction of the associated Legendre operator L,. Functions {Yk(6,4)= eik4Ph(cos solve the eigenvalue problem (4.7), and are easily seen to form a orthogonal basis in the the m-th eigensubspace 8, of As. One can directly 8 = 36, I S2 (problem 4). The one-parameter group of azimuthal rotations verify that , { e z p d H } (stabilizer of the north pole) is diagonalized in the basis { Y h } , the eigencharacters being {... eik4...}, as in (2.9). It remains to show that two other basic elements of 4 3 ) are represented by suitable tridiagonal matrices (2.9).
6)}r!-,
A straightforward, though somewhat tedious way, is to compute 3 generators { H ; V ; W } of 4 3 ) (angular momentum) in polar coordinates on the 2-sphere. In d they are given by
54.4. Angular momentum and Spherical harmonics J, =
- %aY ; J Y = Z a , - za,;
179
J , = Za, - pa,,
Passing to polar coordinates we find (problem 5),
J = z x V , = r u x ( ...)
4;
where (u;u; w) are unit orthogonal vectors: sin0 COSQ
(4.10)
columns of the Jacobian matrix A =
J on the sphere, J , = -sin4 8,
(fig.1). Whence we get three components of a(r;~ 4 )
- cote cos4 ad; J ,
= COSQ 8,
- cote sin4 a4; J,= a4.
(4.1 1)
Fig. 1. The orthogonal unit frame {u;u;w} on the sphere.
Next we apply momentum operators (11) to the basic spherical harmonics
{YX = eik4pk(coss)}p=-,,
of 36,.
The generator of azimuthal rotations about the z-axis, J , is diagonalized in the basis
{ Y h } , J , [ Y 3 = i k Y k . To compute J , ; J ,
we change 0 to variable z = cos6,
8, = & as in (4.8), and introduce operators Q+,.Q- on Sz, sane 0,
rn 4. The z and y-components of J are expressed in terms of Q * Q* = d i 2 a z F L a
J, = %(ei4Q+ 1 - e-i4Q-);
Operators Q
J v = i(ei4Q+ 2
(4.12)
+ e-'+Q-).
* are simply related to the raising and lowering generators { X- ;Y- } of d2
(2.10). Indeed, A
A
X = J , + i J , = ei4Q+; Y = J 2 - i J
1
Now we can easily evaluate J,;J, on spherical harmonics,
= e- id&-.
(4.13)
64.4. Annular momentum and Spherical harmonics
180
and similarly for J,
J,[YkJ = p+')4Qt[pkJ - p - ' ) 4 Q - k [pkA* The new (reduced) raising/lowering operators
Q$ are obtained by restricting Q*of
(4.12) to functions of the form {eik4f(z)}. Now it remains to apply the well-known recurrence relations for associated Legendre functions (problem 6),
Q l ( P i ) = a&+';
and Q i ( P k ) = bkPL-';
(4.15)
with coefficients { a k ; b k } , depending on a particular normalization of { P k } to bring J, and J, (or
?;p (4.13)), to the familiar form of tridiagonal (raising/lowering) operators
(2.9)-(2.13) in basis { Y i } .
Summary: The argument of part 2 provided more than mere realization of
in spherical harmonics. Among other results we established irreducibles { rm} 0
A,
solution of the eigenvalue problem for the spherical Laplacian As: eigenvalues
= m(m+1); rn = 0;1;...; eigenspaces are spherical harmonics 3 6,.
we showed that the basis { Y h = eik4Pk(cos8)}gives a solution of the joint eigenvalue problem for operators As and J,: 0
A Y," -m(mtl)Yk J,[Yh]= i k Y h 0 the azimuthal angular momentum J , is shown to decompose L2(S2)into the m direct sum of eigenspaces: @ Sk;each Sk = { F = eik4f(cos8)}N L2[-l; 11. The Laplacian restricted on $k becomes the k-th associated Legendre operator, As I Sk N L,. So the km - th spherical harmonics factors into the product of the k-th harmonics in and the
{
--bo
km-th associated Legendre function P h (the eigenfunction of
Lk).
the raising/lowering operators r?;? given by (4.13) take Y h into Yk* l . Thus one gets the recurrence relation (4.16) for associated Legendre functions: 0
ph-1
= ( r a - ; i - )... ph;
(4.17)
as well as the formula relating the k-th associated Legendre function to the Legendre polynomial P , (problem 6), P h ( 2 ) = ( l - 2 2 ) q l - *z)m}(m+k).
(4.18)
Proof 3: The shortest proof of Theorem 4.1, however, comes from the general representation theory of chapter 3. To prove equivalence of rmto R I Xm, it is enough
181
$4.4. Angular momentum and Spherical harmonics
to compare the characters of two representations: T and S are equivalent, iff their characters are equal, xT = xs. Since characters { ~ ( u ) are } constant on conjugacy classes of u E G , it suffices to compute them on class-representatives. In case of SU(2) each u is conjugate (equivalent) to a diagonal matrix,
where {e
* "I2}- eigenvalues of u. Similarly each u
E
Sq3)is conjugate to -
-
e4H =
In both ag
N
I'
-sin#
C O S ~
cases operators 7rZpdH are digonalized in the natural basis,
diag{eim4; ei(m-l)'* 1
"'1 *
e-im4}, and we readily find
(4.19)
Since characters of
{P} on diagonal elements coincide with those of { R I X,},
it
follows that the representations are equivalent, QED.
Remark Irreducible characters
{xm = tra,}
are real and orthogonal with respect
to the reduced Haar measure (problem 7 ) on conjugacy classes, ( x m I xk) =
I
d6 = 6mk-
xm(d) xk(6)
Using this relation one can decompose the product of any two characters Im t k I XrnXk=
C
xj*
I The product of characters gives the character of the tensor (Kronecker) product of two representations, xT s = xTxs, and similar relation holds for direct sums: xT = xT x s . As a consequence we get the Clebsch-Gordon (decomposition) formula for tensor products of irreducible Sy2) - representations, j= I m-k
+
Formula (4.19) is a special case of the Weyl character-formula, that will be studied in the next chapter.
182
54.4. Angular momentum and Spherical harmonics Application to the Radon t r d o r m on the sphue. The Radon transform is defined by integrating function f(z)on S2 along all great circles (geodesics) y (see §2.5),
=&
A
%:f-f(y)
f ,<
fds.
Each circle 7 can be identified with its north pole
= <(7), 80 one can think of 3,as
transforming functions {f(z)} into functions {?(<)I commutes with the regular representation R of
on
s'.
Clearly, operator I
Sq3) on S2, 80 by Theorem 4.1 it must
be scalar on each subspace 36, of spherical harmonics, 9 1 3 6 , = c,Z.
We could also say
that Imust be a "function of the Laplacian A on S'". Instead of A it is often convenient {m
to
L=
use operator
44-f,
whose eigenvalues are
= 0; 1; ...}, L 136, = m. Then we can write '% = F ( A ) , where F(m) = c.,
function F (or coefficients
integers To find
it suffices to evaluate '% of a special set of spherical
{c,})
harmonics. We shall chose the zonal ones, Y o = Yo,(@)= P,(cosO). The %image of such Y o is another zonal harmonics, cmY0, whence we find the coefficient c, PO( 0)
=-
c,
YO(0)'
It remains to compute the numerator and the denominator of (20). The former is obtained by integrating Yo along any meridian-circle of S2. In the standard normalization of the Legendre polynomials, Pk=+-[(l-Z
2
)T(k),
(21)
2 k!
we get for even k = 2m (% vanishes on all odd functions!), numerator = &f/ods (see [Erd], ch.lO), while
= ;]p2,(cose)de
= J-(*~)z, 24rn m
0
denominator
(-1)"(2,") = P,,(O) = 22m
Hence follows the ratio, ~ 2 , z=
+ (-1)'"
2rn rn ).
The latter can be expressed in terms of the ratio of r-functions, namely,
F(z)= evaluated a t all integers. So
The standard Stirling formula for
r(Zt1)
z z r ( t / z+I ) ~ '
r(z),yields the following asymptotics of F ( z ) , F ( z ) $&, as z+m.
-
So one could say that the Radon transform is equivalent to a fractional power %
-
1
(-A)T,restricted on "even functions".
s4.4. Angular momentum and Spherical harmonics
183
Problems and Exercises: 1. If operator Q commutes with all multiplications qkf(z)+(z)f(z), and translation on R”, then Q must be scalar. Use it to show that Weyl algebra W , acts irreducibly on Lz(R”) (or polynomials 9). 2. Verify the commutation relations (2.10) for operators {h; X;Y} of (4.4). 3. Orth onal invarianta: group G = S q 3 ) acts on R3, and on its “cotangent space” R,=R eR3,
T
(4.22) u: (z;€Hu(z); 40); 2; € E R3. This action results from a general “cotangent map” defined by a linear transformation 9 E GL, (4.23)
9: ( z ; € ) ~ ( g ( z ) ; = g - * ( ~ ) ) .
Both actions on R3 and 96
ZZ
R6
extend to polynomial functions: 9,= {f(z)} and
if(’; <)I.
i) Show that algebra A, of S q 3 ) -invariants in 9, consists of radial functions {f = C a j r z i } , i.e. A, is generated by a single invariant: r2 = I z 12. ii) Algebra A, of G-invariants in 9, is generated by 3 elements: {I=?; special case of Weyl’s Theorem on invariants [We]).
2.t;
I
iii) Elements {A = Caapzoa@} of the Weyl algebra W, can be identified with polynomials {f(z;<)= Caopza€P},called symbols of differential operators A. Check that the action (4.22)-(4.23) of GL, and S q 3 ) on 9, and W , coincide. Conclude that the only S q 3 ) -invariants of W, are generated by operators (4.4). 4. Check that {f = r m Y i ( g ; 8 ) }are harmonic polynomials in R3, so
M, I Sz = g, - the
m-th eigenspace of A,.
5. Find the gradient V, and the angular momentum operator Z X Vin polar coordinates z = r sinOsin$; y = r sin0 C O S ~ ;z = r cos8, of R3. Compute the Jacobian matrix: A = a(z, Y,2,) = (yrv; rsin0 w), a(r, 0,d) where {u;u;w} are three unit orthogonal vectors (4.10) obtained from columns of A; invert A using orthogonality, and show that in polar coordinates,
-
6. Aasoeiated Legendre functions:
i) Derive the recurrence relations (4.17) for associated Leg5ndre functions (Use raising/lowering operators Q (4.14) related to 4 generators X ; Y , via (4.13)). ii) Apply recurrence relations (4.17) to derive formula (4.18) for associated Legendre functions. Show that operator R, = (1 - z2)k/zak = pkaE, that takes P , into P i , factors into the product of 1st order operators:
+
Rk = (pa (E-i)pt)(pa
+ ( ~ - 2 ) ~ ~ ) . . . (+~pai ) pa;
=6
2;
and the j-th factor coincides with the reduced raising operafor 0; of (4.14). Establish the intertwining relation: LkRk = RkL,; i.e. Rk intertwines the 0-th and the
k - th (associated) Legendre operators, and complete the argument.
7. Derive the reduced Haar measure on conjugacy classes of s y 2 ) : dp(q5) = sin2($)dd.
54.5. Laplacian on the n-sphere
184
v.5.Laplauan on the n-sphere. Spherical Laplacian A, on S"-' is invariant under orthogonal rotations {u E Sqnj}, its eigenspaces consist of spherical harmonics 36, = { Y : of degree k}, the eigenvdues being: A, = k(k + n - 2) = (k + In this section we shall exploit the spectral theory of A,, to find the Green's functions (fundamental solutions) of the heat and wave problems on the sphere, and establish interesting relations with special functions (Gegenbauer-Legendre), and the Poisson kernel in the unit ball B C R".
y)2(y)2.
5.1. Harmonic polynomials and decomposition of Lz(S"-'). The Laplacian A in polar coordinates {qe} of W" has the form,
A = 8;
+ *af
-tr-2AS, (5.1) where As denotes t,he spherical Laplacian on S"-'. We shall introduce subspaces of spherical harmonics in L'(S"-l)}, by taking homogeneous harmonic polynomials of degree k on W",
Mk = {f = and restricting them on S"-'.
a,za:Af = 0},
lal=k
Spectral decomposition of As is analogous to the 2-D
case of the previous section (Theorem 4.1). Namely,
Theorem 5.1: (i) Space L2(Sn-') is decomposed into the direct orthogonal sum of spherical harmonics
{ME};
(ii) {Xik} are eigenspaces of the Laplacian As, of eigenvalues Xk = k(k+n-2), and
also irreducible subspaces of the regular representation R,f = f(zU),of group G = S q n ) on L2(S"-'). The eigenvalues are easily obtained from the polar form (5.1). Indeed, for any harmonic function f , homogeneous of degree k (which restricts to a spherical harmonics Y = f I S), we have
0 = A [ f ] h= {k(k-1)
+ (n-l)k+
As}[Y],
hence,
A s [ Y]= - k(k + n-2)Y; on Y E 36,. The irreducibility part will follow from the general results of chapter 5 (JJ5.3; 5.7). More elementary argument exploits a characterization of {K,} as eigenspaces of As, and the fact, that each X, contains a unique Sqn-ltinvariant
(mi-symmetric function) Yo.
Indeed, such Y o becomes a function of a single angle 8, between rotation. Decomposing As in spherical angles {O; ...} on Sn-',
t
E S"-', and the axis of
185
64.5. Laplacian on the n-sphere A , = a:
+ (n-2)cotOae + +A', sin 0
where A' is an Sn-2 ('equatorial") Laplacian, we see that Y o solves an ODE, Ye0
+ (n-2)cotOYe + k(k+n-2)Y
= 0.
(5.2)
Changing variable, 0-2 = cos0, we bring (5.2) to the standard (Gegenbauer) form, and get a unique solution,
Yo(@) = Cr(cos0), - a Gegenbauer polynomial of degree k, and order rn =
9.
Once all {Xk}are shown to posseas a unique K-invariant (K = S q n - l ) ) , they must be irreducible under the regular action R of G = S q n ) on L2(S), QED.
5.2. The heat and wave kernels on SR-'. Next shall study the wave and heat-
kernels on
s"-'. Our main tool will be the Dirichlet problem: Au = 0;u I s = f ; in the
unit ball B
c R".
where
T
In chapter 2 we derived the Poisson kernel of the ball,
= I z I ; 0 - angle between z and y, and constant
C, = r("'2) Kernel P can be 2*"/2 .
viewed as a formal solution of an operator-valued ODE, U"
-I-q u '
1 + -A[u] = 0;u I r2
= f.
(5.4)
Here A denotes the Laplacian As. Indeed, solving formally (5.4) we get u = ,.(A)[ f], where exponential s(A)is found from the characteristic equation of (5.5),
(5.5)
(5.6)
s(s-l)t(n-l)stA=O,
We shall drop the negative root, since u is required to be regular at {0}, and denote square-root term by L,
L=
{W,
(5.7)
a positive self-adjoint operator with spectral resolution,
To define (5.8) we take eigenvalues { A , = k(k+n-2)) of As, call the corresponding spectral projections Ek:L2(S)+36k (spherical harmonics of degree k ) , and set eigenvalue /Jk=
{m+ 9, =k
in accord with (5.6) and the general notion of operator "+(A)"(Appendix A, and 52.3).
54.5. Laplacian on t h e n-sphere
186
In other words Poisson kernel P(r;w;w'),represents a "function of operator L"
(5.9), +(L),where +(A) =
- 9).
From the Poisson kernel {Pp) (5.3) we c m easily pass to a semigroup of L, { e - t L } . Indeed,
p = Cnr1-n/2
(1/p-r)
-2cosfp
[T l+t2
of (5.9) and dropping r'-"i2 in both, we get the
Comparing it with operator integral kernel of r L ,
Now a substitution r = e-t, brings it into the semigroup-form, = C,
e-"
sinh 1
(5.10)
From (5.10) we also get a Neumann semigroup-kernel,
(5.11)
(5.10)-(5.11) can be viewed as the Dirichlet/Neumann Poisson kernels on the semi-infinite cylinder { ( w ; t ) }= Sn-' x [O;oo). Let
us
remark
that
formulae
Next we shall analytically continue them in the complex range of parameter t, sin( t L)
Ret 2 0 , to get wave-propagators { c o s ( t L ) ; T } .As in $2.3 we formally substitute (it - E ) in both kernels and let €40.So L - 1e i t L = lim CN-I
ei" = lim
€-+o
c,' [2(cosh(it - C) - cose)l( n - 2 ) / 2
(5.12)
C,sint [2(cosh(it - C)
-C O S O ) ] ~ / ~ '
and cos- sin-propa.gators are the real and imaginary parts of corresponding unitary
s4.5. Laplacian on the n-sphere
187
groups. In the special case of S2, it gives, L - 1 ei t L =
1 2rJ2(coa t - case)'
hence
(5.13)
*
which then extends periodically in time t with period 2n. Formula (5.13) gives an S2-version of the 2-D Euclidian propagator (chapter 2),
It obeys, as the former, the finite-propagation-speed principle, but due to timeperiodicity4, the out-spreading wave-front reassembles after t = 2a at s single source, and the whole pattern repeats again. In higher even dimensions ([Ta2], chapter 4) one finds the propagators to be
where M e f ( x ) denotes the mean value of f on the geodesic sphere
, centered
at x of
radius 6, S&x) = {y: x. y = c o d }
M B f ( 4=
f(y)dS(y).
sg(z)
The even-D formulae clearly demonstrate the strict Huygens principle: at each time t, distributional kernels of L-'sin(tL), and cos(tL) are supported on the geodesic sphere St = ( 2 - y = c o s t } , rather than the ball {x.y
1 cost}.
The heat-propagator can be constructed by Fourier-transforming the wave-group (5.12), or cos-propagator (5.14),
Using Pa-periodicity of c o s ( t i ) and the Poisson summation formula of 52.1 for the 1-D Gaussian, 4Notice that spectrum of L, {Ak(L)= k + q } , consists of integers for even n's, and halfintegers for odd n's. So a unitary group of L, {eitL} is either 2 r or 4r-periodic.
44.5. Laplacian on the n-sphere
188
(*-U)2
G(z;t)= xezp(.-$m2+imz) = (27rt)-'xexp( -7) = @ ( z ; t ) -theta-function,
v
we get the sprerical Gaussian for odd-dimensional ,--id -
1,
Sn-',
a
2r am0 a8
Remark One can derive similar formulae for hyperbolic Laplacians on spaces Wn by analytically extending metric tensor (see [Ta2], chapter 4). 5.3. Spherical harmonics. We described eigenspaces (36,)
of L as degree-k harmonic polynomials on S"-l. To describe them explicitly we can choose a single element Y oE 36, and apply all rotations { u E S q n ) } to it. The simplest choice is
Y,= ( C u j z j ) ' =
- a linear form on
W", with complex coefficients { u j } , raised to
the k-th power. One easily verifies,
A[Ya]= k(k-l)(a-a)(a.z)k-2, hence Y , is harmonic, iff a - a = X u ; = 0. Since S q n ) acts irreducibly on 36, (as will be explained in §5.7), the latter coincides with the linear span of all such Y's5, J6, = Span{Y,: a - a = 0).
Another possible choice is an axi-symmetric (zonal) function Y o= Pk(cos8), i.e. an Sqn-1)-invariant Yo. Since regular representation R on Sn-' is induced by the stabilizer subgroup K = S q n - l), we have shown via the Frobenius reciprocity, that each
c R has a unique K-invariant. Function Y(8) = Pk(cos8) satisfies a "radially
reduced" Laplace's condition,
Y" -+ (n-2)cotB Y' + k(k + n-2)Y = 0, or in terms of variable z = cos8, Y = P ( z ) solves, (1-z2)-m$(l-z2)m~~ The
solutions
are
well
+ k ( l ~t n - 2 ) ~= 0, m = n-2. 2
known
{ P , = C p ( z ) : k= 0; I; ...} of order m =
Gegenbauer
(ultraspherical)
9(see [Erd, ~01.21).
polynomials
An important, feature of Yo = Pk(cos8) is described by the following Lemma. Lemma 2: Integral kernel E,(z; y) = Ph(cos8), c o d = x y, properly nomnulized, defines an orthogonal projection Ek: L2-+Jb,. Indeed, any G-invariant (integral) operator K ( z ;y) on S"-', a rank-one symmetric space (see $5.7), must be of the form K(2.y)(problem l), i.e. K must depend only on the 5Later (chapter 5 ) we shall see that {Ya}defines the so called highest weight-vector of an irreducible representation irk of Sqn)on h,.
54.5. Laplacian on the n-sphere
189
geodesic distance between z and y : d ( z ; y ) = c o s - l ( z . y ) . So Ek commutes with the Gaction on L2. It clearly takes L2 into 36,, u(z)=
P k ( z . y ) f ( y ) d S ( y )=+-A[u] = jA,[Pk( ...)I f d S = X p .
IS
Hence, by irreducibility of xk, Ek136, = Const, and the constant is found from the normalizing condition, Const = Ek[PJ(l) =IIPk11' = 1.
Now we can compare the Neumann semigroup, L - l e - t L ,
in two different
representations: in terms of the integral kernel (5.11), and through the spectral decomposition via projections {Ek},
where { pk = k
+ F}are eigenvalues of L. Returning to the original variable r = e - t ,
we get the expansion of the conjugate (Neumann) Poisson kernel of the sphere,
(5.15)
by dividing them with p k . Relation Here we renormalized polynomials {Pk}, (5.15) gives a generating junction6 for the Gegenbauer polynomials { C r } (rn =
9).
The latter in turn yields the Cauchy-integral formulae for {Pk= C r } , p k ( t )= 2*i 1 /(l-21C 1 c2),,,cL+' dC - contour integral in C. +
Y
In
special
Pk(t) = -[(l 1 2kk!
case
n =3
we
get
the
familiar
Legendre
polynomials
- t 2 ) k ] ( kwhose ), generating function,
Further results, details and references could be found in the book [Ta2] by M.Taylor, whose approach we closely followed, and [He12].
'A generating function of an orthogonal family { d l r ( z ) }is defined aa a function F(r;z), that admits a Taylor/Laurent expansion in variable r, with coefficients {dk}, F ( r ; r ) = E r k d k ( r ) . All classical orthogonal polynomials are known t o have generating functions ([BE]), in particular Gegenbauer {C"} are generated by
190
54.5. Laplacian on the n-sphere
Problems and Exercises: 1. Show that the commutator of the regular representation R of S q n ) on S"-' consists of integral operators K ( z ;y) = F ( z .y), i.e. kernel K depends only on the geodesic distance e = cos - ' ( 2 . y), between z and y. Hint: the n-sphere is a 2-point symmetric space, i.e. any pair (z;y) is taken into any other equidistant pair ( 2 ' ; ~ ' ) by an orthogonal u E S q n ) (compare to problem 4 of 91.3).
Chapter 5. Classical compact Lie groups and algebras. In this chapter we shall develop the structure and the representation theory of classical compact Lie groups and associated complez and real Lie algebras. There are four types of classical simple compact Lie groups and algebras: unitary, orthogonal (odd and even) and symplectic. Together with three ezceptional algebras they provide a complete list of all simple Lie algebras, due to the celebrated Cartan classification Theorem. We shall start this chapter with a brief introduction to the basic structural theory of simple and semisimple Lie algebras: Cartan subalgebra, root system, Weyl group. These results will lay ground for a subsequent study of representations of semisimple algebras. We shall give several constructions of irreducible representations, including highest weight realization and the Weyl's theory of fensor powers and Young symmetriters. Finally, we proceed to the heart of the classical compact Lie group theory: the celebrated Weyl character formulae. The last section applies results of preceding parts (sj5.1-6) to study Laplacians on compact symmetric space K\G.
$5.1 Simple and Semisimple Lie Algebras, Weyl unitary trick. We introduce basic types of Lie algebras, solvable, nilpotent, simple and semisimple, and the Cartan list of simple algebras. Then we outline the Weyl unitary trick, that relates representations of compact Lie groups to those of complex semisimple algebras { (5). This makes available the machinery of chapter 3 to study representations of semisimple algebras. It implies, in particular, that any finite-D representation T of (5 is completely reducible (direct sum of irreducible components), also any semisimple (5 is itself decomposed into the direct sum of simple ideals.
1.1. Definitions. We remind the reader the basic notions of sample, semisimple, solvable and nilpotent Lie algebras and groups, introduced in 51.4 (chapter 1). Solvable and nilpotent algebras axe defined through an upper/lower derived series of 8,
(I) 8 3 B1= [8;8] 3 8*=[81;81] 3 ... 3 [8";87 = 8jn+l> ... (11) B > 8 , = [8;8]>9,=[8;9,]>...>[8;8,]=8,+1>... Here
[ Z ; 9 ] denotes S p a n { [ X ; Y ] ; XE Z,Y E 9).All
series {Bkc )'32
terms of the upper/lower
axe ideals of 8, [23;8k] c Bk; moreover the commutant 23, is the
smallest ideal so that %/ 23, is commutative.
Solvable algebras have the upper derived series (I) terminated by {0}, Bn+' = 0, while nilpotent (a subclass of solvable) have the lower series (11) terminates by {O}. Typical examples are upper/lower triangular matrices:
{[' ':.
]}solvable ;
1
{ [ ':. i
nilpotent ( 0's on the main diagonal !).
35.1 Simple and Semisimple Lie Algebras, Weyl unitary trick
192
Obviously, the last term 8" of (I) is a commutative ideal, and all quotients
!Bk/Bk++' are commutative ideals of B/Bk+' (solvable case), while 8, c center%, and any %k/%k+l belongs in the center of the factor - algebra %/Bk+, in the nilpotent case. All ideals {BE} and {Bk} are characteristic, i.e. invariant under all automorphisms (respectively derivations) of B. If B is solvable, then B1 is known to be nilpotent (cf. problem 1, 85.3). Also algebra B is nilpotent iff all operators { a d X } are nilpotent,
ad; = 0.
Semisimple Lie algebras 0 can be described in many different ways:
(i) 0 has n o ,solvable, hence, abelaan ideals;
(ii) the commutator subalgebra [@;GI= 0. The third ch,macterization is given in terms of the Killing form, an inner product on 0,defined by
( X IYfzfir(adXadY).
(1.1)
(iii) algebra 0 is semisimple iff the Killing form is non-degenerate. One can show that ideal and factors 0/$of semisimple algebras are also semisimple. W e will not provide a complete detail of the equivalence proof, but just mention that the argument is based on invariance properties of the Killing form relative t o the adjoint action:
( a d X Y I Z) + (Y I o d X Z ) = 0, all X, Y , Z E B ( A d g Y I Adg Z ) = (Y I Z), all Y,Z E 0 and g E G. The invariance relations (1.2) imply that the null -space
(1.2)
of the Killing form,
W = { Y : ( X I Y) := 0, for all X E 0) is an ideal, and the Killing form on W is identically = 0. Any such Lie algebra W could be shown t o be solvable, which contradicts semi-simplicity of 8, and proves (i). The reader is invited t o check (i)-(iii) for the classical simple Lie algebras, listed below (problem 6).
Algebra I Iis called simple, if it has no ideals. All simple (complex) Lie algebras were classified by Eli Cartan. There are four classical series, and 5 exceptional algebras. The four classical series in the standard designation consist of
(A,) Special linear algebras: q n + l ; C ) = {all (n+l)x(n+l) matrices A of t r A = 0).
55.1 Simple and Semisimple Lie Algebras, Weyl unitary trick
193
(B,) Odd-dimensional orthogonal algebras: 90(2n+l; C ) = {complex matrices AZ. y’+ 2. AG = O}, with the standard symmetric dot product 3 y’ = E x k Y k on 4?+’,
-
(C,) Symplectic algebras: (JAZ I a) (JZ I AC) = 0}, where J =
sp(n.43)= (2n x2n
matrices
A,
satisfying
[_“,,k]denotes the symplectic form on C2,.
+
(D,,) Euen-D orthogonal algebras: 4 2 1 1 ;C). The 5 exceptional Lie algebras are designated as
G,; F,;E,; E,; Es
subscript
indicating their rank, and will be described in the next section. The structure of any Lie algebra could be reduced to its “elemental blocks”. Namely,
Theorem 1: i) A n y semisimple Lie algebra splits into the direct sum of simple ideals: (5 = CB(5, - irreducible components of the adjoint action ad, I (5. ii) A n arbitrary algebra admits a Leui-Maliev decomposition (51.4) into a semidirect product, B = ‘Jz D (5, of the mazimal solvable ideal ‘Jz (radical of B), and a semisimple subalgebra (5. The first statement will follow from the complete reducibility of any (finite - D) representation T of a semisimple Lie algebra, via the Weyl unitary trick.
1.2. The Weyl unitary trick. The Weyl “unitary trick” allows to reduce representations of complex semisimple Lie algebras to those of compact groups. The key element of the Weyl reduction is based on an important notion of Cartan involution. All simple and semisimple complex Lie algebras are known to have an involutiue
wtomorphism u,with properties (i) a [ X ; Y ]= [ a ( X ) ; u ( Y ) ](ii) , o ( c X )= E X , for any complex c, (iii) a’ = 1. The set of fixed points of called real compact form of
(5,
R = { X : o ( X ) = X } forms a real subalgebra R of
0,
(5,
i.e.
(5=R@ql
(1.3)
direct sum of the subalgebra R and the subspace Q = i R , the subgroup K c G of algebra
R c (5, is compact. Decomposition (1.3) is crucial in the Weyl’s method. It
means that algebra (5 is a complezification of the compact real Lie algebra R. Therefore, any complex representation of (5 (respectively analytic representation of its Lie group G) is uniquely determined by its restriction on the compact form R (respectively compact Lie subgroup K ) . Thus we get at our disposal the whole machinery of representations of compact groups, developed in chapter 3. Before we shall sketch the general argument for
85.1 Simple and Semisimple Lie Algebras, Weyl unitary trick
194
(1.3) let us give a few examples.
Examples: 1) Algebra @ = q n ) , with involution u ( X ) = - X*, has real compact form si = su(n); 2) Orthogonal algebra: so(n;C) has an involution: X-,-TX, and compact form st = SO(n;W);
{[
3) Symplectic: sp(n;C)= involution: X+-XI, and the compact form
si =
{[<*
I
] a E su(n);c - hermitian symmetric = sp(n),
called compact symplectic algebra. The existence proof of the Cartan involution and real compact form (1.3) semisimple algebras is based on certain properties of the Killing form (1.1). This form will also yield a simple characterization of compact (real) Lie algebras.
Theorem 2: A real semisimple algebra si is compact iff the Killing form is negative definite ( X I X ) < 0, for all X # 0. To establish (ii) we observe that all Lie algebra adjoint operators { a d X } are antisymmetric with respect to the Killing form, hence Lie group { A d g } are orthogonal. The definiteness (positive or negative) of the Killing form implies that the adjoint map
g+Adg
takes G into the compact orthogonal group O(n), n = dirnG. Since G is
semisimple, it implies that the factor-group G/Z (modulo discrete center Z ) is compact. Hence Z is finite and G is itself compact. Conversely, given a compact (simple) group G, there exists a definite (positive/negative) product X . Y on 8 s.t. adjoint maps {Adg} is orthogonal (the latter is easily achieved by “averaging” any product with respect to the Haar measure on C). Any other bilinear form, including Killing’s, is given then by a suitable symmetric operator
(XIY ) = Q X * Y . The invariance condition (1.2) implies that
Q commutes with the adjoint representation
a!), Q = X I , a multiple of the < 0, since for any antisymmetric
A d g , and since !.he latter is irreducible (simplicity of identity, by Schur’s Lemma. Obviously, constant X matrix A , lrA2 15 0, QED.
After Theorem 2 one can easily find compact real forms (1.3) in 0,any maximal real subalgebra si in 0 with negative definite Killing form {(YI Y)< 0;Y E St}. The
$5.1 Simple and Semisimple Lie Algebras, Weyl unitary trick
195
orthogonal complement of such St, R = { Y :( X I Y )= 0; X E R}, must be a null-space of the Killing form {(YI Y )= 0; Y E R }, otherwise, subalgebra R could be extended by adding Y oE R to it. But degeneracy of the Killing form on R contradicts once again semi-simplicity of (3. So si = 0, and the complex hull of Si, R CEI iR = 0,QED. Finally, the proof of Theorem 1 (complete reducibility for representations, T , and ad,, of semisimple Lie algebras) amounts by the Weyl unitary trick to complete reducibility of representations of compact Lie groups, established in chapter 3 (53.1).
196
$5.1 Simple and Semisimple Lie Algebras, Weyl unitary trick
Problems and Exercises: 1. Find the derived series for the algebra of all upper triangular matrices. 2. Show that the ideals {Bt; Bk} of the upper/lower derived series are characteristic, i.e. invariant under derivations, Der(23) = { A : 843; A [ X ;Y ] = [ A ( X ) ; Y ] [ X ;A ( Y ) ] ;X ; Y } .
+
:]:
3. Find the algebra of derivations Der(B) for the Heisenberg Lie algebra
B={[
O 0a c
alla,b,c}
4. Verify the invariance relations (1.2) for the Killing form.
5. Show that K = {fixed elements of u } is the maximal compact subgroup of G . 6. Check that classical algebras 8 = SL,; S q n ) ; Sdn) are simple, i.e. the adjoint action {ad,:X E 8 ) is irreducible; the Killing form on all of them is non-degenerate.
7. Show that the real orthogonal group I s q 4 ; R) N Sy2) X sy2)/Z21 i) Use a representation of 3-sphere by unit quaternions, $4.1 (chapter 4), S ' N Q;= {( = cr+pt:
I< I
=
la
I '+ I p I ' = 1) N So(2);
ii) define the action of group G = SO(2)xSy2)/{ & I } on S3 via quaternionic multiplication, ( € ; 9 ) :z+€-'zv; € , 9 ,z E Q; iii) verify that the stabilizer of point { 1) E Q;,H
N
S0(2)/{ f I } = SO(3);
iv) use a characterization of S q 4 ) , as group acting transitively on S3with a fixed point stabilizer G , N So(3).
$5.2. Cartan subalgebra. Root system. Weyl group.
197
55.2. Cartan subalgebra. Root system. Weyl group. In this section we shall introduce and study three most important structural elements of semisimple Lie algebras: Cartan subalgebra, root system (roots and root vectors), and the Weyl group. Most of the results will be illustrated with The structure theory presented here examples, d(n) and so(.). will lay ground for the subsequent study of representations of semisimple algebras.
2.1. Any semisimple Lie algebra will be shown to possess a maximal abelian subalgebra 8,called Cartan subalgebra of 0, such that the family of adjoint operators E G} are simultaneously diagonalized
{ad,:H
{ a d H : H E S}”. All other (nonzeros) diagonal blocks are one-dimensional, the corresponding entries a l ( H ) ; a 2 ( H )..., ; (linear functionals on 8) are called roots of 8,and the eigenvectors {X,} - root vectors, adH(X,) = o(H)X,; H E 8. The 0-diagonal block here corresponds to
8 = “null-space of
To find a Cartan subalgebra in 0 one picks a regular element H E 0,whose adjoint map adH has the maxima1 possible rank = dimad,(@).
The commutator of
such H is the Cartan subalgebra. The root system C = { a } can always be ordered, so for any pair a,@we have either a
< p, or a > p, furthermore
C is divided into two opposite halves: positive roots
C+ and negative roots C - (the splitting depends, of course, on a particular choice of ordering in C). Having chosen some ordering we call a positive root a simple, if a can not be decomposed into the sum of other positive roots p y. Simple roots are linearly
+
independent and form a basis of
8,dim8 = #{simple
roots} is called the rank of Lie
algebra 0.We shall not provide the general argument (see [Ser]; [Jac]; [Hell), but rather illustrate all concepts with the example of Lie algebra
1) Cartan subalgebra trh =
6
q n ):
consists of diagonal matrices: h = d k g ( ...Itj...);
C h j = 0.
2 ) Root system C is labeled by pairs of indices: jk ( j# k), where ( h ) = h 3’ - h k . 3k
55.2. Cartan subalgebra. Root system. Weyl group.
198
3 ) Positive roots: { a j k } correspond to j
< k , negative
to j
> k.
This comes from the natural lexicographical ordering of Cartan elements (real
> H‘ = (hi;...), if t h e first j , where hi # h>, has two roots ajk > aj,k/iff j 5 f, and (in case j = j ’ ) k > k’.
diagonal matrices): H = (h,;h,; ...)
hj > h). It follows that
4) Simple roots are of the form {aii+l = aj}. k-1
Any other positive roots are sums of simple roots: a j k = C a .; root crln is the i
highest, while anl- the lowest (negative!).
’
5 ) Root vectors { X , = X j k } are Kronecker 6 -matrices with 1 a t the j k t h place and 0 at the rest. We shall denote by { X j k } ( j < k ) positive root vectors, and by
{Yjk = X k j } ( j < k ) negative root vectors, so jk will always mean pair j < k. 7 ) The commutation relations: [h;X,] = cr(h)X,; [h;Y,]= - a(h)Y,; [X,; Y,] E 8; hold for any h E 8; and root a = jk. Returning to the general situation, Cartan algebra product, the Killing form of (5:
( h I h’) = tr(adhadh,) =
8 c (5
(2.2)
has a natural inner
C a(h)a(h’)- sum over all roots,
(2.3)
aEC
in terms of its root-system. Product (2.3) allows us to identify each root a (a linear functional on
8)with
an element H , E 8, turning C into a system of vectors { H a } in
8. Let us compute roots {ITii} for Lie algebra sl,. Take a set of basic diagonal matrices { E i j = diag(... 1; ... -1;...)}, with 1 on the i-th, (-1) on the j-th place, and the rest zeros, and compute the product (problem 2 )
(Eji 1 h) = %(hi - hi).
(2.4)
So roots H i j itre diagonal matrices E i j divided by 2 (we ignore the unessential factor n in (2.4). In special cases 4 2 ) and 4 3 ) , the positive root system is made of matrices, q 2 ) : H = dia& -$); 4 3 ) : HI = diag(f; -3;o); H , = (0;:; -;); HI, = ($;o;- f ) = H,+H,; here {H,; H,} form a basis of simple roots of 43). Root system { H,} along with all (positive/negative) root vectors { X,; Y,} forms the so called Cartan basis of Lie algebra
(5.
One can easily verify the following
55.2. Cartan subalgebra. Root system. Weyl group. commutation relations for the Cartan basis of
199
43):
x,,
[H,; x121 = ; [H,; y121 = - y1*; [X,, ; y121 = 2 H1; which generalize the commutation formulae (2.10)for 4 2 ) in chapter 4.
(2.5)
Similarly relations are verified for q n ) ,
Let us now compute the (Killing) inner product on the Cartan subalgebra 4j
c 431, ( h I H,)= Wl - b);( h I H,)= % - h3); ( h I HlZ) = %, - h3),
(2.6)
for h = (h,; h,; h3). Introducing new coordinates on h = t,H, t,H,, we find
8: t, = ;(hl - h,), t , = f(h, - h3),
i.e. writing
+
11 H I.II = %;( H , I H,)=
-
i; (H,I H,,)
=
( h I H,)=3 4 ; ( h I H,J = 3(t,+t,); etc.
The inner-product structure on the real vector-space 4j allows one to associate to any root h, E C a reflection :s, 4j -+$, w.r. to the plane orthogonal to ha,
It turns out that all reflections {s,} map the root system C into itself. Thus the family {s,} generates a finite group W of isometries of 8 called the Weyl group of 0.
For algebras 4 .) reflections {s,:a = jlc}, can be explicitly calculated by (2.6) - ( 2 . 7 ) (problems 1,2). One can show that sij transposes the ith and j t h entry of h. Therefore the Weyl group of 4 .) coincides with the permutation group of n elements
w = w,. 2.2. Root systems for simple and semisimple Lie algebras. An alternative way to C = { a } in space 4j = Wm, equipped with an inner product ( I ), a finite system of vectors, invariant under all reflections: s,: p-ip - 2 introduce simple and semisimple Lie algebras is to start with a root system
Symmetries (2.8) generate the Weyl group of root-system: W = W(C). Under some additional (minor) constraints such systems could be completely classified and give rise to root systems of simple and semisimple Lie algebras. The former are called
indecomposible root-systems (i.e. C can not be broken into orthogonal pieces: C'UC'' with (a I p) = 0, for all a E C'; PEE'), while the latter (semisimple) are made of
$5.2. Cartan subalgebra. Root system. Weyl group,
200
orthogonal simple root systems, CIUC,U ...U C,, according to a decomposition of 0 into the direct sum of simple components
Geometrically, all root systems of low
&@k. 1
ranks: r = 1;2;3 are sketched below (fig.2;3). For higher ranks we adopt a description of [SerZ] (chapter 5 ) , in terms of an orthonormal basis {el;...en} c W", and the lattice
A = A, spanned by { e,}. ntl
to be a hyperplane in Rn+l orthogonal to C e,. 1 Then C consists of a.11 vectors in @flA,+l of norm 2, i.e. all {e, - ek: j # k}, the basis Series A, (n 2 1): We take
could be chosen as {e, - e3+1:1
5 j 5 n}, and the Weyl group W =
is made of all
permutations of {1;2;...;n+l}. Series B, (n
:> 1): In space @ = W" we consider all lattice points of norm 1 and
&; C = { a E A,: ( a I a ) = 1, or 2). Clearly, C is made of { fe, fe,: i # j } and { fe,} the basis of C: {el - q;q - q;... ; en-l - en; en}; and the Weyl group W is generated by all permutations of {1;2; ...n}, and all sign changes (multiplications): e,+ fe,. So W = (Z, x ... x Z, ) D W, - semidirect product.
-
For n = 1 algebras A, and B, are isomorphic, i.e. 4 2 ) N 4 3 ) (chapter 4). Series C , (n 2 1): The root system of C,-type is dual to the B,-root system, i.e.
C(C) = {a* = 2 A ; a E C(B)}.So C(C) consists of { fei fe3: i # j} and
).I
(a
{ f2e,},
has a basis {el - q ;e2 - q;... ; en-l - en; 2en}; and the same Weyl group as B,.
D, (n 2 2): The D,-root system consists of all a E A, of ( a I a ) = 2, i.e. all i # j } with the basis {el - q ;q - q; ... ; en-l -en; en-l +en}. The Weyl
Series
{ f e, fe,:
group is made of all permutations and all sign changes: e,+
{ - }, the latter group being isomorphic to 2, W = z, D W,.
N
(ZJn-'
fe,; with even number of
c (Z,)".
So Weyl group
Algebra G,: This root-system was sketched in fig.2 (iv). It could be described as the set of algebraic integers { z = a Q(w)
+ bw + cw2: a, b,c E Z} of
c C, generated by the cubic root of identity
the cyclotomic field
w = ei2*/3, of norm I Z I = 1, or 3.
For other exceptional algebras we refer to [Ser]. In low ranks there are many overlaps between 4 series:
B,: sp(1)N 4 2 ) N 4 3 ) (chapters 1,4);
El]Cl
N
A,
l=1C2
N
B:!:sp(2) N 4 5 ) ; and D,
N
A:%:4 6 ) N 44) (problem 6).
D,
N
N
A, @ A,: 4 4 ) N 4 2 ) @ 4 2 ) (problem 5)
201
55.2. Cartan subalgebra. Root system. Weyl group.
Given a root system C = { a } in Wn = !$ with a chosen basis { a l ...a,,} ; one can construct the corresponding semisimple Lie algebra. We denote by C * the set of positive/negative roots, and set
@ =f fa> O@(Off$@-,) with 1-dimensional subspaces: 6, = span{Xff}; 6-,= span{Yff= X - m } . One has the following commutation relations,
(i) [ h ; X f f=] a ( h ) X f f where ; a ( h ) = (a* I h) = 2-;(ff I 4 (ff Iff) and a* = 2 A = H , denotes the dual vector to a.
(ff Iff)
The root-vectors could be normalized so that if a+/3 E C 0; otherwise The coefficients { N a p } depend on the choice of basis in C. (ii) [ X , ; Y , ] = H,; [ X , ; X p ]=
The construction can also be implemented in terms of generators of
...an} in chosen basis {a1;
8, respectively
(5,
i.e. a
the dual basis {Hl;...H n } , and the
corresponding root vectors {XI;...Xn} (positive), and { Y l ;...Y n } (negative). The generators satisfy Weyl commutation relations:
(2.9)
(2.10) Here numbers { n2.3. = n(ai;aj)}denote pairings of basic roots {a1; ...a n } ,
.(a;@) = 2-(a I a) (a
Iff)'
The set of numbers { n ( i ; j ) }forms the Cartan matrii of C . Numbers n(a;@) are known to take on integer values (0; f 1;f 2 ; f 3). They have a simple geometric interpretation, .(@;a)= 2
hence
I I m
os6;
n(a;P)n(@;a) =4 ~ 0 ~ ~ 8 ; where I9 denotes the angle between roots a and
p.
In particular, angle 19 could take on
only 4 possible sets of values: I9 = { ~ } ; { ~ ; ~ } ; { ~ ; ~ accordingly } ; { ~ ; ~ } we , get 7 possible
202
$5.2. Cartan subalgebra. Root system. Weyl group.
configurations of (non-colinear) pairs {a$} (see fig.1 and the table).
Table 1. 1) n(a;P)= 0
@;a) = 0
9=;
orthogonal roots
2) .(a$) = 1
n(P;a)= 1
9=;
[PI = la1
3) n(a;P)=-1
n(P;a)= -1
9 = &3
IPI=IaI
4) n(a;P)= 1
.(p;a)
9=f
IPI=
5) n(a;P)= -1
n(P;a)= - 2
9=
6) .(a$) = 1
n(P;a)= 3
9=;
lPl= d3laI
7) n(a;P)= -1
n(p;a)= - 3
9=k 6
lPl=
=2
f
4lal
= 4lal
&la1
Let us remark that basis {a1; ...an}in any root system C is chosen in such a way, that the Cartan numbers {n(<j)} are negative, so all angles Bij between pairs {ai;aj} are > This also explains the choice of negative exponentials -n(z;j) in (2.10). Finally, any set of Weyl generators (2.9)-(2.10) yields a semisimple Lie algebra with root system
5.
c. B
ka
&a
B
@
\La
&; By +a
-a
Fig.1: illustrates all possible angles between pairs of root vectors {qp} and their relative length.
85.2.Cartan subalgebra.
Root system. Weyl group.
203
Fig. 2 shows all root systems in rank 2. There are 4 different cases:
(i) A,: algebra 4 3 ) has 6 roots in hexagonal arrangement; roots a,@, a+p correspond t o diagonal triples: (1; -1; 0); (0;l;-1) and (1;O;-1);
iE3
a+ZE
(ii) B, and C,: algebras 4 5 ) and sp(2) are isomorphic, as evidenced from their root diagrams; those become identical when roots a and p are interchanged.
(iii) D,: algebra 4 4 ) is not simple, but breaks into the direct sum of 2 orthogonal A,- diagrams. We have already mentioned that 4 4 ) 4'4@ 4-21, hence complex 4 4 ) N 42)%3 42).
=
( i v ) Ezceptional Lie algebra G, has 12 roots arranged in a king-David star.
55.2. Cartan subalgebra. Root system. Weyl group.
204
Fig.l(a): Root system A, = D, (d4N 4 6 ) ) consists of 12 vertices of cubo-octahedron. The figure shows 6 positive roots, spanned b y a iriple of simple roots: a;o;y. All roots have equal length.
4 (algebra sp(3)) consists of 12 vertices and 6 ceders of square faces. We have shown 9 positive roots, spanned by simple roots: a;P;y.
Fig.3 (b): Root system
The C, - root system is made of the same 1.2 poinis, but ihe “long roots” (vertices) and short roots (ceders) interchange. So a becomes long, while /3,r short.
Dynkin diagrams. Root systems C are conveniently labeled by certain graphs, called Dynkin diagrams. The vertices of diagrams are simple (positive) roots of a fixed Cartan basis. Two roots a, p are connected by a single, double or triple bar, if they make angles
%
and
f
respectively (the orthogonal roots are disconnected).
Furthermore, the lines between uneven roots are equipped with arrows from the short to the long root. Figure 4 below sketches Dynkin diagrams of all simple Lie algebras, classical and exceptional.
a-8
O2
Fig.4: Dynkin diagrams in ranks 2 and 9.
205
55.2. Cartan subalgebra. Root system. Weyl group. Problems and Exercises: 1. Use formula (2.6) to compute reflections { s j k } of 4 3 ) . Show that the Weyl group of 4 3 ) is W,.
1 1
I
2. Check the product formula (2.4); calculate E j k and ( E j k E .,k,) for the roots E j k = diag( 1;... - l...)of 4 n ) . Find angles between roots. Show t i a t the Weyl group of sl(n) is W,.
...
3. Find the Cartan subalgebra, root system and the Weyl group of the Lie algebra 4 4 ) . 4. Check that a Weyl reflection
u,: h - h - 2
o_ (a14
defined on the Cartan subalgebra algebra 0, by u,: X p -X
c 0, extends to an automorphisms
of the whole Lie
.,(P)'
5. Show the real orthogonal group S q 4 ; R )
N S y 2 ) x SU(2)/Z2. i) Use the representation of 3-sphere by unit quaternions, $4.1 (chapter 4),
S32:Q;={<=a+PL: 1 ( 1 2 =
Ip12=1}~So(2);
ii) define the action of group G = SU(2) x SU(2) on ( € ; 9 ) :z+€-'z9;
S3via quaternionic multiplication,
€, 9, z 6 Q;
iii) verify that the stabilizer of point { 1) E Q;, H
N
So(2)/{ f I } = So(3);
iv) use a characterization of So(4), as group acting transitively on stabilizer G , 2: So(3).
S3with a fixed point
6. Show that Lie algebras 4 4 ) and 4 6 ) are isomorphic, by comparing their compact forms: 4 4 ) and 4 6 ; R ) , and choosing an appropriate Cartan basis for both.
7. Compute the Cartan matrix {n(<j)} for 4 classical series: A,;B,;C,;D, and for G,.
8. Use a description of Cartan numbers { n ( i ; j ) } and angles in table 1 to get all root systems in rank 2 and 3.
55.3. Highest
206
weight representations.
55.3.Highest weight representations. The main result in the representation theory of semisimple Lie algebras describes their structure in terms of weights, linear functionals on the Cartan subalgebra of 0. Weights are linear combinations of roots, p = C naa, with integral coefficients, so they_belong to a lattice r spanned by roots. Each irreducible r E G is labeled by its highest weight A. We shall give a detailed description of the weight-diagram of an irreducible representation, and its relation to the Weyl group of 0, emphasizing the example of SL(n). Then we construct some explicit realization of the highest weight representations r A of G = SL(n) in function spaces over G , and its quotients. This construction underlines the fundamental role of 2 algebraic operations: symmetrization and antisymmetrization, in the representation theory of SL(n). It also exhibits as "induced representations" in spaces of polynomial (holomorphic) functions on G.
{d],
3.1. General construction. We shall use the previous notations: O - a semisimple Lie algebra, $ its Cilrtan subalgebra, {H,} - the root system in $; {X,;Y,} - Cartan basis of root vectors, and W - the Weyl group. In case of O = q n ) , $ consists of diagonal matrices, and roots and root vectors are
H - H .j k =di~g(...I....-I...).X. 2, 2 jk = 6 .jk ; Yj k. = 6 k j - K r o n e c k e r d , f o r j < k . (I-
9
The Weyl group W = VVn consists of all permutations {u}, acting on diagonal h E $, and also on the whole Lie algebra O itself by conjugations, X+u-'Xa. The following Theorem reveals the structure of irreducible representations of terms of weights { p } (linear functionals on 0),and weight-vectors.
Theorem 1: i ) Any irreducible representation T of algebra 0 in a finitedimensional space T, restricted on the Cartan subalgebra breaks down into the direct sum of eigenspaces T= CBYp, where B Th I Tp = p(h)l = (Hp I h ) l . The system of eigen-functional properties:
{ p } , called weights of T , has the following
ii) all { p } belong to the lattice spanned by simple roots, i.e. p = x n j a j , w i t h integral coefficicmts {nj}. i i i ) the weight diagram {a} of T is invariant under the Weyl group, acting on the Cartan algebra, furthermore multiplicities of eigenspaces, d(p) = dim Tp = d(pa), are equal for all { p } in the same Weyl-orbit.
$5.3.Highest
207
weight representations.
iu) There ezists a unique highest weight vector to= ((Po):Thto= Po(h)to,for all h, whose weight Po 2 P, for all P in the weight diagram. Vector to is annihilated by all by all lowering raising operators: T x t o = 0 for all X = X , . The span of generates the entire space Y. operators { T Y I T Y 2...(to)}
to
Its proof will consists of several steps. Step 1: Existence of the highest-weight vector
tofollows from a general theorem
in the representation theory of solvable Lie algebras, so called Lie’s Theorem. We observe that the algebra 13, spanned by 8 and all raising operators S~an{!&x,},,~ (called Borel subalgebra of 8 ) is solvable. Indeed, adh(X,) = a ( h ) X , and [X,;Xp] c X,+p; for all pairs of roots a;P. In case of q n ) the Borel subalgebra B+ is made of all upper-triangular matrices ! Lie’s Theorem: Any solvable Lie algebra of operators 8 c End(Y) has an , all X in 113. Consequently, algebra 9 can be brought eigenvector, X t o = X ( X ) t Ofor into the upper-triangular form, i.e.al1 operators { X } can be represented by the upper-triangular matrices in Y. The proof is inductive, either in dimension of 8, or in its the height, the length of the derived series. Let u be an eigenvector of an ideal 2I = [B,B]= B, (inductive assumption
!) and p - the corresponding eigenweight: Yu = p(Y)u,for all Y in 9I. For any X in B we denote by
<x= e x ( u ) , and show that tX
is also an eigenvector of 9I of an eigenweight
p x ( Y ) = p(ezpadX(Y)).Indeed, Y e X ( 4= eXAdezpx(Y)2, = P(Ade,,X(Y)) eX ( u ) . It remains to observe that the number of weights { p } of a finite-D representation is always finite. Hence a weight can not be continuously deformed by the family of operators {X: p+px}
and we get px = p, for all X. The latter means that the
eigenspace Yp = { u : Yu = p(Y)u}is invariant under the entire Lie algebra B.
I
Now we can restrict operators { X Yp}, and reduce the problem to a two -step solvable factor - algebra %/Lerp= %,. Algebra %, has at most a one-D commutator, [~B,;%,I=
SPon{Yoh [X;YOl=,(X)Y,,
and the operator Y o is scalar: Y o = poZ = p ( Y o ) I . But tr y o = po dimY = 1 t r [ X ; Yo] = 0 .
4x1
Therefore, Y o = 0 and we get the commutative algebra of operators B,/[ . .;. .I
u
%/[..;..I
acting on the eigenspace Yp.So weight p must be = 0, and the problem is reduced to a
55.3.Highest weight representations.
208
commutative family of operators, acting on Y, which is well known t o have a joint eigenvector, QED.
Existence of the highest-weight vector readily follows now (for other applications of Lie's Theorem see problem 1). Step 2: Weight diagram. Other statements of Theorem 1 will be demonstrated in the case of algebra qn). By the Weyl "unitary trick" the latter is equivalent to the study of representations of Syn),a compact f o m of SL(n). We denote by D N T"-' the diagonal (Cartan) subgroup of G = SU(n). All operators T , (h E D) can be diagonalized,
T , =[
.'*esm'6~,,,...
]
where h = diag(tel;ele z;...); 8 = (O1;O2; ...); 8, = 0;
and weight m = (ml;m2; ...) represents a tuple of integers. Obviously, any weight m = (ml;m2;...)
is uniquely determined, modulo a diagonal shift: (ml;m2;
...) + (ml+mo;
m2+mo;...).
Moreover, any weight mu = (mu(l); m,(2); ...), with u in the Weyl group of sr(n), belongs t o the weight diagram (spectrum) of representation h+Th of D and multiplicity
d(m) = d(m"), for all m and u. Indeed, W acts by automorphisms of S w n ) and SL(n), u: X+u-'X
u, that leave Cartan subgroup D invariant. So the W-orbit of any m in the
weight diagram contains mu = (ml; m2; ...) with ml 2 m2 2 ... We shall use this ordering of weights, and choose the highest weights m. The corresponding vector
to is annihilated
to= 0. Indeed, for any weight p and root a, operator
by all raising operators: T, ik
T,,:Yp+Vp+,,
i.e. X, raises weights, while T y :Vp+Yp - , lowers them. I claim that a
Yo = span{u = produet(T>,)((o):
a = jk}, is an invariant subspace of T. Indeed, all u's
are the eigenvectors of D, so the Cartan subalgebra takes Yo into itself. It remains t o show that all operators {T,) take Yo into itself. Given any product of lowering operators
Y , Y , ...Y N (Y3= Y p ) ,and a raising X = X,, we use the commutation relation XY1Y2
Applying both sides to
Y1...[X;Y,] ...Y N . ...YN = Y1 ...YNX +Sum ~ S J S N
to,the first term in the RHS vanishes (X<,=O!). To analyze the
remaining terms we observe that [ X , ; Y p ] is either a multiple of Y o - , (if a
.>a).
In the
first two cases we obviously remain in Yo. In the later case we continue the process, i.e. write each of the remaining products
$5.3. Highest weight representations. ***YN 83 Y j + , * . * Y N x , - p +
x,-pyj+l
209
a**(
and argue as above. The procesa will terminate after all terms will transform into products of Y's. This shows that Yo is invariant under the whole group (algebra) G, hence by irreducibility To =
Y.
Step 3 Uniqueness of the highest weight vector is easier to prove by extending representation T to the Lie group G of the algebra (5.We shall prove it for G = SL(n), by embedding representation T into a function-space on
G,i.e. realizing it as part
of the
regular representation.
T of G can be realized on a subspace of functions on G by picking a vector to in V, a functional qo in Y* (the dual space) and mapping u = T g t o into a function (matrix entry) f(g) = (TgtoI qo). Obviously, W: v = Ca .T (to)-f = (Tgu I qo) maps T into a finite-D invariant subspace of the i) Any irreducible representation
3 gj regular representation R on G. Furthermore, since X+TX is a complex representation of
the complex Lie algebra, all entries are regular (holomorphic) functions on G (C has a natural complex structure !), in fact, for an algebraic groups, like SL(n), they all are polynomial functions
.
tobe the highest weight vector of T , qo be the lowest weight vector of the contragredient representation pg = (inverse of the dual operator T $ to Tg). So 9 T e , , ~ ( t o )= to,T',,,y(qo) = qo for all raising X and lowering Y , and Th to= X(h)to, ii) We choose now
TLqo = p(h)qW The corresponding matrix entry f(g) = ( T g t oI qo) has the properties that j(yg+)
= f ( g ) ; for all z E N + ; y E N -
the upper and lower triangular subgroups of G)
f ( h ) = X(h) = p(h-'), all diagonal h. Indeed, (ThtO I SO) =
I q0) = (€0I Ti-l)70) = p(h-l)(tO I k~)? (toI qo) # 0. Here we shall use the
and (TygztoI qo) = (TgtoI qo). It remains to see that
Gauss decomposition of any (generic) matrix into the product: g = y . h z (lowertriangular, diagonal, and upper-triangular). Then f ( g ) = (Thto 1%)
= X(h)(toI 'lo).
But f(g) can not be identically 0, due to irreducibility. iii) Obviously, uniqueness of the function f(g) satisfying properties (3.1) implies uniqueness of the highest (lowest) weight vectors.
To complete the proof of Theorem 1 it remains to observe that weights {m} of
55.3.Highest weight representations.
210
the representation T of 4.) described as tuples of integers (ml; m,; ...) are integral n-I multiples of roots. In other words we have to express met9 = C mjOj (mn can always be n-1 1 taken O!) as C vj(Bj - Bj+’) . I
Corollary: A n irreducible (analytic) representation ?r of a semisimple Lie group G
has a unique vector vo that satisfies: .,(to) = tofor aU z E N + , and ?rh(to) = X(h)t0 for all h E D. Such to must be the highest weight vector and X the corresponding highest weight.
Indeed, the proof shows that Span{?ry(~o): yE
IV-} is an invariant subspace of n.
Example 2: We shall illustrate the foregoing with an example of group SL(3) and two its representations: the natural (smallest) representation T’ in C3 and the adjoint representation r2 on its Lie algebra 43) =C*. For n1 the weight diagram consists of three weights: m = (I,O,O)-+rnatria:(m . ) = =$(2,1); Ik
m = (O,l,O)+rnatriz (mI.k ) = m = (0,0,1)
[ [-’ ; bp
= $(-1, 1);
-[ -,]+? 0 -1
= $(-1;-2). All three of them have multiplicity 1.
For r 2 the weight- diagram coincides with the root-diagram, so it has 6 roots: a = H j k (J’ # k) ( 3 positive: j < k, and 3 negative: j > k), each of multiplicity 1 and the weight a = 0 of multiplicity 2. Remark:a general formula for multiplicities {d(p;A):- A an irreducible representation
5 p 5 A}
in a weight-diagram of
of semisimple compact Lie group G is due to Kostant
(*e [CTI), U F W
sum over all Weyl transformations {u}. Here P(y)denotes a partition function, which measures the number of positive weights “less than y” in the root lattice
P(y) = {y’= E m m a5 y: m, 2 0).
3.2. Explicit form of highest weight representations of 4 .).We denote by D the the subgroups of upper/lower diagonal (Cartan) subgroup of G = SL(n), by N triangular matrices with (1) on the main diagonal, by B = D N - the corresponding Bore1 subgroups (all upper/lower triangular matrices).
*
Any (almost any) matrix g E SL(n) can be uniquely factored into the product of
$5.3.Highest
211
weight representations.
lower-, diagonal, and upper- triangular matrices, g = yhs, y E N - ; x E N + ; h E D, the well known Gauss decomposition of the Linear algebra.
,. We pick the highest/lowest weight vectors Spaces 9 representations T=T, and
'f
to
and
vo
of
(contragredient), and assign to each v E Y = Y(T) its
matrix entry:
4 T g " 170) = fk). Functions {fv=f(g) = (T,v I qo)} form a subspace 9, in space e(G), satisfying : i) f(yg) = f(g); for all y E N - (the lower-triangular subgroup) ii) f(hg) = X(h)f(g); for all diagonal h. Furthermore, the highest weight function fo(g) = (TgtoI Q) is invariant under the right multiplication with s E N + : fo(gs) = fo(g). By the Gauss decomposition, g = yhs, any f E 9, is uniquely determined by its values on N + , and the highest weight function fo(s) = 1 on N,. Thus fo(g) = X(h), in terms of the Gauss decomposition g = yhs. We
also have iii) 9, = Span{Rg(fo) = fo(sg) = X(h(s,g))fo(zg): g E G}. Properties (i-ii) show that T is embedded into the induced representation:
R = ind(X I B-;G), where X I B means an obvious extension of character X from D to the Bore1 subgroup B- = N - D , X(yh) = X(h). Property (iii) completely characterizes 9, in the representation space of R. So we can think of TA, either as a subspace of e(G), satisfying (i-iii), or a subspace of e(P-). In the former case 4, is spanned by all right translates of fo = X(h), in the latter by all operators Rg(fo) applied to fo=lin 9 ( P - ) realization. Proposition 3: Representation T = R 19, is irreducible. To show irreducibility of 9,
we apply once again the unitary trick and the 1-1
correspondence between (analytic) representations of SL(n) and those of So(n). Obviously, T = R I 41, is analytic, so if TI
were reducible, each component of it SWn) would contain a highest-weight vector, by Theorem 1, which contradicts uniqueness of the
function fo.
We shall illustrate this construction with the example of SL(2). The Gauss factorization:
55.3.Highest
212
identifies B-\G
weight representations.
with the group
Space TAconsists of all functions f(z).Weight X = m (integer), i.e. X(h) = hm, for h = diag(h;h-I), and the induced representation R , f ( z ) = X ( h ( z ; g ) ) f ( z g ) ,where the cocycle h(z,g ) and the action of G on {z} can be explicitly calculated:
Thus h ( z , g ) = diag(a+cz;&), element g acts on z by fractional linear transformations 9:z-+# = the resulting representation
$$;
Rgf(.) = ( a t c z ) m f (
).
Space 4, spanned by f o = 1 consists of all polynomial functions of degree m. Thus we get a familiar from chapter 4 realization of representations { T “ } of Sf(2), or
Sl.42) in polynomial functions.
Basic representations {r’}. Returning to Sf(n) we shall first construct a special family of representa.tions
basis of all irreducible
{T’},
{dk)}E:2
acting in function-spaces 4,,that will provide a
in the sense that each space 9, will be built from 4,.
To this end it will be convenient to write weights X of a representation T of SL(n) as tuples of integers m = (ml;mz;...m,),i.e. ~ ( h=)h,mlh,mz... for any h = (h,;h,; ...h,) in the diagonal subgroup D. The n-tuples of integers (ml;m2;...) are lezicographical ordered, with the highest weight (character) X satisfying rn, 2 m z2 ... 2 m, = 0. Such n-tuples are often called signatures of T. We shall alternate two notations: T’ (A-highest weight), or r m(m-signature) for irreducible representations of Sf(n). Often it will be convenient to change the natural coordinates (diagonal entries)
...h,)
6,=hlh,; ...; 6,=hlh, ...h,; ... . Then X(h) = 6116z2...with a tuple of integers: a = (rl;r2;...), rl = m,-m,; ...; rk = rnk-rnkS1; ... . So a third label for irreducible representations of SL(n)will be IT(*)}. (Itl$,;
on D to another set: 6,=h,;
r r
All irreducible representations of SL(n), consequently Syn),will be realized in certain spaces of polynomial functions on G . Notice that matrix entries { z i j } of any g = ( z i j ) ,as well as the minors:
“cut-off by rows: i l , i z ,...ik; and columns: j 1 , j 2 ,...jk),are polynomial functions on S f ( n )
55.3.Highest
213
weight representations.
of degrees 1,2,... Let us introduce the following function spaces on G: 4, = Span{~,~}!j=~ - all entries of 1-st row;
- all minors made of the 1-st and 2-nd row entries
4, = Span{ 4, = Span{ME;:..jk
dim 9, =
= Mjljz...jJ
- all minor made of the first
k rows,
(t).
Spaces 4, are invariant under the right multiplication with elements g E SL(n) as a consequence of the general product formula for minors of any pair of n x n matrices A and B .. . .. . Mala2...ak pip2-*.pk B). M22a2...ak(AB) = (3.2) P l P Z . . . p k ( A )Mj, j,... j k ( 3132"'3k
pl
Abbreviating notations for tuples of multi-indices j = (j1<j2< ...<jk), we can write (3.2), as
XM'~(A)
M/(AB) = P M;(B), (3.3) the form similar to the standard matrix-multiplication A . B, but with Ic-tuples { k j } in place of indices {i;j}, in matrix entries. We fix a k-tuple of row indices i = (ili2...ik), and consider a subspace spanned by all minors with the given i, T i = span{Mi: all j } . It follows from (3.3), that 4; is invariant under right multiplication with B E SL(n). Thus we get a family of representations {dk)}:=: in spaces 4, = 9(l;2,,,k), all subrepresentations of the regular/induced representation R. Representations
{ d k )can } be constructed from the simplest natural action r1 u R 14,of
SL(n) on C" 21 T,,by extending r1 to all antisymmetric (exterior) tensor powers. Indeed, spaces 4fk are identified with kth exterior powers of C", A k(Cn), by mapping the natural basis: e . h e . A . . . A e +MIZ.:.' 31 32 jk J1 i2...i,(g)*
Functions f E 4, satisfy f(yhg) = a k ( h ) f ( g ) ,for all y,h,g, where ak(h) = 6 k = h1h2 ...h k , in particular, 4, can be realized as a subspace of €'(P-). We need to show that representations {dk'}are irreducible and find their signatures { ( Y k } , i.e.4, = Tak, as defined earlier. We notice that each minor Mjlj2...jk is an eigenvector of the diagonal group D acting by the right translation of weight X(h) = hjlh j2...hjk,the corresponding tuple m = (0; ...;1;O; ...1;...), with 1's in the j,, j z , ... j k spots and zeros at the rest. Obviously, the highest weight X has signature m = (1; 1; ...1; 0; 0; ...), the first k entries
$5.3.Highest
214
weight representations.
being 1, while the rest 0. Its signature in terms of r-parameters becomes (0; ...;1;O;...) (1 at the kfhspot). We shall called it ak. It remains to show that M,,...k(g) is a unique function in
Tk
satisfying
f ( g z ) = f ( g ) , for all z E N,. Notice that all highest weight X's belong to a lattice in the vector space 8 (Cartan subalgebra) of 0 = qn), spanned by the positive roots of 6. So one can consider formal sums of highest weights: (X,p)+X+p. It turns out that this operation extends to representation spaces 4, namely the product Indeed, Tx+,, 3 '?FAT,, since all products f = f l ( g ) f z ( g ) satisfy:
TAT, = TX+,.
f (yhg) = X ( h M h ) f ( g )for , all Y, h, 9, and Tx+/l is spanned by fo = 1 = 1 1 (in 9 ( P ) -realization). But fo = 1 is the only
-
highest weight function in TAT,, as all highest weight functions are characterized by the property f(gz)= f ( g ) , or f(z)= Const! Starting with the basis
{Tk} one can construct all other spaces TA,
X = (m,,m,,...)H(Y = (rl,r2,...r k ) , by taking 4, = 9;19;2...TP. Thus we established the following Theorem 4: (i) For any pair of highest weights X and p the product of irreducible subspaces 99 ,, is an irreducible subspace TX+, of the regular/induced
representation ,. (ii) All irreducible representations {x'} are generated by the representations spaces 9, = Span{Xjlj2...jk} -Ak(T,). Moreover, ifX = c r k a k , then T, coincides with the product T, = $1 {7rk}t;;an
...92...
Remark: The reader should not confuse the multiplication of spaces {T,} with the standard operation of tensor (Kronecker) product of representations
{7rA}.
The latter
are typically highly reducible. But the above construction of irreducible representations of
Sf(n) based on "products" of antisymmetric tensor powers
7rk
= A'(r'), is closely
related to tensor product construction of irreducible representations, examined in the next section. Example 5. We conclude with an example of SL(3). Two basic representations of SL(3) are 7r1: C3+C3 (the natural) and 7rz: Az(C3)2 C3+Az(C3). Although both act in a is contragredient of d. Indeed, space A2(C3)N 9, has a basis made of minors M,,(g); M,,(g); M,,(g). Formula (3.2) shows
3-dim space they are different, in fact
7r2
that Mjk(ga)are linear combinations of M,,(g) with coefficients that are 2 x 2 minors of
55.3.Highest aESL(3), so the matrix of
7ri
215
weight representations.
in the basis
{M'=M23;
M2=M13; M 3 = M12} is
Any irreducible representation of SL(3) with signature a = ( p , q ) is realized in the space 9 ,, of polynomial functions
y = (y1,y2,y3) denote the first two rows of matrix g, and where z=(z~,z~,z~); Y , = z2y3- z3y2; etc. are the corresponding minors. Group G acts on space 4,,, by the change of variable: f-f(zg;g-'y), i.e. the left multiplication of a row vector z by matrix g and the right of a column y by g'l. We get as special cases representations tensor powers of
d,Yp(rl),and
T(~'O),
r ( O V e ) = Yq(r2), i.e.
basic antisymmetric representation
T'.
which represent all symmetric
q-th symmetric tensor power of the
Indeed, the corresponding function-spaces are
4, = {polynomials of degree p in z1,z2,z3} N Yp(C3), and
To, = {polynomials in y1,y2,y3 of degree q } N Y'J(C3), but the G-action on the z- and y-spaces are different (contragredient one to the other!).
55.3.Highest
216
weight representations.
Problems and Exercises: 1. Apply Lie's Theorem to show (i) commutator [9);9)] of a solvable Lie algebra 9) is nilpotent; (ii) algebra 9) is nilpotent iff all adjoint operators {ad,:X E 9)) are nilpotent, i.e. ad; = 0, for some n = n(X).
2. Introduce variables { t j = O j of { t l ; ..And1} as
el = A{(n-l)tl 8,
=A{
-1,
.t( n - 2 ) t z
- Oj+l},
and show that
{el; ...en-,} are expressed in terms
+ ...+
+ ( n - 2 ) t z + ...+ tn-l} + tn-l}
~ ~ - ~ = A ( - t ~ - 2 ...-( t , - n-2)tn-, Get as a corollary, n-1
C m .e . = hz vj(e - B 1
n-1
~ + ~ )
1
l'
with integral c:oefficients {vj}, given by the following relations in terms of the differences { m j k = mj-mk} <
u l = C3 .> 1
[
m ~ j
u z = z j.> 2 m 1 j + C j > Z m z j
u,,-~=
ml;n-l+
m2;n-l
+...+ mn-2;n-l.
m13
m12
m23
... ...
ml;n-l
m2;n-] mn-Z; n-
3. A symmetric tensor power of the representation r1 of SL(3) acts in a 6-D space S,(C3) = C3 8 sC3. Show that r1 8 srl is irreducible and calculate its weight diagram. 4. Prove (3.2).
5 . Show that ~ ( ~ ) cAk(r('))-the z kth exterior power of r1 in the space Ak(C"), i.e.Tk and Jk)(tl A ... A tk)= (r1t1)A A ( r ' t k ) , for all products t1A ... A (k. cz A
...
6. Show that M I Z , , , kis he only N+-invariant minor (i.e.M(gz) = M ( g ) for z in N+) among all M . .
{
3132...jk
\
'
7. Show that representations r k and rn-k of SL(n) are contragredient. Hint: spaces A k and h n-k are dual each to the other by the pairing:
(2: = €1 A €2-.. A t k ; f i = V 1 A ek
So ( r i z
fr Vn-k)+€l
A
*+*
A Vn-k
= det(€l;€z;...€n)
3
-
ek
= (€ I 6)
n
I ~ i - ~= (zf iI)fi)detg.
8. Let Ak denote the kth exterior power a matrix A E GL(n), i.e. entries Ak. are k x k SJ minors of A. Find the inverse of Ak.
217
s5.4. Tensors and Young tableaux. 35.4. Tensors and Young tableaux. $5.4 outlines the Weyl theory of tensor powers, invariants and representations of classical compact groups. It turns out that all representations of compact groups are
obtained from the simplest (natural) representation r (e.g. 4 .)acting in C"), by taking all tensor powers {Sm(r):rn=O;l;2...} of z, and decomposing them according to the action of permutation group W,. The latter gives rise to various types of W,tensorial symmetries (statistics), of which symmetric and antisymmetric tensors form two special (extreme) cases. All other (intermediate) symmetries are described by the so called Young tableauz. The theory of Young tableaux and symmetrizers yields irreducible representations of both Sr(n) and W,. In fact, two sets are naturally paired, as SL(n) and W, form a maximal commuting pair (Howe's "dual pair") in the corresponding tensor-power space.
4.1. Invariants. Let us start with a simple example of rank-2 tensors. Space
T2(Cn) = S,
e A,,
breaks into the direct sum of its symmetric and antisymmetric parts,
both subspaces being invariant under
dm)and
irreducible (problem 1). The
corresponding projections:
xs:( 8 q+!j((
87
+78
(symmetrization),
and
xa: ( 8 q+!j(( 8 77-7 8 (anti-symmetrization), provide a canonical decomposition of 4'. A possible approach to a general (higher-rank) problem, due to H.Wey1 [WeS], lies through the study of all invariants of the upper triangular group N + in the tensor algebra T=%Tm (with tensor multiplication 0
Tk8 Smc Tk+m).Thinking of tensors as multilinear forms over C", the product of two forms (k-form I and j-form
4) becomes a
(k+j)-form
*(El;. .&)4(7]1;...qj). Once all N +
invariants are known it remains to identify eigenvectors of the diagonal group D among them, i.e. find the highest weight tensors. One natural set of N+- invariants, we have encountered so far, were exterior products: wk = el A ... Aek, where {el;
... en}
multilinear forms on 43" they become uk((;%
c) = det
[
is the natural basis of Cn. Written as t1
tk
'I1 q2 (1
...
'Ik
f 2 * * * ck
]
k x k determinants formed by the first k entries of vectors [ = (El;. ..En); q = (ql;...q,); ... Next we can take their tensor powers {w; = wk 8 ... 8 Wk},i.e. kr-multilinear forms: Wk(~1;E2;-..Ek) wk(ql;...qk)...wk(cl;...ck),
in vector variables
{El...(k; ql.. .qk;...; C1.. .Ck: ti;q j ;I, E C"},
under N,, as well as their tensor products:
which are also invariant
$5.4. Tensors and Young tableaux.
218
w, = w : ~ W W : L ~..~ wrl, . of rank m l=< C j < k jrj. Permutation group W,
(4.1)
acts naturally on the tensor-product space Tm by
interchanging tensorial components, U : w = ( 1 B -..B (m+ = (u(1)B B (a(,) and this action clearly commutes with the representation T("') of SL(n). Therefore, all permutations of f o r m (4.1) {was = swa: s E W,} are also N , invariants. One of the m . .
main result of the Weyl S L - invariant theory asserts: the space of all N , invariants of
'3' is generated by tensors {was}, i.e. each invariant i s a linear combination of where a denotes a partitions ofm = j r j , a = ( r l ; ...rk), and s E W,.
{was},
C
Obviously, each m-tensor w:k w:k_il.. .wT1 is an eigenvector of the diagonal group
D, of a highest (eigen) weight X(h) = hy1...h r k ; with mi = C ri. Thus by counting all i
j
highest weight tensors (with their multiplicities) we obtain a hecompositions of dm) into the sum of irreducible representations {r,}, of signature (I = (ml;m2;..m k ) , as well as multiplicities of
in a(,).
In special cases (tensors of low rank) this procedure yields a complete and explicit answer. However, to decompose higher rank tensor spaces Tm one needs to classify different types of tensorial symmetries, in other words to study the action of the permutation group Wm on T"'. Two examples of Wm-symmetries were already mentioned, the symmetric and
antisymmetric parts of 9 :, symmetrizer
Ym(Cn) and Am(C"), given by two projections, the
Both subspaces Ym and A"' are invariant under T("'), and the resulting subrepresentations: dm)I Ym and dm)I Am, are irreducible with signatures: asym := (m; 0. . .), and aanl= ( 1,1,...1 ,O,...) (see problem 1). Y
m-lrrnes
But unlike the case of rank 2, the higher rank tensor spaces Tm (m 2 3) are not made up of their symmetric and antisymmetric parts: dimY'"
Ym @ Am,
+ dimAm = n2 < nm = dim Tm.
There exists a variety of intermediate mized symmetries. We shall see that all
219
55.4. Tensors and Young tableaux.
such symmetries could be built of two basic operations of symmetrization and antisymmetrization (4.2)-(4.3).The idea is to construct a family of minimal projections {x} in the group algebra of the symmetric group. The construction will be based on the important notion of Young tableauz and the corresponding Young symmetrizer. 4.2. Young tableaux and symmetrizers. Given an integer m, we form a partition of m into the sum of non-increasing integers (ml 2 m22 . . .2 mk 2 0): a = (ml;...;mk);m = m,
+ m, +.. .+ mk (k 5 n - 1).
A Young tableauz a is defined as a diagram of k rows, with m, entries in first row, m2 - in the second, ... mk in the kth row (fig. 5). (b)
(a)
2
7
10 12
3
6
9
1
2
Fig.5 Young fableaux of signatures: a = (5;4;2; l ; l ) , m = 13, in the standard allocation (a); and
(I
= (5;4;3), rn = 12 with
a cyclic permufation s
= (1;...;12) (b).
An alternative set of parameters to describe the ath tableau, are k-tuples of integers {r1;r2.. .;rk: r j = mj-mj+l},rk being the number of columns of largest size “k” rk-1 = #columns of size “k-l”, etc. So Young tableaux label naturally conjugacy classes in W,.
serve to build irreducible characters { x a } of W,
At the same time they will
via Young symmefrizers. From the
general theory of chapter 3, we already know that characters of finite groups correspond in a 1-1 way to conjugacy classes.
Each tableau a could be filled with numbers (1, ...m } in m! different ways, according to all permutations OEW,. Our convention will be to fill in the tableaux column-wise, starting from the upper left corner. This would be called the standard allocation, all others can be obtained from the standard one by permutations s E W, (fig. 5 ) .
A simple example of Young tableau is given by a rectangular k x r - box (rn = kr) with an arbitrary distribution of integers { 1;2;...; m } ,
55.4. Tensors and Young tableaux.
220
P = { p } made of all
Given a diagram a we can introduce two subgroups of W,:
horizontal (row) permutations, and Q = { q } made of all vertical (column) permutations, in the standard allocation. If (0,s) denotes any other Young tableau, based on a,then the corresponding subgroups P', &' are conjugates of P and Q:P' = sPs-',Q = SQS-'. In examples (a)-(b) of fig.5 we get (a) the columnar subgroup:
Q 1: W, x W, x W, x W, x W,, is generated by permutations
of {1;2;3;4;5} N W,; {6;7;8} u W,; {9;10} and {11;12}
N
W,; and (13)
N
W,- trivial;
the row subgroup: P N W, x W, x W, x W, x W,, permutes subsets
u u (5). (b) subgroup Q cz W, x W3 x W, x W, x W, permutes {2;3;4) u {5;6;7}u {8;9;10}.. u
u
{ 1;6;9;11;13} {2;7;9;12} {3;8} (4)
while P N W, x W, x W, permutes {2;5;8;11;1}u
...
One can associate with any allocated Young tableau (a+) the corresponding
Young symmetrizer x=xas
x
= s - l ( C(-l)%P)S, (4.4) sum over all q E Q,p E P, where (-l)Qdenotes the signature of q. We call the reader's attention to the order of two operations: columnar anti-symmetrizers {(-l)qq} of
diagram a, followed by row - symmetrizers { p } .
x
(4.4) could be understood as an element of the group algebra e(VV,), supported on the product of two subgroups': QP c W,, or alternatively as an operator in tensor space Tm, representing a suitable combination of tensorial Symmetrizer
permutations p , q E W ,.
3
Examples: 1) Tableaux[ in T4has two subgroups: Q,P N Z, x 7,; generated by pairs: {q, = (12);q2 = (34)}, and { p , = (13); p , = (24)). So its symmetrizer Xa = (1-Q1)(1-92)(1 t p , tP2)l as an element of group algebra e(W,). The corresponding operator on tensors T4, abbreviated by diagrams with vectors {zi;yi} filling in appropriate slots of
bk]
tableaux a , becomes
xa:
bkbq
A Y I @xz A Y Z t 2 2 A Y I @ X I A y2 t z1A y 2 @ %A y1 t 2 2 A Y Z @ X I A YI.
'Notice that the product-set PQ is not the direct groupproduct P not commute in W,.
X
Q,as two subgroups do
221
55.4. Tensors and Young tableaux.
In other words x is made up of two operations: anti-symmetrization of columns followed by symmetrization of rows. Later we shall see that symmetrizer x maps T4 onto an irreducible subspace of SL(n) of signature a = (2,2,0.. .).
[i
2) Tableau 5] has P I IW3 xZ,; with elements { p l E W3(1;3;5); p 2 = (24)}, P, x Z, generated by {ql = (12); q, = (34)). So
and Q N
Xa
cPl),
= P--~l)(~--qZ)PtPz)(
-
a combination of 8 3! = 48 group elements in W,.
w3
In general, Young symmetrizers will provide a family of projections onto irreducible components of dm). To state precisely the result we shall adopt the following terminology: i) Lezicographical ordering of Young diagrams: a = (m1,m2,...) is said to be greater than a' = (mi;mi;. ..), a
> a', if the
first coordinate m j # m; satisfies m j> m i
so one has m, = mi; ...mj-l = m>-,; but m j > m:.
ii) we call elements {s = qp:q E Q;p E P } simple, and say that s,u E W,
are
simply related, s u, if s = u(qp),for a simple qp. We shall see (Lemma 3, corollary 5 ) , that such relation defines an equivalence in W,. N
Theorem 2: i) Any symmetrizer x = xaU, considered as an element of group algebra e(W,) has x*x = p x , with constant p = p ( a ) , depending on a only, so ix becomes an idempotent (projection) in e(VW,). ii) Two symmetrizers
x = xas
and
x' = xatU
are mutually disjoint
x*x'
= 0, if
a # a' ( a > a'), or for a = a', i f u and s are not simply related, u # s ( q p ) . So mutually disjoint projections x;x' correspond to different diagrams a, a', or to unrelated s and u. iii) If a = a' and u;s are simply related, u = s(qp), then x*x' = ps(C(-l)'qp)u-'. The proof is based on a simple combinatorial Lemma.
Lemma 3: If (a,.) and (a',u) are two Young diagrams with either a > a', or a = a', but s and u non-simply related, then there ezist two indices j , k that belong to the same row in (a,s)and the same column in ( c Y ' , ~ ) . Proof of the Lemma: We take indices in the first (longest !) row of (a+)and call them il,i2,... If the first row of a', mi
< mlr then at least two of i's
must necessarily lie in the
same column of ( a ' , ~ )so, the Lemma's conclusion would hold. Assuming mi = ml and
222
$5.4. Tensors and Young tableaux. no i's belong to the same column in a', we can bring all of them by a vertical permutation q1 of a' to the first row, and within the first row we rearrange them by horizontal p 1 in the same order as in (a+).Thus 1" rows of (0,s) and (a',u) become identica1,i.e.s = a q l p l (modulo remaining rows). We continue the process with the remaining rows, each time seeking the Lemma's conclusion to fail. This yields a t the end a = a' and s = u q l p l q 2 p z . .qkpk. Since each p j is a permutation of the j t h row, it
commutes with all remaining terms of the product, and the latter becomes u ( q l q z . . . ) ( p l p z . . .). So the only way for the Lemma's conclusion to fail is when a s
= a' and
being simply related to u, QED.
Corollary 4 Given any pair {(a,s);(a',a)} with either a >a', or a = a', but s unrelated to 0 , there exists a row transposition ~ , E P and , a column transposition qo E Q, so that sp,s-' = aq,a-'. In particular, for each non-simple s # q p , there exist a pair p,, q,, s.t. s p , = qos. Another result could be deduced along the lines of Lemma 3.
Corollary 5: F o r a n y p a i r of elements p E P, q E Q in a fixed Y o u n g tableau a , there exists another p a i r p' E P ; q' E Q, so that simple element qp = p'q'. Hence, simple relation, s N u, is indeed an equivalence, although product QP is not a subgroup of W ,! Proof of Theorem 2: We observe that symmetrizer
x
of Young tableau (a+)has certain
invariance properties w.r. to the left multiplication with elements q E sQs-'and right multiplication with p E SPS-', X(QUP) = PXP
The same holds for
x2 = X*X,
or
= (-1)Qx(.), all
9, P.
x*x',
p X ' ( q 1 p ) = ( - l ) q x * x ' ( t ) , for all q E sQs-';
p E UPU-'.
(4.5)
Now we take a pair qorpo of the corollary and get: X2(qotpo) = - x 2 ( t ) , by (4.5), and x2(qotp,) = x 2 ( t ) , whenever 1 is non-simple by the corollary. Thus
0; for non-simple 1; = x2(e); for simple t = q i Similarly one verifies other statements of the Theorem. As a corollary of Theorem 2 we get a family of idempotents (projections) { x = xos} in
the group algebra e(W,).
Each of them is minimal, so
x
projects group algebra e(W,)
onto an irreducible subspace V = Vas = x * e . Indeed, the argument of Theorem 2 shows
55.4. Tensors and Young tableaux.
223
that x*f*x = p(f)x, for any f in the group algebra e(W,).
x
subprojection $ of
satisfies x*$*x=+=px,
hence $ = p x
In particular, any (a multiple of
x).
Furthermore, all irreducible representations { r = ra*' = R I V) are equivalent, so r = ra in R, d ( a ) , and the norming
(depends only on signature a). The multiplicity of
x = xas are related by p(u)d(a) = m! = I W, I.
coefficients p = p(a) of symmetrizer
Indeed, trace of the convolution operator
x:+x*+,
in the group algebra C(W,),
trx = x(e)rn! = m!, is equal to p . d ( r ) , but the degree of each irreducible
T
coincides by the Frobenius
reciprocity with its multiplicity m ( r ) in the regular representation R, d ( r ) = m(r)= #{disjoint projections
x ( , ~ )of
Minimal projections
algebra
e
x ( , ~ )of signature a).
give rise to central projections
xu on
primary
components 5, of the regular representation,
4EcW, x ( , , I ) = ' l i c s ( c ( - l ) Q Q P ) s - l .
xa =I Projections
{x"}
P
a
are orthogonal, and the family
{x,},
is complete in the center Z(W,)
of the group algebra. Indeed, their number is equal to the number of partitions of rn,
which is the same
BS
the number of conjugacy classes in W,
irreducible representations of W,!
or equivalence classes of
Thus we get at once a complete characterization and
explicit construction of irreducible representations { 7") of W,.
Next we shall apply Young symmetrizers to our main problem: decomposition of tensor products a("') of S f ( n ) .
Theorem 6 (i) Each operator x = x a s projects T"'(C") onto an irreducible subspace TaSof ?r(m)l and the restriction d"')I "range X " N ?ral an irreducible representation of SL(n) of signature a. (ii) The multiplicity of ?rd in d"') is equal to d ( a ) = degree of P. Moreover, the central projection x" of W, projects Tm onto the primary subspace 5, of ?ral invariant under both groups. (iii) The joint action of W, x SL(n) on 5, factors into the product of irreducible representations r" 8 T", and T, N Space(P) 8 Space(?r"). In other words, if r("') denotes the action of W, on 5, ( T ( ~ ) =R @ I , - the n-th multiple of the regular representation), and d"')the action of SL(n) on F , then the tensor-product action of W, x SL(n) is decomposed ag
65.4. Tensors and Young tableaux.
224
~ ( ~~1 r . ( ~R )@=r("') sum over all partitions a: m =
N
p T~ 8 K",
C mj.
The last statement means that groups SL(n) and W,
form a dual (mazimal
commuting) pair in P,by analogy with pairs: {So(3);SL2},or {So(n);SL,} in L2(Wn),
studied in chapter 4. So the algebras spanned by SL, and W, commut ant s:
Com(r(m)1 SL(n))= Alg (W,) Proof: i) Irreducibility of subspaces 05,,
in
'dn)are
mutual
= Alg (T"')).
and Corn(W,)
is established by finding all Weyl ( N - )
invariants in Ttrsand identifying the unique highest weight vector. Let Young symmetrizer of partition a. I claim that
x
x = xas be
a
maps all but one element of the Weyl
basis of invariants {was = s up1wp2...wLk] into 0,
li
0; if a # a' or a = a' but u Xas(Wa'u) = was; if'a=a' and (J = s
#s
Indeed, the basis of Weyl invariants written in the tensorial notation consists of WcI
= (el h %..
or their permutations
elements {el;
11-1
.h ek)
@
swa, s E W.,
p1
.
8 (el h . . h e k - 1 ) '8
,'. ..8
? I:
'l,
The corresponding Young tableau is filled by
...,; e k ] , according to the following allocation scheme, ml times
m, times
[q mk times Obviously, all
IOW
permutations p E P of diagram a
leave w a fixed while each q
multiplies it by (-1)q. Thus xa(wa) = wa, as claimed. For any diagram a' different from a, or unrelated s8 and u there will be a pair of basic elements { e j ; e j ] , that lie in the same
column of diagram (a',s-'u). So the corresponding determinant (wedge product) will be
0. We have thus shown each subspace 5,, = xas[Tm] to have a unique highest weight vector, hence follows irreducibility. ii) Next we take the primary subspace 05, = xa[Sm] and want to establish irreducibility of the joint action of W,xSL(n) on 5,. This would imply the proper factorization of space 5, and representation minimal projection
(dm).dm)) ITa into
xa E e(W,),
the product
7a@7ra.
But each
maps Tm onto an irreducible subspace of G = SL(n).
From the general result for commuting pairs (W,;G)
we get a tensor product
decomposition of part (iii), QED.
h a r k 7 Irreducible representations of G = W,
can be realized in a more direct way, using partitions (Young tableaux) a:m = E m j (ml 2 m, 2 ...). With any a
225
55.4. Tensors and Young tableaux.
we associate a subgroup H = H, 21 Wml x ... x Wmkof G , and a consider two natural 1D representations of H,: trivial 1, and signature Xg=sgns. Let T , and Tb, be the corresponding induced representations of G: T = ind(1 I H;G); T’ = ind(sgn I H ; G ) . All partitions { a } split into dual pairs: atta*, by interchanging rows and columns (so mf = #{mk 2 j } ) (fig.6). a
Fig.6 Dual Young tableau2 {a;,*}.
The key result (Weyl; von Neumann) gives the intertwining numbers of representations {T,} and {Tb,}:
Hence, T , and Th. have a joint irreducible component 7, of multiplicity 1. The degree of P can be derived by careful analysis of intertwining operators, and is equal to m ! n ( t i- e j ) i<j
el!. ..ek!
where ti = mi t i-1.
Remark 8 In the previous section we realized irreducible representations of SL(n) in function spaces TA (of small degree). Tensors and Young tableau give another realization in “large spaces” Tm.There exists an intermediate construction, where the role of W, is played by the group
SL(k) (see [Ze]). Namely, we take a vector space 96 of all k x n
matrices with the natural actions of two groups H = SL(k), and G = SL(n) by the left/right multiplication:
( h , g ) : X = (zij)+h-’Xg = TAJX). This action extends to all polynomial functions
Ym = U ( X ) = C c p X P : P = (Pij), CPij = m), of degree m, and gives rise to a representation
“ ; g ) f ( X ) = f(h-’Xg). The polynomial space 9 , Ym(96) ~ is the mth symmetric tensor power of 96, so zm is the mth symmetric power of
T’.
The reader is asked to show that representation
T”
of G x H
226
55.4. Tensors and Younn tableaux. is irreducible, groups G and R form a dual (maximal) pair in any space Trn(problem 5), hence there is 1-1 correspondence between signatures of I€ and G.
Problems and Exercises: 1. Show that both are irreducible subrepreaentations of r(')(Aint: identify Yz with the space of symmetric matrices and A' with antisymmetric, with the action of G on both spaces by g: X-+TgXg. Find all eigenvectors of the diagonal subgroup). 2. Show that T3 has a decomposition with signatures a = (3,0,. ..) (symmetric); a = (l,l,l,O ...) (antisymmetric); and 2 copies of a = (2,1,0...) (Hint: count Weyl invariants, observe that w 2 ( ( , ( ) w l ( 9 ) - w2(q,()wl(() = w2((,q)wl(() for any triple of vectors (,q,( in C"!).
3. Compute symmertizers of Young tableaux:
(1 ')
and
')
4. Show that xrn splits into the direct sum of irreducible products ra @ xa, where ra and
are irreducible representations of H and G respectively of the same signature a = (ml 2 m2 2 2 mj 2 ...). (Hint: use a version of Gauss decomposition for k x m matrices and show that the highest weight vectors for both groups are the familiar products of minors:
...
rn
rn
M12...p(X PM12...p-1(X) P-'...Ml(X)rnl? constructed from the first p columns of X, then first p-1 columns, etc.).
5. Show that the natural action of groups SL(n)xSL(m) in space 96= Matnxrn is irreducible; groups SL(n) and SL(m) form a maximal dual pair in any polynomial (symmetric tensor) space Pk(96).
6. Use Young tableau and formula (4.5) to classify irreducible representations of symmetric groups W,; W,; W,. Find their characters and degrees. Compare the results for W, with f3.3 (example 3.7), identify Young tableau of { 1; sgn; x';7r3+;ir3-}.
55.5. Haar measure on compact semisimple Lie groups.
227
$5.5. Haar measure on compact semisimple Lie groups. We compute the Haar measure on compact Lie group, reduced to conjugacy classes of C,taking as a model group Syn). This result will lay the ground for the celebrated Weyl character formulae in 55.6, that lie at the heart of the representation theory of classical compact groups ($5.6).
We shall assume G to be a subgroup of GLn(C), so its Lie algebra (5 = T , tangent space at {e}, could be identified with a subalgebra of M a t , = gqn). The tangent space at any point g E G is then T, = go. Each element X E 8 (tangent vector at {e}), extends to a left-invariant vector field < ( g ) = gX (matrix product). To construct an invariant volume element on G, we can choose a basis of left invariant vector fields, or the corresponding left-invariant differential 1-forms: w j = C a j k ( s ) d s k ,where {xl;...;xN: N = dirnG} are local coordinates on G (vector fields could be identified with 1-forms via a left-invariant metric). Then we define an invariant volume element, as a differential N -form W
=
N 1 Wk
= det(ajk)dzlA
... A d X N .
A canonical way to construct {wk} is to consider a Lie-algebra-valued differential 1-form, called gauge potential = g-'dg = g-'ak( g)dZk. At each point g in the group, R takes on values in the cotangent space at { g } , and is clearly left-invariant, (gog)-ld(g0g) = g-ldg. Choosing a basis {el;... e N } in 8 and expanding R in this basis, we get a basis of scalar G-invariant 1-forms, = CWkek; needed for the volume element (5.1). It is an easy exercise to directly verify (via the group identity g*g = I) that for unitary/orthogonal groups Syn), Sqn) matrixdifferential g-'dg is skew-symmetric, dgjk = - dijkj, so dg takes on values in the Lie algebra s 4 n ) or d n ) . The above procedure provides a general construction of the Haar measure dg on matrix Lie groups. However, for the purposes of analysis we often need an explicit form of dg (or w ) in a suitably chosen coordinate system on G. Here we shall do it for the unitary groups yn),Syn), and more general compact Lie groups G, based on the structure theory of 55.2. These results will be needed in the section on Weyl character formulae for irreducible representations of compact groups G.
228
$5.5. Haar measure on compact semisimple Lie groups. ~~
Since charact,ers depend only on conjugacy classes, it suffices to compute the reduced Haar measure on conjugacy classes of G. We denote by H the space of conjugacy classes: K , = {g-'hg:g E G}, and choose coordinates h = (hl; ...;hm) on H, as well as transversal variables y = (yl;...) on each class K , 11 G/Gh (G modulo the stabilizer of h, G, = {g:g-lhg = h}). We would like to decompose the Haar measure on
G, as . d g = p(h)dhd,Y, (5.2) where dh(y) denotes a G-invariant measure on K,; dh - some natural measure on H , and the density factor p ( h ) measures the relative size of conjugacy classes {K,:h E H}. Our goal is to compute factor p ( h ) in (5.2). We shall do it first for group SU(n). Each unitary matrix u is conjugate to a diagonal (Cartan) matrix
['
i0,
h=
eie2 eie,
1,
u = vhv-'.
(5.3)
Matrix h is unique, modulo all permutations of diagonal entries, i.e the W e y l group action on SU(n). Thus a generic conjugacy class is identified with a point in a "positive cone" in the torus, H = { 8 , > 8, > ... > 8,) = U"/modulo W,, the so called Weyl chambe?. Factor v in (5.3) is also non unique, v E SU(n)/Un, since stabilizer of a generic diagonal element h is T". So one can think of G (or rather a dense open subset in G), as decomposed into the product: G N H x G/Tn. To find invariant integral in coordinates (h,v) we differentiate the relation: u = v-lhv, d u = ( - v - v v v-')hv and multiply it on the right with u-' = v-'h-'v du u-1 = v-'(dh
h-1
+ v-'dh v + v-lh
+ h dv v-lh-1-
dv,
dv v-I)..
Introducing right-invariant cotangent vectors 6u = duu-', written, as 6 u = v-'{6h ( A d , - 1)6v}v.
+
the latter can be
(5.4)
@'
Equation (5.4) represents the Jacobian map of the coordinate transformation: We need to compute the determinant of @'. Modulo unessential v-
@:(h;v)-u.
21n genera1 a Weyl chamber A c 5, in a semisimple algebra 8 with a fixed system of positive roots C, = {a},consists of all { H E a:(aI H ) > 0, all a} (fig.7).
55.5. Haar measure on compact semisimple Lie groups.
229
conjugation (which has no effect on detdj’), the Jacobian map consists of an identity block in h-variables (dim = n - l ) , and a h-dependent linear block Adh - I , in the complementary variables 6v = { 6 ~ ; ~ ) . Variables {vij} can be chosen so that the diagonal part 6vjj = 0, which follows from the decomposition of the Lie algebra into the sum of its “diagonal” (Cartan) . matrix entries { b U j k : j 5 k}, as part, and “off-diagonal” (root) parts $ @ 8 Regarding generators of independent coordinates on G,and writing hk = eiBk,we get the formula for the Haar measure on G
4.)
Here dv represents the invariant volume on the quotient-space SU(n)/Un, the determinant of (Ad, - I) being squared, since each off-diagonal entry v j k represents a complex variable. Finally, remembering that all off-diagonal elements { 6ui = 6vij} are eigenvectors of the adjoint map Ad, (root vectors in @),
(Ad,-I)bvjk=
(hj/hk-1)6vjk
= ( e i(ej- ‘k)
- lYvjk;
we can write the determinant factor in (5.5) explicitly in terms of parameters 8, /(i(’j-’k) j
-I)[ =
n
(:(‘j
- ’k) - l x e - i(ej- ’k)
j
- 1) =
n
4sin2f+).
(5.6)
j
In this form the result easily extends to all simple/semisimple compact Lie groups G , the role of diagonal matrices being played by the Cartan subalgebra $ c 0, and pairs (ij)being replaced by positive roots { a } of 0. Precisely, we decompose G into the product H+ x G / H , where H = exp$ is a Cartan subgroup (maximal torus) in G, $+ denotes a positive Weyl chamber in 8, a cone made of all elements { X E $:(a I X ) > 0}, for all positive roots a E C+ (see fig.7), and H+ = exp$+- the image of the Weyl chamber in H. The key to a decomposition lies in the fact, that generic u E G can be conjugated to a Cartan element h, u = vhv-* determined uniquely, modulo Weyl-group3. The relation (5.4) is derived the same way as for SU(n), and determinant is evaluated in terms of positive roots. The final result can be stated in terms of variables h = expie, B E 8,and v E G / H ,
3A Lie group analog of diagonalization of unitary matrices.
230
$5.5. Haar measure on compact semisimple Lie groups. ~~~
~
In this form the result will be used in subsequent sections.
Fig.’l: The Weyl chamber of 4 3 ) (shaded) consists of all weights A, thai make posiiive producis wiih ihe posiiive basic roots: (A ).I > 0; (A I P ) > 0.
Problems and Exercises: 1. a) Compute the Haar measure for other classical compact groups: S q n ) and Sp(n). b) Compare (5.6) with formulae of 84.1 for groups SU(2) and Sq3).
55.6.The Weyl character formulae.
231
$5.6. The Weyl character formulae. Our main goal in this section is to derive the celebrated Weyl character formulae for irreducible representations of compact semisimple Lie groups. Most of the discussion below will concentrate on groups SU(n);U(n) (respectively SL, and GL,, but whenever possible we shall indicate connections and extensions to other compact groups.
6.1. First Weyl formula. Irreducible representations { T " } of group G = SU(n), or U ( n ) were described in 55.3 in terms of their signatures (highest weights) a = ( m l ; m 2... ; m,). The latter means an ordered n-tuple of integers4: m, 2 m, 2 ... 2 m,. We denote the character of P by xa = ha. Since function x = xa is conjugate-invariant on G, and any element U E S y n ) is conjugate to a diagonal matrix h = diag(h,;. ..;h,), it suffices to evaluate xa on the diagonal (Cartan) subgroup D N I F 1 , or T" (for SU(n) and U(n),respectively). We shall start with two special cases: the symmetric Yk(r)and antisymmetric A k ( r ) tensor powers of the natural representation r in C". Their characters will be denoted pk and q k respectively. In both cases we know explicitly all weights { p } and weight vectors: 0
For symmetric space Lpk(C") realized by polynomials of degree k in n variables,
the weight vectors are monomials { z p :/3 = (k,;...;k,)}, and the corresponding weights are tuples of multi-indices /3 of norm l/3l=k. Each has multiplicity 1, hence Tk(zl;...;z,),
Similarly for antisymmetric A'((c") weights are labeled by ordered tuples i = (i, < i, < ... < ik), ha = h . ...h. , with the corresponding weight vectors: e . A ... A e. . ak 21 zk Hence, Qk(h)= = . hi,.. .hik. 0
Chi a
21<
c...
<2k
In general, each character ~ ( h=) xa = trra of SU(n), restricted on the diagonal (Cartan) subgroup D = {h:h= (eiel;...;e"n)} becomes a trigonometric polynomial a,eim . , with integer coefficients a, = multiplicity of the mth weight in the weight diagram of ra. Unfortunately the multiplicities {a,} are not easy to compute, so we
C
4We
remind the reader that different n-tuples a yield different representations of U(n), but for
Syn), only the differences {(...;mi- mi+,;....)} matter. Indeed, shifting all m j by any p , m j - + m+j p , will multiply 7ra by a central character: {e*PzZ) for z = zZ, which is trivial on Syn). So signatures LI
can always be chosen, context.
90
that C m j = O , or m, = O . We shall use both choices, depending on the
$5.6. The Weyl character formulae.
232 shall proceed differently.
Let us observe that each character is invariant under all Weyl transformations
u E W,which in case of SU(n);U(n)coincides with the symmetric group W,. Indeed, all Weyl elements u belong to G itself, so aha-' = V ,for any diagonal (Cartan) element h, which yields arn e i m . 8 = X(h). X(V) = a,eim * ou =
c
Therefore all Fourier coefficients {a,} with m in the same W-orbit w = {mu: uE
W} are equal, a , = amu= a,, for all m, u. * as the sum of orbital terms: = C eim" , i.e. m E u. a , ~ , summation , over all W-orbits w in the weight diagram, with integer
Thus we can rewrite
x=~
x
x,
coefficients a,. Next we shall apply the orthogonality relation for irreducible characters:
(x1 x) = / x
xdg = 1, dg - normalized Haar measure on G = U(n).
G
Due to conjugate-invariance of f =
(6.2)
I x I ' integral (6.2) can be reduced from G to
the set of conjugacy classes. The latter as we already know are parametrized by points of the quotient space
D/W (diagonal subgroup D=T"-',
modulo Weyl group W), or by
' 2 . . . 2 8,) in D. a "positive cone" Dt = 3JY-l = (6: dl 2 6 The reduced Haar measure was calculated in s5.6,
where 0 = (el;...en) parametrize the Cartan (maximal) torus D c G (any u E G can be i8
h = h ( u ) , with eigenvalues { e k } , ordered lexicographically: 8,>8,>. .., and v E G/D implements the conjugation u = v-lhv). We conjugated into a diagonal matrix
shall recast it in a more convenient form using the well known expansion of the vandermond determinant,
A(h)=
n
j < k
i8'
(e
3
-e
i8
k)
= det
L
1 e
... . . . . . . .. . . .
1 ,ion
. ..ei(n - 1)'n
The determinant can be expanded in the usual way
55.6. The Weyl character formulae.
233 n-1
sum over all permutations 0 . In case of Syn)product ( f i e i e k ) F can be pulled out 1 from each term of (6.4) and get it into a familiar sum of exponentials (cf. chapter 4), exp$(n-1)8,(,)
+ (n-3)eO(,)+...+ ( ~ - n p ~ ( ~ ~ .
(6.5)
Exponentials of each term (6.5) represent a particular positive weight p on D, (transformed by u E W), namely the one-half sum of positive roots: p = 21 H j , = ;(TI - 1;n-3; ...;1-n).
C
j
On su(n) weight p can also be represented by a tuple (n-1; n-2;. ..l;0), due to the zero-trace cancellation. Thus the reduced Haar measure c~ be written as
Next we substitute character
x = Ea,xw, written as the sum of orbital terms eim.O, xw -
c
mEw
and the Haar measure (6.6) into the orthogonality relation (6.2). Let us observe that the density of the reduced Haar measure (6.6) factors into the product AS& with A(h) being equal to an alternating orbital sum,
x a,
So the integrand of (6.2), x A . combines the products of symmetric and antisymmetric orbital sums: x w ~ w lcorresponding , to various orbits w in the weight diagram of a , and a special orbit w' = w(p) of weight p. +
Let us compute the product of two such orbital terms:
xwkwJ,where orbit
w = w(A) comes from a positive weight A, and w' = w(p). We find
Orbits {w(h'+p)} with different weights X may coincide, but picking the highest weight a in the original w, the orbit w(a'+p) will be "highest" among all {w(A"+p): h , ~ }and unique! In particular, taking the highest weight a of a representation, the product X w ( a ) L J = %(.+p) bwxw;
+
c
zw(a+P),
breaks into the sum of the "highest orbital term" and some lower terms {xu} with integer coefficients b,. It remains to observe that different orbital terms {g,,} are orthogonal on torus D = U"-l with respect to the usual "flat" Haar measure dt9,
234
$5.6. The Weyl character formulae.
where 1 w I = n!,for the highest-weight orbit w. Therefore,
must be equal 1. But the latter is impossible unless all lower orbital terms {xu} cancel each other, i.e. all coefficients b, = f a,’ are zero. This means that the product x . A consists of a single term j;. where a is the highest weight of the representation A, 4atp)’ and p = :usurn of positive roots”. As a consequence we get
The Weyl formula was written here in a general form, valid for any semisimple compact Lie group? with product (. I .) given by the Killing form on the Cartan subalgebra of (5. From (6.7) we shall deduce degrees of irreducible representations { A * } . Let us observe that (6.7) represents the ratio of two vandermond determinants. Indeed, passing to variables { h, = e l k } ? =
where a = (ml; m,;. ..mn). Here and henceforth it will be convenient to replace signature a = (ml; ...mn) with another n-tuple l , whose components { l j= m j t 71-j}, so l and a differ by p. We shall also introduce a column vector
Ze =
[ fi?
the lthpower of
h’.
(6.9)
hi1
Then (6.8) can be written as (6.10)
Once character x is known the degree of the corresponding representation A is obtained by evaluating x at {e}. However, formal substitution 6 = 0 (respectively h, = h2 = ... = hn = 1) in (6.8), results in indeterminacy To resolve it we shall make a
8.
s5.6. The Weyl character formulae.
235
substitution: h, = z"-l; h, = znW2;...hn = 1, in terms of parameter z. The numerator of hmn+n-2.
(6.8) now turns into the vandermondian in new variables wl= hrn+n-l; w,= > ... w, = h,m", while the denominator is equal to the vandermondian of {zn-l; ...;1 ) . Thus numerator =
(z
mj+n-i
-2
nk+n-k
); denominator =
i
n
(zn-j- zn-k) .
i
Pulling out ;suitable power of z from both products we-rewrite the ratio as
Passing to the limit as z+l we obtain the degree of xa
(6.11)
Notice that the RHS of (6.11) consists of the Killing products of positive weights a s p and p with all positive roots p of $n). In such form (6.11) extends to all semisimple Lie groups and algebras. 6.2. The second Weyl character formula. The derivation will be based on a remarkable matrix identity due to Cauchy.
Lemma 1: T h e determinant of the n x n
matriz depending o n 2n parameters
(6.12) where A(h), A(z) denote vandermondian difference-products
n ( h j-
h k ) and
Proof of the Lemma proceeds by induction in n. Step 1". Subtract the 1" row from all the others, the resulting j k f h entry becomes zk(hj-
hk)
( 1 - h j z k ) ( l - hlz,)' Pulling common factors ( h j- h,) from j t h row and the n x n determinant as
II ( h j - h l ) I 1
j>l n
1 from k t h column we express 1 - hlzh
55.6.The Weyl character formulae.
236
Step 2'. Now we subtract the 1'' column from all the rest. The resulting j k i h entry is then 'k - '1 (1 - h j ~ k ) ( l -
hjzl)'
Pulling out common factors: (zk - zl) from the kth column and row, we reduce the n - t h determinant to the (n - 1)-st,
det,
1 from 1 - hjzl
the j f h
ll ( h j - h l ) k F l ( ' ~-21) del, j=1 n (1 - h j z l ) i l ( l - ' 1 ' ~
= I,>
whence the result easily follows.
The Cauchy determinant K(h;t) = det 1 can be expanded in two different [1-hj.x] ways. On the one hand we shall see that
XI
1I
1,
(6.13) ~ ( h2);= j$. ..$en 2'1 ...Zen summation over all tuples of indices l,>e2> ...>en. On the other hand the RHS of (6.12) will be expanded into a similar series in "determinantal powers of {te}", but with different coefficients: symmetric characters { P k ( h ) } (6.1). Precisely, the RHS of (6.12), divided by A(h),will be shown to consist of (6.14) as above the summation extends over all symbolic notations p'e;p'e - ... for rewrite it in the following compact form
.t = (el>&> ...>en). Using (6.14) (k = 0,1,. . .) we can
(6.15)
Now it remains to compare the coefficients of t-terms in expansions (6.13) and (6.15), and to remember the 1-st character formula (6.10), to get
(6.16) a n n x n determinant whose entries are symmetric characters:
p k ( h ) = c hP, and signature a = (ml; ...mn) IPkk is related to a n n-tuple l = (el;...en) through the weight p = ( n - 1;n - 2; ...0) -
55.6. The Weyl character formulae.
f x sum of positive roots: el=ml t n-1;
rn, = ez+n-2;.
237
..;mn=en.
Warning! Some indices m = Q,-n+j in (6.16) could turn negative. The convention here is that the negative-order symmetric characters p , (rn < 0) are assumed to be 0, so they won't enter expansion (6.16). Formula (6.16) blends once again two fundamental tensorial operations of symmetrization and anti-symmetrization to produce in an intricate and somewhat mysterious pattern all irreducible characters of SU(n). Curiously the role of two operations is now interchanged compared to the previous sections (55.4-5). There we built all irreducible {T*}, based on symmetrization of all antisymmetric tensor-powers {.rr(') in Ak(C"):1 5 Ic 5 n}, as generators (spaces Sx). Here irreducible { T * } arise from "anti-symmetrizers" (6.16) of various symmetric tensor-representations. To show (6.13) we note that each column of matrix expansion in powers of
Zk,
1
has a geometric series -hjz*]
whose coefficients are powers 6.9) of column h
](i+z,i+z:iz+
I).+
...I; (i+rzi+@ +...);... ; ( i + z n i
Then we expand (6.17) in multi-powers of
(6.17)
...zmm) and collect terms with the identical
(tl;
h-determinants. This results in a double-determinant expansion (6.13). Since h-terms of
(6.13) are precisely the numerators in the 1-st character formula, the Cauchy determinant
1"
could be called a generating function of irreducible characters
{xa}.
To derive
6.15) we shall expand the RHS of the Cauchy identity (6.12). First observe that for each
t
= t k the product (6.18)
with symmetric characters {pm(h)} (6.1). Multiplying expansions (6.18) for different
z = zl;zz;
...;t , we get
,ji+=
c
c 2m;
PmlPmZ***Pm,
(6.19)
T€W
)k=l
where m = (ml; ...; mn) denotes an ordered tuple of integers (signature), and summation extends over all such tuples. We multiply (6.19) by the vandermondian difference-product
A(z) =
c
(-1)' z ' ( ~ ) ,
S€W
with the ''&xm positive weight" p = (n-1;. ..;O) to bring the RHS into the form
(6.20) Inner sum (6.20) consists of alternating orbital terms corresponding to various (non ordered) weights
e = (el;.,.;en)in the form e = p + mu. Notice that only non-degenerate
$5.6. The Weyl character formulae.
238 P s (i.e. t j #
ek, all j,k) will give a nonzeros contribution
to (6.20). Using the symmetry
of each product pml...pmn with respect to permutations r: m+m7 = r ( m ) , we can bring (6.20) in the form (6.21)
Here we abbreviated the product notation pml.. .pm as p(,,,). The outer sum in (6.21) n
extends over all (non-ordered) &tuples, that are related to m-tuples via p and
T:
..en) the unique highest tuple in the orbit of e , (i.e.
e = mr+p.
We denote by C' = (el;.
el>...>&),
and by u the permutation that takes C to
e',
and rewrite (6.21) as the sum
over all ordered t-tuples (6.22)
Here we utilized once again the total W- symmetry of the inner sum. Each of two inner sums (6.22) collapses into a n x n determinant in variables {rk} and { h k } (or p k ( h ) ) , whence follows (6.15).
Remark 6.2: Characters have simple transformation properties under the = xT operations of the direct sum and tensor product of representations: xT = xTxs. The second Weyl formula gives a formal expression of arbitrary xT
+ xs; xa in
@
terms of products of symmetric characters pk. This expansion can be "lifted" to representations. In ot,her words we get an expansion of any irreducible representation ?ra of SL(n) in terms of symmetric representations {rk = Yk(r)}E0. Namely, (6.23) each entry in the formal determinant (6.23) stands for a symmetric representation rk (in tensors of rank k) of signature (k0;...0), and f signs mean the direct sumldifference of representations.
Example 8.3: A representation of signature a = (4,1,0) of SU(3), respectively .t = (6,2,0) can be represented by the tensorial determinant
l o 0 *"I
(6.24)
Here ?yo denotes the trivial representation, two slots (3,l) and (3,2) in the matrix (6.24) are filled with zeros, since they have negative orders: -2,-1. Notice that 7r5 c s4@ d,since Y5(Cn) is naturally embedded in J4 my1. The degree of aa can be computed in 2 different ways, by the 1-st character formula (6.11),
239
55.6. The Weyl character formulae. (6 - 2)(6 - 0)(2 - 0) d(sQ) = (2 1)(2 - 0)(1 - 0)
-
= 24;
on the other hand from 2-nd formula (6.24)we get
d(ra)= d ( x 4 ) d ( d )- d(r5)= 15.3 - 21 = 24.
Additional comments. Our exposition in 55.6 followed [We31 and [Ze]. We shall conclude this section with yet another interesting formula for irreducible characters and degrees {d(cr)} (see [Ze], [Ta2], [Sim]). It can be stated in terms of the highest weight matrix entry,
I u,),
+,(s)= +(,
where u, is the highest-weight vector. Functions {+,}
have the property,
= +u+o,
+a+o
a consequence of the tensor multiplication rule: 77, @
= xu+@
@
... (lower weight T’s).
If we choose a system of basic roots {el; ...en}, n = rk(O), call 4 . = +a ., and write any ’
(I
J
as a linear combination of {ej} with integral coefficients, (I
then clearly,
=
Crnjej;
*, n+Tj. =
The function
+,
plays a role in the study of quantum partition function for hamiltonians,
associated to Yang-Mills potentials. But it also has an interesting connection to the character
x,.
Namely, x,(z) = d ( a )
+,(s- l”9) dg;
G
also the degree of ra is given in terms of
+, (6.25)
There is no direct link of (6.25) to the classical l-st and 2-nd Weyl formulae, so its meaning remains obscure.
$5.6. The Weyl character formulae.
240
Problems and Exercises: 1. Verify Weyl formula for characters p k and qk of the symmetric and antisymmetric tensor powers, Y k ( i r ) and A k ( ~ ) .
2. Write the 1-st and 2-nd Weyl character formulae for the orthogonal and symplectic groups W n ) ,
Mn).
3. Compare the 1-st and 2-nd Weyl formulae with the results of chapter 4 for characters of SU(2) and Sq3). Verify formula (6.23) for Sl42), using the Clebsch-Gordan decomposition of f4.4. (Caution: in chapter 4 we used a slightly different labeling of SU(2) representations, so the present "integral" signature ck = (ml;m2) would correspond to weight m = $ml-mz) of chapter 4). 4. Find characters of all irreducible representations of JIn;C), considered as real Lie algebra (any such is the product of a holornorphic and antiholomorphic representations ira
55.7. Laplacians on symmetric spaces.
241
55.7. LaplacMns on symmetric spaces. Symmetric spaces are defined as Riemannian manifolds, whose curvature tensor is invariant under all parallel transports (see Appendix C). E. Cartan posed the problem to classify all such spaces. By two ingenious arguments he gave the problem a group theoretic formulation. His first approach was baaed on the holonomy group of A. The holonomy group K a t point z E A consists of the parallel transport operators on tangent space T, along all closed path {y} through 2. Clearly, different points z E A give isomorphic groups K. The Riemannian metric is always invariant under K, and for locally symmetric A, the curvature tensor is also K-invariant. Hence, it follows from the Cartan structural equations (Appendix C), that each holonomy u E K induces a local isometry in a neighborhood of z,that leaves z fmed. This yields some algebraic relations between Lie algebra 0 of group K, and the metric and curvature tensors, g and R on A, g(AX;Y)+g(XAY)=O, for all A E R ; X , Y E T z . [A;R,(X,Y)]
= R(AX,Y) + R(X,AY); and R ( X , Y )E R.
Cartan showed that any R satisfying such relations and the standard symmetry condition of the Riemann tensor, defines a locally symmetric space. Thus he waa able to reduce the problem to (i) classification of all possible holonomy groups of symmetric spaces; and (ii) derivation of the curvature tensor on symmetric spaces in terms of their holonomy groups. The second Cartan’s method is based on another important observation: the curvature tensor is invariant under all parallel transports iff each geodesic symmetry y(s)-i7( - s), at 2 E A defines a local isometry on A. Here y = y(s;() denotes a geodesic through z in the direction ( E T,. The latter gave meaning to the terminology “locally symmetric”. This result becomes particularly significant for “globally symmetric spaces”, where each local symmetry extends through the global symmetry of A. Hence, any globally symmetric space possesses a transitive group G of isometries, and could be represented as a quotient K\G. Subgroup K c G stabilizes point z in A, and itself consists of fixed points of an involutive automorphism 0 (that results from the geodesic reflection). Group G becomes semisimple, when one drops a “trivial (Euclidian) factor”. The problem is then reduced to the study of involutive automorphisms of semisimple Lie algebras.
7.1. Compact symmetric spaces. In this section we shall be mostly interested in compact symmetric spaces A = K\G i.e. quotients of compact semisimple Lie groups
G, modulo a subgroup K , which stabilizes an involutive (Cartan) automorphism 0 of G, K = { u E G:0(u) = u } , 8’ = 1. Automorphism 0 splits Lie algebra (5 of group G into the direct (orthogonal) sum of the subalgebra Si of K , and a subspace !+ identified I, with the tangent space at zo = { K } , other tangent spaces5 { T z : z= xi}, being given by the adjoint action of G, T, N Adg(?), applied to 8. Furthermore, automorphism 0 takes on value f 1 (0 R N I and 0 @ 21 - I),so the resulting Lie brackets between the Si and !$ take the form:
242
55.7. Laplacians on symmetric spaces.
Space !# contains a maximal abelian (Curtan) subalgebras Z 2: Fir, whose dimension n = dim% is called the rank of A. The image of Z under the exponential map O-G, forms a maximal geodesicdy flat torus 7 = expZ 2: T" in A. Space On of all flat n-tori ( 7 ) is itself a smooth manifold, whose dimension depends on dimG and its rank. Group G acts on On,turning it into a homogeneous space K,\G, KO- stabilizer of 2 in G.
7.2. Restricted root system. The abelian part Z of !# could be embedded into a larger Cartan subalgebra $ of (5, in fact, $ can be chosen as the sum Z@1I(Cartan parts of !# and A). Algebra Z breaks space (5 into the sum of eigensubspaces @@, which lie in !# @ A (but not in either of two summonds). We want to define a restricted root system C = {a}in the !#-component of 0. To this end we square adjoint operators {ad&:HE %}, and note that squares map !# into itself by (7.2). Hence, !# is broken into ( p a@!#-,), where adh I !#*, = a(H)'. As in the case of entire O, the direct sum: @ a the restricted root system C(!#) can be ordered, i.e. split into its positive and negative parts C+lJC-. It has a basis of positive roots {a1; ....on}(n=rk!# = dim%), the (restricted) Weyl group W(!#), etc. In one respect, however, root systems C(O) and C(Sp) differ significantly, the former was shown in 55.2 to be multiplicity free, i.e. dim@, = 1, for all a # 0, whereas the latter has typically higher multiplicities { m a2 1). In fact, each root a of !# can be extended through a root of (5, and m, measures the number of such expansions. Examples below will illustrate the general concepts.
7.3. Classification. Symmetric spaces (compact and noncompact) were completely classified by Cartan (see [Hell), who discovered their close connection with semisimple Lie algebras and groups. Here we shall list the basic examples of so called irreducible globally symmetric compact spaces.
'One can associate with any compact pair {@R} a noncompact (dual) pair {a*$} (in the terminology of [Hell), where 8 = R 8 8, 8*= R @ iq c BC - a complexification of 8. The standard example of a dual pair in Lie algebras 8 = 8 (n) 3 R = 0 fl (n), and @* = al(n; R) 3 8 fl (n). The @Killing form on 9 (better to say, on its dual iq c 8 ), defines a G-invariant Riemannian metric on A, whose curvature tensor is obtained from the Lietracket,
R(X,Y).Z= -[[X;Y];Z], for X,Y,ZE $ ! = 8,. (7.2) Notice that the curvature tensor is always anti-symmetric relative to the metric {g(z)}. So operator R(X,Y)belongs to the Lie algebra of the holonomy group K of {z}, and the triple bracket (7.2) does obey the basic properties of the curvature tensor (Appedix C). Indeed, [[!&!$];!$I c !$, and operators {adi,,,l:X,Y E q} belong to the adjoint Lie algebra R, acting orthogonally on 8.
65.7. Laplacians on symmetric spaces.
243
Table of symmetric spacea.
I Space
lnvolu tion
Rank
6) SYn)/SO(n)
& X + X (complex conjugation)
T
(ii) SYZn)/Sp(n)
r=n-1 where J, is symplectic matrix [-I in C Z ~
e ( X )= J , X J ,
l,
=n -1
1'
the matrix of the indefinite P P+cl (p;q)-form: C z .y . - C z .y . 1 p+l " in c P + ~ ;
e(x)= Ip,XI,,,
same I,, as
r = rnin(p;q)
above in WP'q;
The are a few other examples, related to exceptional simple Lie algebras. More general symmetric spaces can be decomposed into products of the irreducibles ones: ALl x AL, x ... The simplest example is the product of 2-spheres (S2x SZ)/Z2,a quotient So(4)/So(2) x Sq2) (problem 5).
7.4. Laplaciana on manifold and symmetric spaces. We recall the definition of the Laplace-Beltrami operator A on a Riemannian manifold 96. Let (gjj) denote the Riemann metric tensor on tangent spaces, {Tz},and (g'j) - the dual metric on .. cotangent spaces {Ti}, matrix j = (ga3)= (gij)-'. So the metric (arc-length) element,
C
ds2 = . . gijdz'dzj, a3
in local coordinates z = ( x i ) on 96, while the volume element of g has the form,
d m . 4GiJ The g-gradient and g-divergence operations on functions { f} and vector-fields d T / = A = d m d z = Jdx, where J =
244
55.7. Laplacians on symmetric spaces.
{ F } are given by
of = ( C g ' j a j f ) = B * a f ; v . F = ) C a k ( J f k ) = ~ . ( J F ) . k
j
One can easily check that operations V and V - are dual (adjoint) one to the other relative to the L*-inner product with the volume element, (Of IF)= j V f . F d V = - / f V - F d V , (7.3) the dot-product in (7.3) clearly means the (gij)-product on tangent-spaces {Tz}. Let us remark that any coordinate change, z=$(y), takes tensor (gij) into g4 = 'AgA, and +A-'~J(~A-'), where A = 4' denotes the Jacobian of map 4. Hence one could check that all the above definition do not depend on a particular choice of local coordinates (z). The g-Laplacian is a product of operations g-div and g-grad,
Once again the definition is coordihate-free. For the purpose of analysis on symmetric spaces we shall need the so called multi-polar (or generalized radial) coordinates on 96. A natural framework for
introducing such coordinates is a manifold 96 with a compact Lie group K of isometries, like symmetric space 96 = K\G. We choose a transversal manifold 4 c 96, that cuts across each K-orbit S at a single point t, and assume (without loss of generality) that 4 is orthogonal to all orbits {S= S,}. Manifold T will play the role of multi-radial directions, while orbits {S,} will form a family of spheres of multi-radii t. The following examples will illustrate the concept: 1) R" with the natural SO(n)-action, where 4 = R+ (half-line), while S = S, are standard spheres of radii T 2 0; with S q n ) acting by mi-symmetric rotations. Here 4 is a 2) sphere S" c great circle (in fact, semicircle) through the "North pole" of S", while family {S,} is made of horizontal (transverse) spheres of "spherical radii" r(t) = t - angle, respectively Euclidian radii r ( t ) == sint (see fig.8). 3) Hyperbolic space
W", is usually realized
as a one-sheet hyperboloid
{ ( r' zI . ) : ~ :C.3 = I} in Wntl. Once again Sqn) acts on W" by mi-symmetric rotations, and foliates it into a one-parameter family of orbits {S,} of radii r = sinh t.
245
55.7. Laplacians on symmetric spaces.
Fig.8: Orbit-spaces of the n-sphere and n-hyperboloid relative lo So(,)-action.
We remark that all 3 represent model examples of rank-one symmetric spaces with symmetry-groups: M, = R” D S q n ) ; S q n t l ) ; S q l ; n ) , and the maximal compact subgroup K = S q n ) . Here radial manifold T is one-dimensional. But
S = K\G
higher rank symmetric spaces provide examples of multi-D radial part.
4) Group S q p ) x Sqq), p 5 q, acting on the flat space‘ 96 = Matp q, (u, v): X-bu- lXV, has the radial (transverse) manifold, made of diagonal matrices
{x, =[
11
...
]
t~
t = (t,;...t p ) ;t , > t , > ... > t ,
]
(7.4)
Given a manifold 96 with compact Lie group K , acting by isometries, and a transverse radial manifold 4, we pass to new variable { ( t ; O ) } on T x S , (orbit
S N K,\K), and find, ds2 = dt2
+ C g i j ( t ; O )d0’dOj;
where g1 denotes the restricted metric on K-orbits {S = S t } . Hence, the volume density
J =
,/-,
will depend on t only (due to K-invariance of the metric g1 on St). In
fact, factor J ( t ) will represent the Riemannian rn-volume (rn = dims)of the orbit. The Laplacian in “multi-polar” coordinates,
+ a,. g’(t; o p e ;
A = +atJa,
(7.5)
and its radial part,
‘Let us notice that space %VMat,-, space SO(p+dlSo(p) x So(d of type IV.
represents a linearization (tangent) of the symmetric
246
$5.7. Laplacians on symmetric spaces.
Here A , denotes the natural Laplacian on 2. We shall demonstrate both formulae with a few examples. 7.5. Rank-one spaces: Rn; Sn;H". Here the polar variable r measures the geodesic distance from the given point (pole) to z = (r;O).The metric tensor in polar coordinates is dr2 + r2de2;
R"
dr2+sinh2rd02; H"
Hence the density factors and the radial Laplacians become,
J ( r )=
i
r"-'; on Rn
sin"-'r; on S" sinh"-lr; on H"
; A, =
i
8:
+ +r;
on R"
+ (n-1)cot ra,; on S" 8: + (n-1)coth dr; on H" 8:
7.6. Higher rank spaces. Turning to a general symmetric space 96 = K\G, we choose a maximal abelian subalgebra Z in 'Ip (dim% = n), pick a system of positive roots in 2,{ a E C+(?$)}, or zE+(v), in the compact case, and consider a Weyl chamber in Z (fig.71,
A
= {H: (aI H) > 0, for all a E C,}
Exponential map, exp:'Ip+%, takes A into a flat (totally geodesic) manifold 4 c 96, that cuts transversely through all K-orbits. So here parameters { t = (tl;...;t,)} are coordinates of element H E A in a fixed basis of simple roots { a l ... ; an}, t j = ( H I aj).
To understand the role of the Weyl chamber, as "radial parameters", we first turn to the linearized case, i.e. space 'IpcC, with the adjoint action of I<, {adK ISp}. Here a generic K-orbit S punches the Weyl chamber at a single point a = a ( S ) .This follows from a characterization of K-invariant functions (polynomials) on 'Ip. Namely, coincides with the joint level set of Theorem (Chevallet): (i) Any K-orbit on the K-invariant functions/polynomials on 'Ip;
(ii) Any K-invariant function f is uniquely determined by its its restriction on the Cartan component 2 of 'Ipl and f 1 % is W-invariant (here W means the Weyl group of 8, generated by all symrnetries/reflections of the restricted root system). Thus we get a 1-1 correspondence between K-orbits in 'Ip and W-orbits in 2.So a generic K-orbit will hit Cartan part 2 at exactly {W(a)}-locations, with only one point
55.7. Laplacians on symmetric spaces.
247
.(a) in the Weyl chamber, and the latter give the requisite transverse (polar) directions. We shall illustrate the statement for the rank-one space S". Here 'p N R", K = S q n ) acts on 'p by rotations; Cartan Z is identified with the 1-st coordinate axis; K-invariants are all even polynomials of r = I z I , f =
C a m I z I Im; W
consists of 2 elements,
f :zl- f zl,hence W-invariants are all even polynomials in zl,and the correspondence f-f
I Z is clearly
1-to-1. Moreover, any even (W-invariant) f = zlZmextends to a K -
invariant by integration over K,
1,
= Const rZm.
z12m-+
Similar arguments apply in the general case.
Passing to a curved homogeneous space 96 = I<\G the exponential map, exp:V+$, will take the Z<-orbits in into the K-orbits in 96, and bend Cartan part q, and its Weyl chamber, into a flat torus (or a totally geodesic Euclidian leaf, in the noncompact case). Of course, the density-factors and the resulting radial Laplacians will look different in both cases. Precisely, Symmetric spaces of complex semisimple groups Z<\G. Here radial sector 2 coincides with the real part of the complex Cartan subalgebra !$ c 8, function (7.7) where p =
tz
(Y
- the half-sum of positive roots. The radial Laplacian here is
a>O
where A , is the standard "flat Laplacian" on the real Cartan (Abelian) sector 2. Compact groups and symmetric spaces. The radial part of group G is made of its
Cartan subalgebra
8 (the Weyl chamber), and the density takes the form, J ( H )=
n
a E iC,
U
2sinq=
det(a)eiP (*I;
(7.9)
U€W
The 2-nd formula was essentially derived in 57.5, and the first could be established in a similar way. For symmetric space 96 = K\G each root cr will come with its multiplicity, so
65.7. Laplacians on symmetric spaces.
248
Hence the corresponding radial Laplacians takes the form
A, =A, Here 3, means
a
+
m,cota( ...)3 ,
(7.10)
-
partial derivative a 3,in the direction a.
In case of noncompact symmetric space all {sincu} in J are replaced by hyperbolic {sinha}, as in (7.7). By the same pattern all {cot a} in (7.10) are replaced by hyperbolic {coth a}. Once again, for compact groups expression (7.10) simplify as in (7.8) to (7.11) Here * indicates the half-sum of positive imaginary roots of O (compact Lie algebras have purely imaginary roots!).
Flat spaces. We think of the adjoint action of K on the !$-component in the Cartan decomposition, O = R @ Sp, (5 is assumed to be a noncompact Lie algebra, R its maximal compact subalgebra. Let us remark that any Cartan pair {RCO} can be associated with a triple of algebras and the corresponding symmetric spaces: hyperbolic, spherical and flat-type. Namely, (i) noncompact algebra O = R @ !$ (here the Killing form ( X 1 Y )is positive on !$, due to “maximality” of R c 0),yields hyperbolic-type symmetric spaces = K\G, with noncompact symmetry group G. ii) the “dual“ compact Lie algebra 6 = R @ i!$ c 0, (the Killing form negative everywhere) yields a compact-type space % = K\G, with compact symmetry group G of algebra g . N
N
iii) the motion algebra !lJl = !$ @ R, where !$ is considered as an abelian algebra, yields the flat symmetric space 8 = K\M, with the motion symmetry group, M = !$ D K ( K acting by the adjoint representation on !$). Three model examples: O = 4 1 ; n ) ; 6 = so(n+l); and !lJl= Rn D JO(n), with R = so(.) and !$ N Wn, exemplify the general situation. Another illustrative example is furnished by the pair: 4n;W ) 3 so(n;R). Here !$ consists of real symmetric matrices SM,(R), the dual compact algebra is 6 = su(n) = so(.) @i!$, while the flat (motion) group is the semidirect product: SM,D Sqn), with S q n ) , acting by conjugation X+u-’Xu, on the abelian group SM,.
55.7. Laplacians on symmetric spaces.
249
In the general flat case, the Weyl chamber A c 8 makes up the “cone of radial directions”, the density-factor and the radial Laplacian become,
7.7. Spectra of Laplacians on symmetric spaces. We shall solve the eigenvalue problem for Laplacians on symmetric spaces K\G, in terms of the weight structure of {St;?} and the representation theory of group G , developed in the preceding sections.
Theorem 2 (i) Eigenvalues {A,} of the Laplacian A on symmetric space 96 = K\G are labeled by the restricted weight lattice {a = x k j a j ; k j 2 0) in the Weyl chamber A c 2. T h e a - t h eigenvahe is equal t o (7.12) where p = ~ ~ r n ,- athe half-sum of all positive roots taken with their multiplicities {ma}.
(ii) T h e multiplicity of A, is equal to the degree of a n irreducible representation of group G .
?ra
The last statement requires some clarifications: in part (i) we talked about restricted roots in the Cartan sector Z c !$, while (ii) evidently refers to weights of @ itself, i.e. vectors of 8.But the latter breaks into the direct sum ?I @ 2 of the Cartan sectors of R and 9,hence weights { a € % } can be identified with elements of 8, that “vanish on the R-part”. Indeed, we shall give a precise meaning to this statement in terms of the regular representation of G on 96, induced by the trivial character 1 of subgroup K . The derivation of (7.12) will consists of 3 steps. 1”. We show that invariant Laplacian A on 96 is the image of the Casimir element L in the center of the enveloping algebra7 (lI(0).The latter Laplacian on G itself and on all quotients K\G. Since Casimir L, and its commute with the regular representation R of G on L2(96), the eigenspaces precisely the isotropic (primary) components of R,
so called
gives the image A of A are
Rrr @-?ra@da. a€G
2”. We show that representation R on symmetric 96 has simple (multiplicity-free) spectrum, i.e. each ?ra enters R with multiplicity 1, hence the intertwining algebra of 7Enveloping algebra a(@)means the associative hull of the Lie algebra 8, i.e. all noncommutative polynomials in variables {XI; ...X } (a basis of vector space a), where all products obey the Lie bracket relation, XiXj-XjXi= cck:Xk, where {cb.}are structure constants. ‘3
‘3
55.7. Laplacians on symmetric spaces.
250
the representation R, is generated by a single element A. In other words, any intertwining operator QR, = RuQ,u E G , is a function of A, Q = f ( A ) . This implies that L 2 ( q=
936,;
where eigensubspaces (36,)
are at once the irreducible subspaces of
G (the reader may
compare it with the case of S2-Laplacian and the orthogonal action of
Sq3)on S2 in
chapter 4).
3". Finally we use the radially reduced Laplacians above, and the Weyl character formulae of the previous section to compute the eigenvalues (infinitesimal characters) of the Casimirs L on irreducible spaces of ra. 1" Lemma: Regular representation R on any compact symmetric space A = h'\G A
has simple spectrum, i.e. multiplicity of any ineducible T E G in R, rn(r;R)5 1. The general argument (cf. [GGP]) exploits the Cartan automorphism 8, and so called Kspherical functions on
G. The latter consists of all functions on G, bi-invariant under all
right and left translations with h',
LK = {f:f(uzu) = f(z); u,u E K ; z E C } C L = L(G). Let r be an irreducible representation of group G (or algebra L) in space Y, and To be a subspace of K-invariants in Y. Then a simple argument,' due to Godement (problem 2) shows that Yo is invariant under the spherical subalgebra
and the restriction {r
LK,
rf I yoc v0,for all f E L ~ , I To:! E LK} is irreducible. But the spherical f
algebra on any
semisimple Lie group G (compact or noncompact alike) equipped with a Cartan automorphism is always commutative (problem 2). Hence, dimYo = 1, and Frobenius reciprocity (53.2) immediately implies simple spectrum of R.
2". Casimir element. Let G be a semisimple Lie group with Lie algebra 0 and B ( X ; Y )= tr(adxady) - the Killing form on 0.We pick a basis { X I ;...X n } in 6 , form a ..
symmetric tensor (metric) gij = B ( X j ; X j ) ,its inverse (9") = ( g i j ) , ' , and define an element A of degree 2 in the enveloping algebra a(@), ..
A =tg'JXiXj.
'The argument applies to all pairs { G ; K } , not necessarily compact G , and yet more general pairs of algebras L 3 A = XLX, where x is an idempotent (projection) in L,
x2 = x.
55.7. Laplacians on symmetric spaces.
251
One can verify (problem 6), that 0
A belongs to the center of 1(6), i.e. adx(A) = 0, for all X E 0.
0
on simple algebras (5 element A is unique and independent of a particular
basis (since the Killing form is unique)!
A particular choice of the Cartan basis { H j : X , ; Y , } (55.2) yields the Casimir element of the form,
{H;}are dual vectors to {Hi}: ( H i I H;)= 6 i j . The corresponding differential
where
operator A =
cgijaxpxjon G commutes with
all left and right translations (here
{ax}
denotes left-invariant vector fields on G). Let us also remark that coefficients {gij} give rise to a left-invariant pseudo-Riemannian metric on G (Riemannian in the compact case), and A becomes the Laplace-Beltrami operator of this metric. Furthermore, any G-action on manifold A, yields a representation of its Lie algebra 0 by vector fields (1-st order differential operators) {dx:X E (5). Hence the enveloping algebra
a(0)acts on A by higher order differential operators:
a 3 p ( X )=
cumxyl...X F + p ( a ) c =
u,a;:
...a;;.
Clearly, Casimir element A, considered on a symmetric space AL = K\G, turns into a G-invariant Laplacian on A. Let us remark that a G-invariant metric tensor (or differential operator) on AL is uniquely determined by it values at a single point xo E A, and its value (symbol) at xo must be a adK-invariant polynomial on !#. Here K denotes the stabilizer of point xo in G The eigenvalues {A,}
of A on L2(AL) are labeled by highest weights { a }
(irreducible components) of the corresponding regular representation R = RA. By
' , of R, is scalar, Schur's Lemma, A restricted on any irreducible component Y
A 1 To = A,I. 3". To compute eigenvalues {A,} it suffices to evaluate element A E a(@),on the irreducible character x,, a eoo-function on A,
4xal
=LXa.
Here we shall utilize the Weyl character formula of the preceding section and the explicit formula for radially reduced Laplacians (7.8); (7.11). We recall the Weyl
$5.7. Laplacians on symmetric spaces.
252
character formula,
(7.13) The denominator of (7.13), J ( H ) , represents the Haar density restricted on the maximal torus of G. Formulae (7.8); (7.11) show that radially-reduced Laplacian/Casimir on Lie group itself is conjugate to a flat Laplacian on the maximal abelian subgroup (torus) in G.Precisely, -+ (P 1 P ) ] J , with weight p=iC a. By (7.13) the 'reduced Casimir" applied to a character xa, a > O turns into the standard (flat) Laplacian on Tn, evaluated on the trigonometric polynomial in the numerator of (7.13), ARM
=
I
'1
I
- (P P ) ) [ ~ d e t ( a ) e ~ ( aI+ p = ((a-+ p I a -+ P ) - (P P))xa,
Since all abelian characters { e the result,
I '):oE W} on Tn have equal norms,
we get
(7.14) The difference between regular representations of group G on itself and on the symmetric space K\G is that the latter selects a particular set of roots, "orthogonal to R". Also multiplicities of eigenvalues {A,} on G and on A, are different. The former are always equal to deg2(a") by the Frobenius reciprocity, so spectrum of the groupLaplacian, spec(Ac) = {X,=IIcrtp112-~~pl12:#(Xa) = d ( ~ ~ ) ~ : a E 6 } , the latter have dimension equal to degree of a", since (Lemma 1") spectrum of R" is multiplicity-free. This completes the proof of the Theorem.
Remark: In all the above compact cases (group G,or symmetric quotient A) eigenvalues {A,} of the Laplacian (Casimir) are in 1-1 correspondence with irreducible representations (highest weights) { a } .In fact, the lexicographic ordering of weights { a } , via a choice of the positive (Cartan) roots in $ (ot Z),yields the same ordering of eigenvalues: + for all a,P in the Weyl chamber.
mi
Indeed, by (7.14) the difference,
$5.7. Laplacians on symmetric spaces.
253
+
xg - xa = IIP -t- PIP -lIa -t- PIP = (P + a 2P I P -a)2 0, since ( p 17)> 0, for any positive weight 7 . We shall conclude the discussion with 2 examples. 7.8. Examples.
The n-sphere S" E S q n + l ) / S q n ) . The n-sphere Laplacian was analyzed in 54.5. We shall reexamine it in the light of previous discussion. Here subalgebra A 2: SO(n),subspace !# consists of matrices: with columnar vector b E W"; isotropy subgroup K = S q n ) acts on Cp by rotations:
-
'Xbu = xu(b).
Maximal abelian subalgebras Z C ! # are 1-dimensional (rank l!), and the expimage of Z becomes a closed geodesics (great circle 7 ) in Sn. Lattice A consists of integers k, and the basic positive root { 1) has multiplicity n-1, hence p The k-th eigenvalue of the Laplacian A,,,
9.
the eigensubspace 36, being made of all spherical harmonics of degree k , restrictions on Snc Rn+', of all harmonic polynomials ($4.4-5 of ch.4),
Furthermore, the %,-component of the regular representation R of group G = SO(n+l) on space L2(Sn), is an irreducible representation rk of weight k = (k;0; 0; ...) (cf. Theorem 5.1 of s4.5). So
R II
9
rk;
each r k entering R with multiplicity 1. .Symmetric
space
S y n ) / S o ( n ) . Here subalgebra R consists of real
antisymmetric matrices, while subspace p is realized by real symmetric matrices. Indeed, any su(n)-element 2 = X t i Y , whose real part X E so(n) = A, and imaginary part Y ESM,. Group K = S q n ) acts on 8 by conjugations: 11: X + u - ' x u . Cartan subgroup Z c !# consists of diagonal matrices, Z 2: Wn-' (rank = n-l!), and expZ spans a geodesically flat torus 72:Tn-' in A. Positive roots in Z
65.7. Laplacians on symmetric spaces.
254
@nq 2: 03”-’ are of the form Hij= diag(0; ...1;...- I;...); on the i-th and j-th place, the basis consists of
H 3. = diag(...1;-1; ...);j= 1; ...n-1; while weights are made of all integral combinations of the basic roots: a = C k 1.H 3. = diag(kl; k,-kl; ...;- k,-,). i The half sum of positive roots: $p = $(n-l;n-3; ...;-n+l), while the inner product ( H I H’) = tr(adHadHI)=
C (hi-hj)(h:-h>). i<j
So the eigenvalues of A are equal to (7.16) where the components of weight a = ( k i j ) are given by
Instead of (n-1)-tuple Q = (kl;...kn-l)we could use the entries of the diagonal matrix, representing it, Q = diag(ml;...m,): where m j = k j - k j - l ; satisfies E m j = 0; and ml 2 m2 2 ... 2 m, (condition of “highest weight”, element of the Weyl chamber). Then components kij = mi - mi, and we get
A, =
c
{[(mi - mi)
+ (j-i)]’ - ( j - i ) 2 ]
i
Let us remark that the rank of symmetric space A = SU(n)/SO(n) is equal to the rank of group G itself. So both spaces have the identical set of weights. As a consequence, we find that the regular representation Rh, consists of all irreducibles R E 6, but unlike Rc, each R enters Rh with multiplicity 1. So multiplicities, rather than “spectrum of irreducible { R } ” , mark the difference between regular representations on A and G).
55.7. Laplacians on symmetric spaces.
255
Problems and Exercises: 1. Show that any n x rn matrix X has a unique representation of the form: X = uXtu, with u E S q n ) ; u E Sqrn), and diagonal matrix X, of (7.4).
2. Use Cartan automorphism 6' on group G to show that spherical algebra LK is commutative. Hint:
(feh)'
Cartan 0 define an automorphism of functions on G f + f e = f ( z e ) , = fehe, while the inversion, z+z- yields an anti-automorphism:
',
f-i,( f * hj = h;j. But on spherical functions {f}, both 6' and -are equal: fe(z) = f(z-')!. 3. Let L be an algebra with an idempotent (projection) element x, and let A = xLx denote the corresponding "spherical" subalgebra of L. If representation T of L in space Y is irreducible, and subspace Yo= T (Y)is the image of TX,then Yo is invariant under A and the restriction T, Yo (f E is an irreducible representation of A.
I
4
Hint: assume that Yo contains a proper A-invariant subspace Yh,take the &-span of Yh, Y' = {T,(Yh):f E L}, and show that TX(Y')c Yh,hence Y' c Y,contradiction.
#
a
Apply this result to a pair G 3 K-compact, where a1 ebra L = L'(G), x any (irreducible) character of K (e.g. x = lK), and A = L - p L * x - the spherical subalgebra. 4. (i) Show that the radially reduced Laplacian on rank-one spaces R"; S"; H" has the form A = a J a , where J denotes the radially reduced Haar density.
5
(ii) Conjugate A with operator,
fi,L+L*
= fi&,and derive the resulting flat SchrGdinger
Jj
L* -
a 2- 2[?(J',' T
+Y 4 iJ ' Y ]
(iii) Use general form of L' to compute radial part of the Laplacian on rank-one symmetric spaces R"; S"; H". (iv) Show that potential
(n-l)(n-3)
; on
Rn (7.17)
("-1)'
(n-l)(n-3) +
; on
H"
lsinh'r
Remark: Dimension n=3 is exceptional for the flat (Euclidian) and hyperbolic cases. Here V turns into a constant, so the radially-reduced Laplacian is conjugate to a flat Laplacian (2-nd derivative) on R+, hence solutions (Green's functions) of such problems are expressed in terms of elementary functions (cf. $2.3).
5. Apply problem 7 of $5.1 and results of chapters 3,4, to decompose the regular representation R of SO(4) on the 3-sphere, and derive the spectrum of he Laplacian on S3. Show that R 2 : girk@irk, where {nk:k=O; 1;...} are standard irredtcible representations of SU(2) of S4.2, and S q 4 ) = SU(2) x SU(2)/{ f I}. Hint: identify R with the double left-right action (u,u):f+f(U-'zu), on L*(G), G = SU(2), and use a decomposition of L2(G) into the sum of matrix algebras @ Matd(*) (chapter 3).
256
55.7.Lapbcians on symmetric spaces. 6. CaSimirS and invariant polynomiah on 0.: (i) The enveloping algebra %(a)can be identified with all polynomials on 8'- the dual to Lie algebra 0, by taking symmetrized products, C : X = x,...x,-+x = - $ C X ~ ( , ) . . . X xi~E(8~(sum ) ; over all permutations). U
Indeed, any Symmetrized element PX(€)=
One should think of {px(()} differential operators.
x E a(@)gives rise to a polynomial function,
C.1 ...
I €)***(xm I €1; € E a**
aa Vnoncommutative" generalizations of symbols of
(ii) Furthermore, the co-adjoint action of G on both spaces, a(@)and (intertwined by C), ,E: d g ( X ) - + P X ( d C o ( O ) (check).
?(a*)coincide
(iii) The center 3(8)of algebra a(@)(Casimirs) consists of all G-invariant polynomials {px(()}. In case of semisimple algebra 8, the latter can be further reduced. (iv) (Chevallet): G-invariant polynomials {f(<)}on 8, when restricted on the Cartan subalgebra 5, C 8, become W-invariants, where W denotes the Weyl group of the pair {a;$}. Conversely, any W-invariant on 5, gives rise to a G-invariant polynomial on 8.
Comments: The material of $85.1-6has become by now fairly standard, and could be found in many textbooks on the representation theory of semisimple Lie groups and algebras ([Serl]; [We3]; [Hel-P];[ge]). The bulk of results in chapter 5 is a lasting contribution of Hermann Weyl and Eli Cartan. Our presentation was largely influenced and owes a great deal to books [We]; [Ze] and [Ael-21 (particularly, in $5.7).
Chapter 6. The Heisenberg group and semidirect products. $6.1. Induced representations and Mackeyi group extension theory. The induction procedure, introduced in f1.2 (chapter l), and studied in detail in chapter 3 (compact and finite groups), allows one to construct in a canonical way a representation {T} of group G, starting from a representation of its subgroup H. In this section we shall extend the results of chapter 3 to infinite semidirect products. The general theory will be illustrated by examples of the Euclidian motion, afine and Poincare groups. After the standard induction procedure we turn to an important modification, called holomorphic induclion. The latter casts a different light on the representation theory of classical compact semisimple groups G. (chapters 4-5). It turns out that all irreducible representations of such G are holomorphically induced, via the celebrated Borel-Weil-Bott Theorem. Its meaning and contents will be further explored in 66.3.
1.1. Induction: We remind the basic constructions and results of $1.2 and 52.2. Induced representations arise in the context of group G acting on a homogeneous Gspace 96 21 H\G, g:z-+xg.Here H = Hzo denotes the stabilizer of a fixed point x,, E 96. The G-action on space 96 defines a H-valued cocycle, h:% x G -H, h(z;gu) = h(z;g)h(zg;u),for all z E 96 and g , u E G.
(1.1)
Given a representation S of subgroup H in space Y, the induced representation T = ind(S I H;G) of G,acts on a suitable space of 'V-valued functions on 96 (continuous, L2,etc.) by operators
T g f W = s,(z,g)[f(~g)l; 9 E G, f = f(x).
(1.2)
The Lz-function space on 96 in (1.2) is taken with respect to a G-invariant measure dx (in case it exists). Otherwise, we pick any measure dp(z), whose translates ,{dp(xg):gE G} are absolutely continuous with respect to dp,
and modify (1.2) by the factor
dm,
T g f = dGjsh(z; g)[f(zg)i.
(1-3)
Factor fl makes representation T in (1.3) unitary with respect to L'(96;dp) @ Yinner product. Cocycle h clearly depends on the choice of coset-representatives {y(z):zE 96) in G, as 7(z)g = h(z,g)y(zQ),but the resulting induced representation T Szof H induce equivalent representations of G. (1.2), (1.3) is independent. Let us also remark that equivalent representations S'
N
258
56.1. induced representations and Mackey's group extension theory. Another description of induced Tg is given in terms of an associated G-invariant
vector bundle, over space 96 = H\G, with fibers Y,
N
Y. Group G acts on W and on base 96, and two
actions commute with the natural projection P :90-96. Furthermore, element g takes fiber Y2+YZg, and thus defines a linear map u(z,g):Yz+YZp; that satisfies a cocycle condition (l.l),a consequence of the group multiplication. Hence, stabilizer subgroup
H, is linearly represented on the fiber-space Y,, h+u(z;h) = SZ(h). The G-action on the vector-bundle yields the induced action by linear operators on vector-spaces of cross-sections
T(96;W ) = {f(z) E V2}, equipped
with the standard
(L', L ~etc.) , norm', (Tgf)(z)= 4 2 , s)-"f(zg)l;
f E T(%;W ) .
(1.4)
A third realization of Tg is given in certain subspaces of V-valued functions on
G , like C(G) 8 Y; L*(G)8 Y, etc. Namely, we consider functions, that transform according to S , under left translations with H , C(G;S)= {f(g):f(h-'g) = S,[f(g)], all
E H , E GI-
(1.5)
Obviously, space C(G;S)is invariant under right translates with g E G , so we get a representation
Tgfb) = f(zs); all z, 9 E G.
(1.6)
In other words T is realized as a subrepresentation of certain multiple of the regular representation on G , T c RG 8 Id(S). One can easily check that all three definitions of T,(1.2); (1.6) and (1.6), are equivalent by writing an intertwining operator, W:C(G;S)+ e(96;Y). Indeed, each function F ( g ) on
G , that transforms according to S , is uniquely determined by its values a t coset representatives { T ~ : ZE 96). So we get W:F(g)+f(z) = F(y,). The second construction also demonstrates independence of T of the choice of coset representatives {yE}!
1.2. Commutator algebra and the irreducibility test for Ind. Our next goal is to characterize the commutator algebra Com(T), in particular to find conditions for Com(T) to be scalar, i.e. T - irreducible. 'All fiber-spaces are assumed to possess a G-invariant metric, and space 96 is assumed to have a G-invariant volume element.
56.1. Induced representations and Mackey's group extension theory. BY a general functional-analytic result2: any linear operator
259
w on space ~(s;v)
is given by a (distributional) operator-valued kernel A(z,y):96x 96+8('Y)- the algebra of linear operators on 'V,
For each z E 96 we denote by ax a representation of stabilizer Hx. Group G acts on the product % X 96, g:(z;z)+(zg;zg), and splits it into the union of G-orbits { w } : one of them is the main diagonal wo = { ( z ; ~ ) } ,all others have a marked point (zo;y), hence { w = w(zo;y)} are identified with Ho-orbits in 96 (Ho-stabilizer of zo). For instance, the orbit-space of the orthogonal group on the product of (n-1)-spheres is ~ n - 1x s"-'/Sqn) N Sn-'/Sqn-1) = [-I; 11. The commutator algebra of the induced representation can be characterized by the Mackey Test (53.2). Theorem 1: The commutator algebra of T = ind(a I H;G) consists of all operatorvalued integral kernels A(z, y) on 96 x 96, satisfying,
for all (almost all) z;y E 96, and g E G.
Formula (1.7) implies that any kernel A(z;y) is uniquely determined on orbit w c 96 x 96 by its value a single point (e.g. (zo;y) E w). Moreover, the diagonal value {A(z;z)} belongs to the commutator Com(az I H,) N Com(a I H), while off-diagonal entries A(z;y) lies in the intertwining space Int(az I H,n Hy;uYI H,nHy) for any pair of points z; y E 96. By analogy with the finite case (chapter 3) we can formally decompose the commutator
+
algebra of T into the direct integral (see subsection 1.7),
ComT = Com(a I H ) Znt(u I H integration over the orbit-space ($ x %)/G
n
n
H,;aW I H H,)dp(w), (1.8) % / H . As the result we obtain the following
irreducibility test for induced representations.
Theorem 2: Representation T = ind(a I H;G) is irreducible iff (i) u is irreducible, and (ii) all intertwining spaces: Int(az I H z n H&' I H,nH,) = 0, f o r any pair 2kernel A(z;y) is derived by observing that point evaluations, f+f(z) = (f 16), are bounded linear functionals on space e(%;...). Any such functional is given by a finite (Borel) measure, dp. Hence, (Wf)(z) = (f I W*[6J)= Jf(y)dp,(y). The density factor, A(z;y)
=w, gives the requisite "distributional measure", & I
80
p , - W[6,1,.
56.1. Induced representations and Mackey’s group extension theory.
260 z#
Y.
The argument, outlined in 52.2 in the finite/discrete case, extends with some technical modifications to infinite topological spaces (manifolds) 96. We shall mention a special case of the normal subgroup H c G. Here space 96 = H\G becomes the factorgroup, all conjugate stabilizers { H z } z€ % coincide with H , G-action on H , h-tg-’hg, induces the “dual” action’ on the dual object g:u-*ag = u(g-1 ...g).
a,
-
Note that elements h E H transform u into an equivalent representation: oh cr. So ug depends only on the class of g in H\G, i.e. ug= uz ( z = H g € % ) . Formal expansion (1.8) amounts to Com(T) = Com(u I H I + Int(u I H a w I ~ ) d p ( w ) ,
(1.9)
If u is irreducible and all {u2:z# z}, belong to different equivalence-classes in i?, then T is also irreducible! This observation will be essential in the study of semidirect products G = H D U.
1.3. Characters of induced representations. Character of a finite-D representation T was defined by XT(g) = trT,. In chapters 3-5 we studied irreducible characters on compact (Lie) groups, and found them to form a family of nice (differentiable) functions, orthogonal in L2(G).For m D unitary representations tr T , does not strictly speaking make sense, so characters should be understood as distributions on G, defined via pairing to suitable test-functions {f(z)}, (xT I f) = t r ( ~ ~ Precisely, ). functions {f} that produce a trace-class operator T , (see Appendix B). For induced representations T = Ind(u I H ;G), the corresponding group-algebra f (g)Tidg, are given by integral kernels (1.10)
where H , denotes a stabilizer of point z E 96 N H\G, and coset representatives {yY} map z+y. Picking a suitable class of functions {f} on G (typically smooth, compactly supported), one can show integral kernels, {Kf(z,y)} to be also smooth and rapidly decaying. Hence, by the standard functional analysis (Appendix B) operators {T>:fE Cp} belong to the trace-class (or Hilbert-Schmidt class) on space L2(96)(as well as spaces of holomorphic functions, like Hardy 36,). Furthermore, for representations in ’The same holdtr for any automorphism a E Aut(H), a:a-+at = a(ha)!
56.1. Induced representations and Mackey's group extension theory.
L2-spaces, we have
261
I
trT; = K;(x, x ) dx.
So character
xT
can be defined as a distribution on space eF(G),
It often happens, however, that such distributions { xT} have piecewise continuous (or smooth) densities with respect to the natural coordinates on Lie group G. We shall see it to be the case for semidirect products: Euclidian, affine, Heisenberg groups, as well as noncompact semisimple groups, like SL, (chapter 7). In $3.2 we established a general character formula for induced representations (1.11)
summation over all fixed points {xj} c 96 of element g E G, equivalently all elements ( ~ 3 ~ of ) G that conjugate g into H. Formula (1.11) can be extended to Lie groups G. We assume that all g E G (or a sufficiently large set of them) has only finitely many fixed points {xl; ...xm} c 96, all of which are nondegenerate, in the sense that Jacobian g * ( x j ) of smooth map g : z - d , at { x j }has no eigenvalue 1. Then for such elements g one has, (1.12) the summation extends over all fixed points {xi} of g , and { s ( x j ) } denote coset representatives of points {zj}c H\G in group G. Formula (1.12) is closely connected with the Atiyah-Bott trace-formula for elliptic operators on vector bundles [AB]. To proceed further we shall need a brief diversion into a general subject of decomposition and inversion/Plancherel formula on noncompact groups. 1.4. Direct-integral decomposition and Plancherel formula. A direct integral X S d p ( s ) , of Hilbert spaces {%,}, labeled by points of a measure-space (L';dp), +f2 consists of all %,-valued functions {f(s)}on R, with the standard L2-norm, Inllf(s)lL$s);
We say that an operator (or representation) T in Hilbert space X is decomposed into the direct integral $ T S d p ( s ) , if there exists a family of Hilbert spaces {%,}, together with a family ofnoperators {T" in X,}, n, and an intertwining map
n,
56.1. Induced representations and Mackey's group extension theory.
262
f E 36 (or a dense subspace &, c 36) into a 36,-valued on 0, 4:f+T(s), so that operator T goes into a multiplication with an operator-valued function T', that transforms each vector
function ](s)
'3: T[f]+( T s f) ( s ) for , s E 0. We can also assume map 4: 36-t [36,dp(s), to be unitary, so
This notion clearly generalizes the usual direct sum decomposition, and it turns into the latter for atomic measures dp = E a j 6 , . . But in general (36,) f
are not
subspaces of 36! The following examples should clarify the concept. Example: The regular representation R of group Z" on 36 = Lz(Z"), Raf = fv-,
is
decomposed into the direct integral of 1-D representations T e = ei*' ' I in spaces
Wy 1:CIS 1"'
Here map 9: L*+
+
R=
+
+
Xe do. T" ... is the standard Fourier transform on Z", T;ie
'
"do, and Lz(Z")
N
and properties (i-ii) are easily verified. A similar decomposition of R into the direct sum/integral of characters (1-D irreducible representations) by the Fourier transform holds on any locally compact abelian group, e.g. R". It was also proven for fairly general classes of noncommutative groups and more general representations, the role of characters being played now by unitary irreducible representation of G.
The general noncommutative Plancherel formula, due to Segal and Mautner [Seg] extends the classical commutative result,
(1.13) integration over the dual (character) group
e.
of an abelian group G with respect to the
properly normalized Haar measure d t on On Rn dp is equal to (2s)-"-rnultiple of the standard Lebesgue measure. A noncommutative analog of (1.13) yields a similar representation for any function f from a dense subspace of L 1 n L 2 (in our case eF(G)) in terms of irreducible characters on G. Namely,
86.1. Induced representations and Mackey’s group extension theory.
where (x,If)=trf^(?r), and the generalized Fourier transform integral T(T)=
](T)
263
is defined via
jGf(Z)T;ldx.
Integration in (1.14) extends over the dual object of G, (the set of equivalence classes of unitary irreducible representations T ) , and d p ( ~ is ) called the Plancherel measure of group G. From the inversion formula (1.14) one can easily get the Plancherel Theorem for L2-norms of f and Indeed, applying (1.14) to a convolution f*f*, where f*(z)= f ( x - l ) , the integrand becomes
P.
1
2
l HS
tr[f^(~)f^= ( ~ )f *^ (] ~ ) so
(Hilbert-Schmidt norm)
(1.14) turns into (1.15)
in other words the L2-norm of f is equal to the L2-norm of (generalized) Fourier modes of f. Let us also remark that inversion formula (1.14) at a single group element go = e yields values f ( g ) at all other points {g},
(1.16)
Finally, formulae (1.14) and (1.15) lead to a direct-integral decomposition of the regular representation on G (or %), into the sum of primary components, (1.17) where multiplicities {d(?r)}could be finite or infinite. To show the latter we consider the family of Hilbert-spaces { X T = HS(T,):
T
E
c},
A
where {V,}
L2(G)into
are the representation-spaces of
4
T’S,
and define the map T : f - f ( ~ ) , from
X,dp(r). Group G acts on each space K,, 9: m - ~ J ( g h
and the resulting action is a d(T)-multiple of
T.
1.5. Semidirect products. For the sake of presentation we restrict ourselves to
semidirect products with a commutative normal subgroup H. We shall restate the
$6.1. Induced representations and Mackey's group extension theory.
264
principal result of §3..3. Theorem 3: Irreducible representations T of semidirect products,
G = HDU,with
commutative H, are parametrized by pairs { ( w ; ~ ) }where , w - an orbit of U in the an irreducible representation of stabilizer U , of the dual group H, and U E point (character) z E w. Furthermore, T is equivalent to induced representation, T N ind(x 8 u I H D U,;G).
oz-
Proof essentially follows the lines of the finite-D case (53.3). We take a n irreducible T of I1 D U , and decompose T I H into the direct integral of characters (e.g. A,(")
= e"
*
',
on R" ),
T IH =
A,(h) o m(z)dp(z); A, E 9, where m(r) denotes the multiplicity (finite or infinite) of A, in TI H, and d p ( z )
-
the
corresponding spectral measure. Irreducibility and U-invariance imply that d p is supported on a single U-orbit w
c H, and coincides with the unique U-invariant measure dw(z) (provided the
latter exists). The representation space I= I ( T ) can then be identified with sections of a Uinvariant vector bundle over w , in each fiber-space I, (zE w ) one gets a representation uz of stabilizer U,. Hence T I U turns into an induced representation ind(u I U,;U). The same holds for the entire representation T o n G,
T
SI
ind(A, @ uz I H
D U,;G).
(1.18)
Conversely, any induced T of type (1.18) is shown t o be irreducible by the Schur's Lemma.
over w ) that Indeed, any operator W in space Lz(w;Y') (Lz-sections of the vector bundle I' commutes with T I H must be a multiplication with a (operator-valued) function,
(Wtb)(.) = Nz)[tb(z)lThen irreducibility of inducing u and the relation TC'WT, = W, for all h E H, implies
W = c l , t o be scalar.
Now we shall illustrate the general results with a few specific examples.
+
Affine p u p : 5: = R D R; is made of transformations { ( a ;b):z+az b; a > 0 } of R. H = R, whose dual H N R contains 3 orbits: w+ = {A > 0); w- = { A < 0) and wo = (0). Hence there are 3 types of irreducible representations of G: A
It has a normal subgroup
1) a pair of m-D unitary representations T+; T-, corresponding to orbits w
;
2) one-parameter family of 1-D representations {Tpy E R) of the commutative quotient R+
N
G/R.
The former are realized in space Lz(R+;$) by operators,
56.1. induced representations and M a c k e y i group extension theory.
T$,
b)+(z)= e
265
ibza4(za),
and l-D characters T[a,b)= u'p. The group-algebra operators {Tf} are given by integral kernels, a0
*
f(f;b)e ib/ydb.
Kr( z; y) = -a0
We leave as an exercise to the reader to compute characters of T
* , and to prove
the Plancherel/inversion formula, f(e) = i t r ( T j + T j ) , or
for a suitable class of test-functions {f} on G. Here
?( * )
denotes operators
{Tf}.
In
other words the Plancherel measure is concentrated at two co-dimensional "fat" points of the dual object:
= {+}U{-}UW.
Euclidian motion group: En = W" D S q n ) , has normal subgroup H = W". U = S q n ) c En acts on by rotations, so orbits are spheres w = w,. = { 1 z 1 = r > O}, and point (0). Irreducible representations Tpt" act on spaces L*(S;Y), where S = Sn-' z w; Y = Yu - the space of g E 0,(stabilizer U , N Sqn-l)), Subgroup
.
u
T ( a , u ) w= elPZ *
u)[4(2")1.
(1.19)
Here z is a unit vector on S, u E S q n ) , a E W", and x+xu denotes the orthogonal transformation u applied to the unit x. So Tr)" is induced by the representation ~
eirz a of
W" x Sqn-1).
The trivial orbit (0) c W" has stabilizer K = S q n ) . So the
corresponding representations are those of the compact factor-group S q n ) = E,/W". One can easily compute integral kernels {Kf(z;y)} of the group-algebra operators Ti and the irreducible characters.
In the simplest case E, we get a one-parameter family of co-D (induced) representations { T P : r> O}, acting in space L2(S'), and a sequence of l-D representations of Sq2) IE,/W2. The characters of {T'} are computed to be xr(a; u ) = 2 4 u ) J o ( r I a where J o is the Bessel function of order 0 (problem 2).
I 1,
(1.20)
The Plancherel/Inversion formula on E, has the form,
(1.21)
266
56.1. Induced representations and Mackey's group extension theory.
with the Plancherel measure dp(r) = rdr, supported on an object
pz= R+UZ.
m-D %+-part" of the dual
En we get
Similarly for
(1.22)
The proof of both formulae is left to the reader (problem 2). The Poincarc: group P, of special relativity is a semidirect product of the Minkowski space M",
with indefinite metric (+---), and the Lorentz group
(I = Sql;n - 1) of isometries of M". The K-orbits in MI" fall into 4 classes: point: (0) two-sheet hyperboloids: wr = {x:2. z = zo2- Exi' = r
> 0} < 0)
one-sheet hyperboloids: wr = { x:x. x = zo2- C :x = -r the light-cone: wo = {x:r = x . z = xo2-
E :x
= 0)
Fig.1: Orbits of the Poincare group in M", {O}, light-cone and 2 kinds of hyperboloids.
The stabilizer of (0) is group Sq1;n-1) itself. Its representation theory, in case n=3, will be developed in the next chapter 7. Two-sheet hyperboloids w have the compact stabilizer K ON S q n - l ) , whose representations were studied in chapter 5. They are labeled by highest weights a. The corresponding irreducible representations of
Pn consists of { T r 7 a } ,r > 0 (orbit parameter). They act in space L2(w)@ Ya and are induced by characters xp(x) = e
irz
0,
tensored with a. Such representations were first
introduced by E.Wigner [Wig2], and proved to play an important role in the quantum field theory. The one-sheet hyperboloids have non-compact stabilizers, KO= Sql;n-2). Their representations are still induced by characters x - , ( z ) = eirzl, tensored with irreducible representations
(0)
of
KO= S q l ; n - 2 ) . In
the simplest case, n = 3,
56.1. Induced representations and Mackey’s group extension theory. stabilizer K O= S q 1 ; l ) is W, so
0
267
E W, but in higher dimensions it involves the
machinery of chapter 7. The stabilizer of the light-cone is the motion group
K ON
= p a - 2 D S q n - 2 ) (problem 4). Its irreducible representations have form ( p > 0; a E Sqn-2)), and the corresponding representations T of Pn are o= induced by such d s . A
1.6. Holomorphically
induced
The
representations.
standard
induced
representations T = ind(S I H ; G ) ,were realized in several different ways: on sections of G-invariant vector bundles ($;Y),over quotients 96 = H\G,
where each stabilizer
{ H , = gi*Hg,),acts the fiber {Yz:z E 96). Alternatively, they were realized in certain subspaces of functions on group G itself (1.7), namely, C(G;S) = { f ( h z )= S(h)-’f(s):h E H ; z E G} C C(G).
For Lie groups G such subspace is characterized in terms of generators { X } of the Lie subalgebra 8,acting on Cm(G;S) by 1-st order differential operators,
+
(8, S,)f = 0; where a, denotes a left-invariant vector field of element X on
(1.23)
G.
The action of H by left translations on G foliates the latter into a union of fibers (H-cosets) and the representation space of T consists of all functions on certain differential system (1.23) along the H-fibers.
G,that
satisfy
One would like to extend this construction to situations when (1.23) becomes a system of Cauchy-Riemann equations:
& f = 0,
on certain complex lane bundles (i.e.
manifolds with analytic local structure holomorphic transition functions between fibers).
So the representation-space will consists of holomorphic sections. More generally, we shall see that homogeneous spaces 96 = H\G of Lie groups can combine both “real” and “complex” parts, so the corresponding functions/sections { f (z,...zk;zl...zm)) will depend on Ic real and m complex variables. We consider a complex Lie algebra gCwith a real form 0,and denote by z+z, conjugation in 0, relative to 0,. Assume that 0, contains a complex subalgebra 9,
8 c 0,with the following properties: 0 9na = Gc - a complexification of real 8,and the
and a real subalgebra
{adX(Y):XE 0
%+
!iicmc0.
adjoint action of
8 on 0,,
a} leaves 3 (and g ) invariant;
5 = !Dlc
- complex subalgebra, whose real part IUI = IUI,
c 0. Clearly,
268
56.1. Induced representations and Mackey’s group extension theory. Hence we get similar inclusions for the corresponding (real) L’ie groups:
HcMcG. Proposition 4 Homogeneous space H\G has the structure of a fiber bundle with complez fibers N H\M (of complez dim = m), over the base 3 = M\G (of dim = k). Clearly, quotient H\G is naturally projected onto M\G = SJ,
p : ‘H-cosetn--rUM-cosetn. 21 H\M,
and claim that the latter is isomorphic to a complex quotient N\M,, of two complex subgroups N M , c G,. It suffices to compare the tangent spaces of both. We observe that !llc = 9 !ll,and 9 n n = 8, We take a fiber space, p-l(z)
+
hence !Ill/ N @ tangent(H\M) 1? !ll,/9. Let us illustrate the above constructions for the unitary algebra 6 = su(n). Its complexification 0, = q n ;C) has the (Cartan) conjugation:X+ - X*. We pick a pair of Borel subalgebras 9; 3,made of upper/lower triangular matrices:
(1.24)
whose sum !XIc = m+g, comprises entire 0, (hence, !ll= ..in)!),
and whose
= R e ( 9 n 3 ) = { h = diag(ial; ...zan)}. So subgroup H i s the maximal torus Tn-’c G = SU(n), and the quotient
intersection is the diagonal (Cartan) subalgebra,
H\G = H \ M = Tn-l\Syn), acquires the natural complex structure, inherited from space SL(n)/N, called flag manifold (problem 3). In the simplest cave, 0 = 4 2 ) , the complexification Oc = 4 2 ; C) contains the upper and lower-triangular (Borel) subalgebras: %={[
q};3
={[ ;
-a]};
their sum 9l + 3 == 9R, = 8,; while the “real part” of the intersection,
A similar construction can be given for any simple/semisimple compact algebra 0 (see chapter 5). Here 9; 3 are two Borel subalgebras of 6, made of all
56.1. Induced representations and Mackey's group extension theory.
269
positive/negative root vectors. Their intersection is once again a Cartan subalgebra $ of G, and the quotient T\G (of compact Lie group modulo maximal torus), has a natural complex structure (problem 3). Compact Lie groups, and their quotients T\G provide one extreme, when the entire homogeneous space turns into a single complex leaf. Another extreme arises when algebra % = 8 = !lJlc, hence 8 = 3, = !Ill,so H = M c G, and we don't get any complex leaves, the entire quotient being the standard H\G! In the context of Proposition 4 we consider functions/sections {f(z;z ) } on space 96 = H\G, holomorphic in variables z = ( z j ) , and define induced representations in those. Precisely, a character (representation) S of the real subgroup H, extends to a holomorphic representation u(X) of the complex algebra %. Then we replace condition (1.24) with
[az + a(Z)]f = 0; for all Z E 3. and call the corresponding function-space C(G; u I N ) . Here az indicates the Cauchy-Riemann a -derivative
(1.25)
in the complex (fiber) variables. Indeed, in the simplest case: 0 = W2; GC= Cz; % = Span(X+iY), {X;Y}-basis in W2, and a(X+iY) = A, space H\G = {O}\R2 N C, and (1.25) turns into the standard 3 - equation,
B f = ;(az + 3,)f= - Xf, whose solution is f = e-"
x holomorphic function.
The resulting representations of G in spaces C(G;u I N ) are called (partially) holornorphically-induced, and denoted by ind(S 1 H ; a I N;G). Next one needs to introduce a G-invariant product in representation spaces C(G;a I N ) . In some cases it proves to be possible, for instance, if the complex component of H\G is compact (like SU(n)), or is equivalent to a bounded domain in C". In such cases, Hilbert space L(G;u I N ) turns into a closed subspace of L2(G;Y',)the representation space of ind(a I H,G). We shall not delve deeply into the subject of holomorphically induced representations here, but refer to [Kirl] for further details. Let us mention, however, an important result, due to Borel-Weil-Bott. They proved, that the holomorphic induction on compact Lie groups G, by characters u of a Bore1 subgroup N (1.24), yields all
56.1. Induced representations and Mackey’s group extension theory.
270
irreducible all irreducible representation
?r
E G.
To give a precise statement, we recall that irreducible ?r E G are labeled by weights (characters) {A} of the Cartan subgroup H N T ” c G , the highest weight X is unique, i.e. the corresponding eigenspace YX is 1-dimensional. Weight A, rather eigenspace YX, gives rise to a holomorphic line bundle 8 =$(A) over the complex manifold 96 = T”\G with fibers N Y,. Hence, we get a finite-dimensional vector space L(96;%)of holomorphic sections of 8. It turns out that dime = deg(nX).Furthermore, Theorem (Borel-Weil-Bott): The naturd (induced) action of G on 96 and B(X), considered on space P. of holomorphic sections is equivalent to 77.’ Proof exploits the highest-weight theory of chapter 5 and consists of several steps. 1) The Weyl “unitary trick” allows to identify representations
T
of G with “holomorphic
representations” of GC - its complexification, so that matrix-entries of
T
on GC become
analytic continuations of entries {f(z) = (T,( 1 7 ) : ~E G}. 2) We realize
T
on by functions (matrix-entries) on GC (or its real non-compact form
GR), that satisfy
f(Cz) = f(z); for all C 6 N f(hz)= X ( h ) f ( z ) ;for all h E H
(1.26)
where H is the Cartan (diagonal) subgroup of GC (GR), while N - is made of negative root-vectors (lower-triangular matrices). The embedding of space ‘V = Yx into functions (1.26) is given by the lowest-weight vector ‘lo E ‘V,
t-f&)
= ( T Z ( 0 I ‘lo).
Condition (1.26) can be written in the infinitesimal form as
= 0, for all negative root-vectors Y = Y , E 9a H f ( z )= (A I H ) f ( z ) ; for H in the Cartan subalgebra where
aY;
denote the right-invariant vector fields on GR (7),
b
(1.27)
(generators of left
translations). 3) An elegant observation of Borel-Weil-Bott was to note that condition (1.27) has an
equivalent form in terms of holomorphic differentiation a j n the complex domain GC where
,
z ,= X , + iY, (a- root of 8).Namely, a-Z f = 0; for all Z = Z = i(X I H); for all H E is, (compact Cartan part) (I
a;Hf
(1.28)
-
56.1. Induced representations and Mackeyi group extension theory.
271
One first observes that space (1.28) is finite-dimensional (it sufices to prove it for the restrictions {f} on the compact quotient T\G
c E\Gc,
then apply a unique analytic
continuation4 from the former to the latter). Hence, space (1.28) has the highest weight function 4(z). Since,
a- 4 = 0, for all z
implies lay4 = 0, for all Y (negative root-vectors)l
on GR, hence (analytic continuation) on Cc, and 4 is an eigenfunction of Cartan 8, it follows that
4 is equal to the standard matrix-entry 4o = (irZto I vo), that couples the
highest and the lowest weight A. Thus we get “space(1.28)” = “space(1.27)”, and two (induced) representations are equivalent. Finally, it remains to observe that space (1.28), when restricted on compact G c GC, defines a holomorphic line-bundle $
= $(A),
and all analytic sections of $,
80
r x becomes a holomorphically induced
representation in “sections” &($), QED.
We shall illustrate the holomorphic induction and the Borel-Weil-Bott Theorem for group SU(2). Here a unitary character S(0) = eime extends through a holomorphic representation/character (1.29)
It turns out that character o = u, gives rise to a complex line-bundle 1, over the quotient-space 96 = SU(Z)/T, whose first Chern-class’ is equal to m. The space of holomorphic sections of L , has dim = m, and the resulting holomorphically-induced 4The simplest prototype of the triple GC;GR;G are, multiplicative groups: C’; R,
and
T = {e”} (unit circle). Representations (characters) ( ~ “ ( 6 ’ ) = erme} of T have a unique extension to
holomorphic characters
{x,(z)
= z”} of real group R, and complex C*.
’Chern classes describe the degree of ‘twisting” of a line (or more general vector) bundle L over 96. The topological structure of any vector bundle is determined by a family of transition coefficients {g,v(r):U,V-a pair of neighborhoods of z in 96). Coefficients {guv} take on values in a etc.), which depends on specific geometric features of L structure group G of L (GL, SL, (Riemannian, Hermitian, holomorphic, etc.). They satisfy the cocycle condition: g~v(z)gvw(z)gwu(r)= 1, for all triples U , V , W3 r; and gvu = g d v
so, su,
So { g u v } defines a G-valued 2-nd cohomology class on 96. But for line bundles (dim[fiber] = l), group G = C’, 80 multiplicative G-cocycle can be turned into the standard (additive) 2-cocycle: -v v
The resulting a E IfZ($$) is the first Chern class of L. In fact, Chern classes always define integral cohomologies, a E If2($$), in other words, differential 2-form a = a; dz;A d r j on 96, integrated over any 2-cycle, (closed 2-D submanifold) S C 96, yields integral values, #sa E Z. The second cohomology group of the sphere S2 = su(2)/T is well known to consist of integers, hence a = m.
272
56.1. induced representations and M a c k e y i group extension theory.
representation is equivalent to the familiar ?-spin representation nmI2 of $4.2. The details will be outlined in 56.3 (example 3).
In conclusion let us remark that the induction procedure in any of its modifications: standard, holomorphic, or, more general, partially holomorphic (“combined”) seems to provide a universal method of constructing irreducible representations. We shall further explore its meaning and significance in 56.3, based on Kirillov’s orbit method and geometric quantization.
$6.1. Induced representations and M a c k e y i group extension theory.
Problems and Exercises: 1. Establish formula (1.24) for characters of irreducible representations (Hint: let go E G and zo be its fixed point. Show that the map (z;h)+s(z)-'hs(z), from $ x H - + G is a diffeomorphism, that takes invariant measure d z d h on S x H, into the measure dp(g) on G , whose density relative to the Haar measure dg at the point go is equal to I det(1 -g;) 1. Use also formula (1.15) for induced representations and the known relation, that gives Jacobian g,: as the ratio of two modular functions on G and H,
-Ad*) - det(gg). AH(h)
The modular function on a locally compact group G is a ratio of the left-to-right invariant Haar measures: On Lie group G, A(g) = det(Ad,). 2. i) Derive the character formulae for representations {T'} of E, and {TP'u}of En. ii) Derive the Plancherel-inversion formulae (1.21)-( 1.22) for motion groups.
3. Establish the complex structure of the quotient-space 96 = SU(n)/T"-*, parametrization by the flag-manifold Sa!(n; C ) / B , .
using a
4. Find stabilizers of orbits of the Lorentz group SO(1;n-1) in M". Show that stabilizer K O of the light-cone coincides with the motion group En-l (Hint: write stabilizer algebra St, of point (1;-1;O;...) by block matrices,
the off-diagonal 2 x ( n - 2 ) blocks {!a}; {Ta;Ta} forms a p-component of the Cartan decomposition of St, (95.7); show that the commutator [Q;'p] is zero!).
273
36.2. The Heisenberg group and
274
the oscillator representation
$6.2. The Heisenberg group and the asdlator representation. The Heisenberg group plays the fundamental role in many areas of harmonic analysis, differential equations, number theory and quantum Physics. It gives a mathematical formulation of the Heisenberg uncertainty principle of quantum mechanics, and reveals close connections to the harmonic oscillator. The latter in turn gives rise to the fundamental creation-annihilation structure of the quantum field theory. In this section we shall develop the representation theory of the Heisenberg group, based on Stone-von Neumann Theorem. Then we apply it to spectral theory of the harmonic oscillator in R", establish connections of the Heisenberg group to symplectic/metaplectic groups and the Weyl algebra, and construct the oscillator representation.
2.1. The Heisenberg group with multiplication given by
W,
consists of all triples {g = (x,y,t):z,y E Rn; t E R}
gag'= (z,y;t)(z',y';t')
Its Lie algebra $,
N
(2.1)
= (xtz';y+y';ttt'tz.y').
R*"+' is also made of triples
{X = (x,y;t)}
with the Lie
bracket
[X; X']= (0,O;2.y'-y
*
x').
Group W, (respectively, algebra 8,) has a 1-D center Z = { ( O , O ; t ) } N W, and the commutative factor-group G/Z N W2", so H, forms a 2-step nilpotent group: [G; GI c 2.
In fact, W, is the simplest among all noncommutative (nilpotent) groups, the
cocycZe6
$(g; 9') = x y', from G x G-tZ, (2.2) measuring a deviation from commutativity. Group W, can be viewed as a semidirect product, G = H D N , of two commutative subgroups: H = ((0, y; t ) }N RnS1, and
N = {(z,O;O)}N W", with N acting on H by unipotent automorpisms,
6A function 4(z; y): G x G-R, on group G with values in R (or in more general commutative group 2) is called a 2-cocycle, if it satisfies: 4 ( a ; b ) - 4 ( a ; b c ) + 4 ( a b ; c ) - q 5 ( b ; c )= 0 , for all a , b , c EG. Here we use an additive convention { f } for the group operation in Z, and multiplicative for G. In other words, the coboundary, &(a;b;c) = 0 , identically. A 2-cocycle is trivial (coboundary), if it can be expressed through a single-variable function $(a), 4(a; b) = $(a) - $(ab) $(b) = a$(a; b), for all a, b E G.
+
Any 2-cocycle gives rise to a central eztension of G by Z, i.e. group H with center Z ( H ) N Z, so that H I 2 N G. The elements of H are pairs { ( z ; t ) : zE G;t E Z } , and the multiplication is defined by $1 (2;t ) * (Y;s) = (zy; t+s+d(z; Y)). The cocycle condition ensures associativity of the group multiplication in H, while trivial cocycle 4 yields the trivial central extensions, direct product, H N Z x G (problem 1).
56.2.The Heisenberg group and
the oscillator representation
275
It is often convenient to write H, in the complex form: C" x R with the product ( 2;
Here cocycle
4
t ),(w; T ) = ( z t w ; t t 7 t s 2 .m).
of (2.2) is replaced by an equivalent (antisymmetric) cocycle
(problem l), qY(a;b)= sz - m = '2;( z
-Z
.w).
(2.4)
One might wonder, to what extent the Heisenberg group is unique among all 2-step nilpotent groups with 1-D center. The answer proves to be positive, within the class of bilinear central eztensions, i.e. extensions defined by bilinear forms, g(a; b) = Qa .b, a, b
E R". Trivial
C$
clearly correspond to symmetric (quadratic) forms Q. So nontrivial
possible central extensions, should correspond to the quotient-space of "all bilinear forms
{Q}", modulo "symmetric {Q}".In other words central (bilinear) extensions of R" are labeled by all antisymmetric matrices
{Q}.But any such Q can be brought by the change
of basis into the form,
or the direct sum of such matrices. This means that space R" is decomposed into the sum, RZm@ Rk, where Q is nondegenerate on the former (like
4' of (2.4)), and annihilates
the latter. The corresponding Q-central extensions is obviously isomorphic to the sum
H, x Rk (Heisenberg plus abelian).
Sometimes one considers the factor-group W,,
modulo a discrete subgroup
Z = ((0,O;k))of the center 2, and calls it the Heisenberg group. In this form space W, is identified with C" x T, and the multiplication is given by (z;t).(w;s) = (z+w;tseiS'"); Finally, let us remark that
t , s - unit complex numbers.
W, can be realized by 3 x 3 upper triangular matrices
IL
with the standard matrix multiplication. Similarly, W, is realized by ( n + 2 ) x ( n + 2 ) matrices, with row-vector I ; column-vector y' and the n x n-identity diagonal block in the middle. Lie algebra 8, has 3 types of generators:
pi = (ei,O;O);q j = (O,ej;O); Z = (O,O;l),
276
56.2.The Heisenberg group
and the oscillator representation
where ei denotes the i-th basic vector in R". They satisfy the canonical (Heisenberg) commutation relations (CCR),
2.2. Canonical commutation relations. An important example of the Heisenbergpair is given by operators of multiplication and differentiation,
Q:f(z)+izf(z); P:f+X(af)(z); in R,
Qj: f(z)+izjf(z); Pk:f+X(akf)(z);in W",
(2.6)
which obey the relations,
[P;Q] = iX, or [Pj;Qk]= i M j k .
(2.7)
We shall see that (2.6), in fact, serves a model example for the Heisenberg relations. Namely, any CCR (2.7) can be realized by a pair of operators (2.6), by a Theorem of Weyl and von Neumann. Historical Remarks: Heisenberg commutation relations first appeared in the context of Quantum mechanics. The states of a quantum system are usually described by Hilbert space vectors { q ~E 36, e.g. 36= L2(R)}, while the observables are given by certain (typically symmetric, but often unbounded!) operators in 36. Examples include the so called position and momentum operators,
Pj:$,-.ihaj$, on 36
Qi:$(z)+zi$(z);
Lz(R"),
angular momentum operator (see J4.4),
M..= z.a .-z.a. $3
8
3
3
8'
and the most important of all the energy (Hamiltonian) operator 7 (e.g. Schrijdinger operator with potential V),
H = - h2 T A + V(z) = :Pz
+V(Q).
An expected value of an observable A at a state $ E 36 is given by the quadratic form
2 = (A)$
( A $ I $1,
while the "mean quadratic deviation",
-4
q A )=
I
= ( A-
mJIll
(2.8)
measures the error in observation. So the precise knowledge of observable A at state 3 (zerwerror!) is attainable only for special states: the eigenuectors of A. In the physical parlor they are called bound states of hamiltonian A. Thus measurability
(or
observability) of 14 becomes paramount to its diagonalization. 'Hamiltonian H completely determines the evolution of quantum system, namely the initial state $, E 36, evolves at time t into a state $ ( t ) = eitH[$,,].In other words, quantum evolution is given by a unitary group, generated by H.
86.2. The Heisenberg group and the oscillator Obviously,
277
representation
any pair of commuting observables { A ; E } can be simultaneously
diagonalized, i.e. observed to any degree of precision. However, noncommuting observables, like position {Qi} and momenta { P i } , cannot be diagonalized. We shall see that a Heisenberg canonical pair { X ; Y } has no nontrivial 1-D, even finite-D representations (problem 2)! It was observed experimentally, that the position and momentum of an electron can not
he accurately measured at once, the product of errors always remained greater than the Plank constant h. This led Heisenberg to state his famous Uncerfainty principle of the quantum theory in the form of the commutation relation,
[ P ;Q ] = ih.
(2.9)
Indeed, one can easily check that for any pair of observables { A ; E } errors (2.8) satisfy,
, any state 4. c ( A ) c ( E ) % ( [ A ; B ] )in
So the Heisenberg relation (2.9) provided a mathematical formulation of experimentally observed uncertainty between P and Q:
(2.10)
c ( P I 4 Q ) L h.
Relation (2.10) has a simple reformulation in terms of the Fourier transform (see $2.1): for any function
4 in the weighted Sobolev space:
I
I z$ I 2dz < 00;
I
I($ I 'd< < 00,
In fact, for any constants a, b we have
2.3. Representations of 4;Stonevon Neumann Theorem. Irreducible representations of W, fall into two classes: 1-D characters of the commutative factor group G / Z N W2",
and a one-parameter family of
m-D representations T A (A E W)
realized in Hilbert space
% L2(Rn) by operators
(2.12) One can check (problem 3), that''2
are induced by 1-D representations (characters)
x x ( b ; t ) = eiXt, of subgroup H = {(O,b;t)}, as explained in $6.1, T X= ind (G;xX H).
$6.2.The Heisenberg group and
278
the oscillator representation
Let us remark that, the structure and realization of { ~ ' s }(2.11), , (2.12), in the form of induced representations, can be easily derived from a semidirect product decomposition of
H,, and the Mackey's theory of 56.1 (problem 3).
However, we shall produce a direct argument based on the Stone-von Neumann Theorem. Let us observe that infinitesimal generators of {T'} (representation of the Lie algebra &) consists of multiplications, and differentiations:
; = P 3.; Z+iXI, (2.13) q,.+TX(q 3.) = iXxj = Q ~pj-+a. 3 the center being represented by scalar operator Z+iXI. Conversely, an infinitesimal representation (2.13) of Lie algebra 8, integrates through the Lie group-representation (2.12).
Theorem (Stone-von Neunmann): A pair of antisymmetric (unbounded) operators P and Q in Hilbert space 36 satisfying the Heisenberg commutation relations: [P;Q] = iXI, can be realized as a multiplication and a difierentiation:
Q$(z)= ~XZ$(Z), P11,(x)= a $ ( ~ ) , on scalar or vector-valued functions 11, E L2(W)(or L2 @ Y). In other words there ezists a unitary intertwining map W: 36-+L2, that takes Q into iAz and P into 8,. The irreducible action (representation) of P,Q an 36 corresponds to dimT = 1 (scalar functions), so any action is equivalent to a "dimY-multiple" of an irreducible action. The proof exploits spectral decomposition of a n antisymmetric operator Any such
Q (Appendix A).
Q can be realized by multiplication in the direct integral space: 36 = R ~ L d = z {Y, -valued L2-functions +(z) on R},
4
Q:+(z)+iz .).(ol Here we assumed X 1, without loss of generality. Operator P generates a one-parameter unitary group U t = e P t . The Heisenberg commutation relations imply
U,QU;'
=Q +itl.
(2.14)
The latter means that conjugation with U t shifts "spectrum of Q" (spectral subspaces) 8 by the amount { t } . Consequently, all spaces {Yz:z E R}, labeling values of {$(z)}, become isomorphic (equidimensional),
TIN Y, and operators {a,} define
a family of
Y-
unitary maps (cocycle) { u ( z ; t ) } ,50 that
(Ut+)(.) = 4 2 ;t)[+(z+t)l. '%f Q had discrete spectrum {Ak}, the relation (2.14) would mean that the eigensubspace E X is shifted by U t onto EX+1. In fact, continuity of t implies that Q has continuous, indeed absolutely continuous (Lebeague) spectrum!
$6.2. The Heisenberg group and the oscillator representation
279
Cocycle a is easily verified to be trivial: a(z;t)= u(z)-'a(z+t), which allows to transform operators { U t } into shifts of V-valued Lz-functions on R, V,rO(Z)
= rO(z - t ) .
via map W : $ ~ ( z ) - + u ( z ) [ $ ( But z ) ] . the latter is obviously generated by operator P = a,,
QED. We have stated Theorem 1 for a single Heisenberg pair {P;Q}.The result easily extends to n-tuples {P,; ... P,; Q,; ...&,}, satisfying CCR (2.5). Here one simultaneously diagonalizes all { Q j } ,and analyzes the action of n-parameter unitary group
{V, = ezp(t,P,+ ...t,P,): t = (tl;A,,)E R"} on joint "spectral subspaced' of Q's. Hilbert space 36 then turns into {V-valued Lz-functions on R"; with translationinvariant (Lebesgue!) measure}, operators { Qj } become multiplications by independent variables { z j } on L2(R"),while { P j } turn into differentiations {aj}. To apply Theorem 1 to irreducible representations of W,,
we observe that any
such T restricted on center 2 must be scalar, T 12 = eiXt. If X = 0, then T factors through the representation of the commutative quotient-group G / Z , so it becomes a character (2.11). For nonzeros X generators (Lie algebra) of
CCR. So Theorem 1 applies, and we get subgroups translations and multiplications on L2(Rn),
W,
obey the Heisenberg
{(a;O;O)} and
{(O;b;t)},acting by
TO$ = $(z+a); Tb$ = eWJ* "+t)$(z), whence follows (2.12). Figure 1 below illustrates the dual object (set of all irreducible representations) of W,: it consists of a series of infinite-D representations {Tx : X E R\O}, and the hyperplane of 1-D representations (characters) of W,/Z { p ( g ) = e2''"W;g = (w;t)}.
I -A
Fig.2: The dual object of the Heisenberg group i s made of 1-parameter family of co- D representations {Tx}, and a twoparameter (or Cn-parameter) family of characters { p } .
2.4. Characters of TXand the Plancherel formula: Representations TXextend in the usual way to any of convolution algebras on G = W,: e,(G), Ch(G), L'(G), etc.,
280
56.2. The Heisenberg group
and the oscillator representation
I
f+Ti
= Cf(g)Twg.
Formula (2.12) implies that operators { T i } are given by integral kernels on R",
q"; Y) = 7 (Y-z; xz; 4, where
(2.15)
f denotes the Fourier transform of f(a,b;t)in the second and third variables, N
f (...; [;A) =
f (...;b; t)e'(€ ' b+At)dbdt.
(2.16)
Rn+1
Evidently smooth compactly supported functions {f} on G yield nice (compact; Hilbert-Schmidt; trace-class) kernels Kf (Appendix B). As before we define the character of T x as a distribution on G, xx = trTA, via pairing to nice test-functions f,
(xAI f ) = tr T Y f ) . The latter is computed by (2.15)-(2.16),
I
t r K f = K f ( z ;z)dz = where
I
f (0;xz; X)dz = (?)"?(O;
RnN
0; A),
denotes the 1-variable Fourier transform in t:
f+T( ...;A) = If(...;t)e%t.
Thus we get t,he character formula for irreducible representations { T x }of
W,, (2.17)
We can interpret (2.17) by saying that distribution xx is supported on a (oneparameter subgroup) center 2, and is equal to ''&function of 2" x "G-invariant density { ( ~ ) n e i X t } ' Integmting r. (2.17) in X we immediately derive the Plancherel/inversion formula on W,, , all f(e) = tr T X ( ~ )d p ( ~ ) for
f E CF(G),
(2.18)
with the Plancherel measure (2.19) supported on the set of infinite-D irreducible representations {TA:X E W\{O}}. The inversion formula (2.18) yields as usual the Plancherel formula
where ?(A) = T ; ( f ) means the noncommutative (operator-valued) ''Fourier transform" of f , and tr (?(X)*?(X)) - its Hilbert-Schmidt norm.
1 ?(X)Ibs=
36.2.The Heisenberg group and the oscillator representation
281
We remark that all integral-operators {Kf}(2.15), with f E e,"(G), belong to the Hilbert-Schmidt class for all A. Indeed,
I K flhs=
7 (y-s;Xs;A)I
2
d s d y < oo!
2.5. The harmonic oscillator. The quantum-mechanical harmonic oscillator is a
Schrodinger operator with quadratic potential. We shall take it in the simplest form (from which more general cases could be easily deduced):
H = :(-A
+ Iz I
2),
+ s2), in 1-D.
in L2(Wn) or H = $(-a2
One is interested is spectral theory of H : its eigenvalues and eigenfunctions. The harmonic oscillator happened to belong to a rear and beautiful species, called solvable models. In other words one can write down explicitly all eigenvalues and eigenfunctions of H . Theorem 2: Spectrum of operator H is purely discrete. In 1-D it consists of a sequence of eigenvalues: A, = k ;: k = 0;1; ...; the corresponding eigenfinctiona being the classical Hermite functions on W, (2.21) Here {hk} denote the classical Hermite polynomialsg, h,(") = 1 e 2 P / 2 ( e - 2 2 ) ( k ) , (2.22)
+
fi
W" has eigenvalues {Ak}, labeled b y n-tuples of integers, = (k,+ ... + k,) +i;and the corresponding eigenfinctions are
The multi-D oscillator on
k = (k,;...k,),
products of 1-variable Hermite functions, $,.(x) = $k,(q)...$kn(zn).
Normalizing factors
{&}
render system {&}
orthonormal in L2(W),
11 G k llL2 = lWe shall establish Theorem 1, aa a simple application of the Heisenberg CCR. But this time we choose a different realization of generators { p ; q } of H,, namely by the so called creation/annihilation pair10,
(2.23) or similarly defined creation/annihilation n-tuples a.=.l(aj+zi);a~=-L(-ak+zk);
J &
Clearly, daggered operator {a:}
J;
15 j ; k < n .
(2.24)
are adjoint to {ak] in L2(R"). One can easily verify that
'Hermite polynomials form one of three known families of classical orthogonal polynomials, along with Laguerre (chapter 8), and Jacobi (chapter 4) (see [Erd];[Leb]).
86.2. The Heisenberg group and the oscillator representation
282 pair {a;.’}
obey the Heisenberg CCR, [a; at]
= ;[a + 2;a - 21 = 1;
or [ a j ; a l ]= 6 j k I .
Q = 2;P = iV, operators a and
But unlike, the position and momentum,
af
adjoint in L2.The harmonic oscillator can be represented in terms of the pair
H = %-a2 + 22) = at,
+ ;= .at- f.
are not self{a;at},
(2.25)
Finally, triple {a; at; If} obeys the commutation relations,
[ H ;at] = at; [R;a] = -a. In this regard
{at;,}
(2.26)
behave, like the raising/lowering elements { X ; Y }of the Lie algebra
4 2 ) (chapter 4). The oscillator, however, is not a Cartan (diagonal) element of pair {at;.},
since [at;a] = I , rather than :If, as in 42)!. The nature of the triple {at;a; H} in
relation t o 4 2 ) will be elucidated below. Commutation relations (2.26) readily yield all spectral results of Theorem 2. Indeed, if X is an eigenvalue of H and cl, - the corresponding eigenvector, then all $k = ( a t ) k [ $ ] ,and
+-,,
If[$] = w,
= a”[?f~]are also eigenvectors,
Since operator H is positive,
14) = it has the lowest eigenvalue A,
1 V+ 1 2 + 2 2 14 12)dz > 0, for all 4 E L Z , 2 0, and eigenfunction $,
called the ground-slate, and the
ground-state energy, i.e. the lowest-energy states of the quantum system. These must be
annihilated by the lowering operator, a[cl,o]=
(a + z)$~= 0;
which immediately yields the ground-state 2
&(z) = e-z 12- the Gaussian.
‘“The terminology came from the quantum-field theory, whose main task is to account for the creation-annihilation of quantum particles (or better to say, “particlestates” of quantum-fields) in subatomic interactions. The mathematical structure for the creation-annihilation processes is based on the notion of (multi-particle) Fock-space, discussed below. I t typically has a (unique) vacuum-state (vector) w, and the corresponding space 36, = span{w}, as well “1-particle”, “2-particlen, ... spaces: 36,; X2;... The creation operators {at} send 36,-;.36,; 36,+36,; ... ; the particles being “created from the vacuum”, while annihilation operators go in the opposite direction, a:36n+36n-1+...+360, thus diminishing the particle number. The simplest model that accommodates such features are raising/lowering element,s {X; Y} of 4 2 ) (chapter 4).
56.2,The Heisenberg group and Substitution of
283
the oscillator representation
do in H (2.25) yields the lowest eigenvalue, H[+J = (a,+
+ i)+o= :lo*
m\
hence all other eigenvalues,
Ak = ;+ t ; t = 0; 1;2; ... Let
+k
= (at)k[$J denote the t-th eigenfunction of H . To get a Aermite-representationof
(lk we apply yet another identity,
= 6 - z = e"2/2(j
-,t
[e-Z2/q,
in other words the raising operator at is conjugated to a derivative-operator 8, via multiplication with the Gaussian. Hence, (J)k
= ,z2/2 ( - qk[e-z2/2].
Applying the latter to the ground-state +o we get the Rodriguez formula (2.22) for +kr
QED.
2.6. The oscillator representation and the metaplectic group. The Heisenberg Lie algebra $, acts naturally on Wn by differentiations and multiplications (positionmomentum operators) (Theorem l), and thus gives rise to the Weyl algebra W = W , (an associative hull of made of all differential operators with polynomial coefficients: A= a,px4aor. (2.27) la+PI < m Algebra W is graded according to the total degree m in variables {z} and derivatives of polynomials A = a ( x ; a )in (2.27),
en),
c
{a}
W o = {const} c W'
c W 2 c ... c W m = { A degA < m} c ...
One can easily check (problem 4) the product and the commutation formulae, W P Wc ~ W P + ~[ ;W P ; W ~ cIW P + ~ -for ~ ; all p , q 2 0.
(2.28)
The lost of 2 degrees in the commutator [ A , B ](A E WP, B E W q ) ,results from the basic relation = const, extended to other generators {x( ;a Y P} of W (problem
[.;a]
4). Hence follows (i) 2-nd degree operators {A} ( m = 2 ) form a Lie subalgebra W 2 of W , with respect to the natural commutator bracket: [A;B ] = AB - BA. In fact, (ii) A subspace of first degree operators {A} (rn = 1) is an ideal of W 2 , isomorphic to Heisenberg algebra $ =
a;,
86.2. The Heisenberg group and
284
the oscillator representation
(iji) W 2 contains a symplectic subalgebra 1' 1 N operators {a$; riaj; zizj}.
sdn), spanned
by all 2-nd order
So Lie algebra W 2 factors into a semidirect product s~'11,with 1' 1 acting by derivation on 8.Consequently, Lie group of W 2 also breaks into a semidirect product:
W,pG, where G denotes a simply connected cover of the symplectic group Sdn). So group S d n ) , and its cover G,act by automorphisms of Lie algebra Q,, and group W,. The latter could be also verified by the direct computation. Namely any
Sdn) 3 g
=[:
~]:(z,y;t)-(~z+cy;~z+dy;l).
(2.29)
respects the Heisenberg Lie bracket on triples {(z;y;t):z,y E R"; t E R} (problem 5).
In the previous subsection we have constructed an irreducible representation of
W, in space L2(Rn),of the form T j f ( z )= ei X ( z. bSc) f
(z+a); g = ( a ,b; c ) E
w,;
whose generators were given by operators,
p j 43.; q3.-Axj; Z+iX.
(2.30)
Formulae (2.30) extends through the representation of the Weyl algebra, in particular, its 2-nd order (Lie) part
W2.So we get
sp(n)-generators acting in L2(Rn) by
operators, q . .+z.d .; qiqj--iiXz.z .. Pipj +La?.; 2~ $3 8, 1 3 2 3
The latter can be lifted to a simply connected cover-group
(2.31)
gdn),and yields the
celebrated oscillator-representation (also known as spinor, metaplectic, Borel-Shale-
Wed), that appears in many different places and finds numerous applications. Another way to introduce the oscillator representation comes from the action of S d n ) by automorphisms of W, (2.29). Let us observe that symplectic automorphisms { u}preserve the center of W,, zu = z, for all z E Z. Hence two representations: Ti, and T i (g E W,), are equivalent, any T Xbeing uniquely determined by its value on 2,
T? = As a consequence we get a family of intertwining operators, { T u x E S d n ) } , determined modulo scalars (like in the Mackey's theory, 56.1). They define a projective representation of G = Sp(n) in L2(R"),
T,, = a(u,u)T,T,; for all u,u E Sdn).
(2.32)
with cocycle a. Any a-projective representation of any group G was shown in $3.2 to correspond to an "honest" representation of a central extension of G,(group G, with a
s6.2.The Heisenberg group and
285
the oscillator representation
center 2 c T, so that Ga/Z E G). In fact, Heisenberg representation (2.12) arises that way (problem 6). Let us also remark that a symplectic extension (2.32) of T’ depends only on sign of A, T = T It remains to compute cocycle a. We shall see that (Y is not trivial on S d n ) , but trivializes on a finite cover of Sdn). For the sake of presentation we consider here only 1-dimensional case Sp(1) SL2(R) (see problem 7 for the general case).
*.
Proposition 3: Cocycle a on SL, is 2-valuedJ a(.) = f I , so the Corresponding central eztension of SL, f o m a 2-fold cover, called the metaplectic group Mp(1).
”=
l1 1
denote the generator of rotations S q 2 ) C SL,. It corresponds to a
quadratic Let e ement i ( p z + q z ) in the Weyl algebra W 2 , which is taken by representation T (2.31) to the harmonic oscillator,
A = ;(-a2
+ z’),
The analysis of the oscillator in the previous part showed its spectrum to consist of halfintegers: {;$:;
...}. Hence, a unitary group, generated by A via (2.31), takes orthogonal
rotations
L
into unitary operators
J
1 The resulting representation, 4+T
44)’
becomes single-valued on a 2-fold cover of S0(2),
isomorphic to Su(2), so cocycle a becomes trivial on Su(2). The corresponding 2-fold cover of SL, “resolves” (trivializes) cocycle a on a subgroup So(2),
a(u;w ) =*- ’(uw)
P(u)P(w)’
for all u;w E sq2).
If G denotes the corresponding 2-fold cover of
(2.33)
SL,, then (2.33), along with the Cartan
decomposition, g = uhv (u,w E K ; h E H) on G , yields the trivial cocycle a on the entire group G, &ED.
2.7. Symmetries and spectral multiplicities of the oscillator. The symplectic group and its spinor representation allow to explain spectral multiplicities of the multiD harmonic oscillator H = ;(-A I z I ’) in Rn. The oscillator has an obvious SO(n)-
+
symmetry, since both the Laplacian A and potential However, the multiplicity of X k = Ic,
Iz I
commute with rotations.
56.2.The Heisenberg group and the oscillator representation
286
#(Ak)
= {(i in):(il+;)+ ...+(i " + f ) = k } = ( k p ) , is much higher than could predicted, based on SO(n)-symmetry (compare the obvious cases of
W2 and W3). This suggests that H might possess a larger symmetry-group. This,
indeed, proves to be the case.
Theorem 4 The symmetry group of the oscillator H in Sp(n) coincides with the unitary group SU(n), and the restriction of S y n ) on eigenspaces is irreducible. Let us remark that both S q n ) and S w n ) are subgroups of Sp(n), the former is given by all block-diagonal matrices
Sqn)= whil: J
=
{[
u
1-
~u u = I } ,
the-latter is made of all 2 n x 2 n orthogonal matrices that commute with I
J.
We associate with any matrix A on the (Heisenberg) phase-space { ( z ; p ) } = R'", a quadratic form
c
fa(";P) = ( A z I P) =
OijziPj9
and the corresponding differential operator LA, given by any possible convention: left {ziaj); right {a,zj} or symmetric (Weyl) {i(ziaj
LA =
Coijriaj
+ ajzi)}, so
(in the left convention).
The oscillator clearly corresponds to the identity matrix,
H=:(-A+ ~
'I'
Z ~ ~ ) C * T
(2.34)
Next we observe that the natural linear action o group GL(2n) on R2" is transformed under the map A+LA to conjugation of matrices, g : A+TgAg.
On the other hand the oscillator representation assigns to any symplectic g a unitary operator T, in L2(Rn), so that
T,-'L
T -L
; g E S d n ) ; A E g42n). ('gAg) So the commutator of H consists precisely of symplectic matrices, that preserve the A g-
Euclidian inner-product form (2.34), { g : T g . g = I } , i.e. group s q 2 n ) n S d n ) = Sqn) (the maximal compact subgroup of S d n ) , see 55.7). Conversely, the commutator of S q n ) can be shown to coincide with {H}. Thus { S q n ) ) and {H} form a maximal commuting pair in the sense of 54.4-4.5 (like the Laplacian
A
on S" and the orthogonal group
S q n + l ) ) . Since the metaplectic representation is irreducible in L2(R") any pair should break it into the sum of "joint irreducible componentsn, $gk. Hence, restrictions k H 18, = lk, and S y n ) I k,-irreducible, QED.
Remark: In the differential-operator form (2.31) Lie algebra w(n) can be represented in
56.2. The Heisenberg group and the oscillator representation
287
terms of the creation-annihilation operators, { a i ; a j t } of (2.24). We set X i j = aiajt, and have the reader verify that { X i j } satisfy the commutation relations of the rujn)-basis, and commute with H . [xij;xkJ = 6 i m X k j + 6 j k X ; , ;
[ H ; X1.1. ]= 0.
After the eigenspaces of H are shown to be irreducible under Syn) the natural question arises, what are these irreducibles {r'}),in terms of their weights, as described in $5.3. It would be difficult to derive the weights directly from the L2(Wn)-realizationof metaplectic T, as the latter does not correspond to any natural (regular/induced) action of SU(n) on its quotients. As we shall see the proper way to interpret TI SU(n) is in terms of the holomorphic-induction of $6.1. This would require yet another realization of T in the complex domain, that we shall explain.
2.8. The Bagman-%gal representation. We shall conclude this section with yet another realization of irreducible representations ITA}of W, and the Weyl algebra in spaces of holomorphic functions on C and Cn, 36=I(C)={F(z):llFI12=
I
'1
IF(Z)~~~-I'I dz .
(2.35)
{z"}r
Polynomials are easily verified (using polar coordinates) to form an orthogonal system in 36, with norms, II 2" 112 = nn!.
1
So we get an orthonormal basis (&zn:n =0;1;-. . Two operators, multiplication z: F(z)+zF(z), and complex differentiation a:F+F ( z ) , are adjoint one to the other, relative to the product (2.35), and obey the Heisenberg CCR, [.;a,] = 1.
So they behave like the creation/annihilation pair { u ; u t } (2.24), while their real/imaginary parts become the position/momentum operators: p = J1( l . t a); q = +( 2-8); z = i A.
d-.
Infinitesimal representation (2.36) of Lie algebra @ could be (exponentiated) through a unitary representation of Lie group W in space %(a)),
(2.36) lifted
In fact, the equivalence of all generators and representations: 2'' on L2(R) (2.12) and U Aon X(C) (2.37), can be established via an intertwining map W:Lz+J6, (2.38) J
-00
288
56.2. The Heisenberg group and the oscillator representation
We shall leave the details to the reader (see problem 8), and just mention that Bargman-Segal representation and spaces { 36(C)} have many remarkable features. One of them is the reproducing kernel { K ( z ; w ) }on CxC, which gives the value of F E f16 at any point z in terms of integrals over C,
F ( z ) = (K(z; ...) 1 J'(...)),2
K(a;w)F(w)e-
I I
2
d2w. C The reproducing kernel on space X(C) is equal to K(z;w) = ex* ' iij. =
Finally, we can go back to the decomposition problem for the metaplectic T , restricted on subgroup SU(n). Let us observe that SU(n) acts on C" N W2" in the natural way, by unitary linear maps, z+z". Hence, its action on space 36 consists of coordinate transformations,
T,: F(z)+F(z"), on polynomials { F ( z ) } . Space of polynomials 9 is made of homogeneous components of various degrees,
each 9k identified with symmetric tensors, and T 19, is precisely the k-th symmetric tensor power of the natural representation x in C". So it has signature (weight) a = (k;O; ...0) (see 55.3)! Further results and details could be found in [How]; [Ta2]; [GSl].
289
56.2.The Heisenberg group and the oscillator representation Problems and Exercises: 1. i) Show that any Z-valued 2-cocycle 4(a; b) on group G defines a central extension H of G by Z, via multiplication formula: (2; 1 ) .(y; s) = (zy; t+s+$(z; y)), 2, y E G; t , s E Z. ii) The extension H is trivial, iff the cocycle 4 is trivial: 4(a;b) = +(a) -+(cab) ++(a) = B+(a;b),for some $(a); iii) Two extensions H,; H,, defined by cocycles 9,; 42 are equivalent (isomorphic), iff 4, and d2 differ by a trivial cocycle a+ (Hint: construct an isomorphism u:H1+H2, via map, (a;O)+(a; +(a)). iv) Apply the above results to the Heisenberg group to show that cocycle (2.2) is nontrivial, and two cocycles: 4 , ( z ; z') = z y'; q&(z; w ) = 3 ( z E ). Hence two multiplication formulae are equivalent.
-
2. Show that the Heisenberg algebra has no nontrivial finitedimensional representations (Hint: commutator [Tx;Ty]has trace 0, so it cannot be X I with X # O!). 3. Apply the Mackey's groupextension theory for semi-direct products to derive all irreducible representations (2.11)-(2.12) of H,.
c WP+q-', for the Weyl algebra (Do it first for generators { z a ; a } of W, starting with the basic relations [ti$,] = 6 i j ) .
4. (i) Check the commutati n relation [Wp;W.]
B
(ii) conclude that the first-degree part W' is isomorphic to the Heisenberg algebra via map, P = a . a + b . z +c+(a; b ; c ) E 8, (a;b E R"; c E R).
s,,
(ii) Show that the 2-nd order operators of the form:
P = C a 13. .a?. + C bijziaj + C c i j t i z j = A a . a + B Z .a + Cz.t, :I
-. aE
with symmetric matrices { A ; B } and an arbitrary C E gqn), form a symplectic Lie algebra via identification:
C
B
'dn)*
5. Show that (2.29) defines an automorp ism of ,, that preserves its center, and any such automorphism is given by an element of Sdn). 6. Show that representation TX(2.12) of H, comes from a projective representation of the commutative group R2" N C", defined by the cocycle
a(z;w ) = e iX3(r.
= eiX(z * Y' - Y '"); where = t+iy;
7. Show that metaplectic group
-
= z'+iy'.
Mdn) is a 2"-fold cover of Sdn), with center ZN
Use commuting oscillators: A 3. = $(-a:+z:);
z2x
...x z 2
j = 1; ...n, in the Weyl algebra W,.
8. i) Check that map (2.38) intertwines representations TXand U x of H, in spaces L2(R") and X(C") (use Lie algebra generators). ii) Find the unitary group ( S q 2 ) c Sdl)), generated by the oscillator $(p2 %realization of the metaplectic extension of TX,and show that
T : ~ ) F ( ~=)eidI2F(zei$.
+ q 2 ) in the (2.39)
iii) Show that the standard Fourier transform T:L2(R)+L2(R) corresponds to 4 = (2.39).
4in
56.3.The Kirillov orbit
290
method
s6.3. The Kirillov orbit method. Many results of the representation theory of H,, outlined in the preceding section can be extended to arbitrary nilpotent, solvable, exponential and more general Lie groups [Kirl,Z], [Pu]. The key idea, crystallized in the work of Kirillov, Souriau, Kostant et al., w a to associate irreducible representations of G to orbits 0 of the co-adjoint action of G in the dual space (5' to Lie algebra (s. We shall outline the construction of representations To,then derive the character formulae, and the Plancherel measure, based on the orbit method.
3.1. Construction of representations 9. A co-adjoint orbit 0 c 8*always carries a natural sympkctic structure, nonsingular skew-symmetric bilinear form on tangent spaces: B = B 6 ( X ; Y ) ;X , Y E T6 - tangent vectors at point t E 0. Equivalently, there exists a nonsingular differential %form:
R = RB = Cbijdt' A d t j (in local coordinates { ( j ) on 0). The latter defines a Poisson-Lie bracket on functions (observables)
(3.1)
{f} on
0 (see
chapter 8):
So the space of smooth functions em(0) turns into an m-dimensional Lie algebra. The tangent space of 0 at point {t}is identified with the quotient 8/8(- Lie algebra modulo the stabilizer of t, G6= { Y : a d ; ( t ) = 0). By definition form B, at a point
t E 0 is equal to B&X;Y )= ( [ X Y , ]I t),for any pair of elements X , Y E 8.
(3.2)
Clearly, B6 depends only on the images (projections) of vectors X,Y on the We shall list a few basic properties of form B: tangent space T6= (5/8(.
B6 annihilates X , Y E G6,so it depends only on classes of X , Y in the tangent space T6 8/8( 0
0
B 6 is nonsingular on TC(symplectic)
0
2-form RB (3.1) is closed, i.e. differential
don,7
1<3
( d j b j k - djb;k
-+ dkbjj)dfi A d t j A d ( k = 0.
(3.3)
The latter follows from the Jacobi identity on Lie algebra 8 (problem 1). We also have 0
bilinear form B is G-invariant on 0 (with respect to the co-adjoint action).
56.3.The Kirillov orbit method
291
In particular, all co-orbits in 0*have even dimensions, d i m 0 = 2m, and the product L?A ... AL? (m-times) defines an invariant volume element on 0. In the standard terminology symplectic form B turns 0 to a classical mechanical phase-space
G.Elements of Lie classical observables (hamiltonians) on 0, dx([) = (XI t).
(see chapter 8 ) , with a large symmetry group family of
algebra 0 define a
In the general setup of Geometric quantization one is given a classical mechanical system (symplectic manifold 0 ) with a Hamiltonian h, or a family of hamiltonians { h j } , closed with respect to the Poisson-Lie bracket on 0, in other words a Poisson-Lie
algebra 0 of classical observables (functions on 0). Then one is asked to construct a Quantum (Hilbert) phase-space J6, and to assign quantum observables (operators in
X)
to classical observables, maintaining all possible Lie symmetries of the system. In other words one has to construct an irreducible representation of group G, associated to 0. The general construction exploits the notion of an admissible subalgebra (or polarization) of 0,i.e. a maximal subalgebra $, that satisfies (E 1 [$;$I) = 0, so [ defines a 1-D (local) representation of subgroup H = exp$, c r ( e t p Y ) = ei(t I
Subalgebras $ satisfying
([$;$I I t )= 0,
'),
I/ E $.
(3.4)
are said to be subordinated to (. Any
subordinated subalgebra $ has dim(@/$) 2 &mO,
since $/0(c T t ( 0 ) is an isotropic
subspace of symplectic form B. In many cases the maximal allowed dimension (codim$=@imO) is attained by some $ c 0 . Moreover, orbit 0 contains the nullspace $ = {q:(q I $) = 0) (the so called Pukanszky condition), so orbit 0 is foliated
'
into subspaces of dimension =$limo. This condition holds for large classes of semisimple and solvable Lie groups (examples 1,2 below). Fig.3 gives a schematic view of a co-adjoint orbit 0 foliated into linear subspaces {Sj } of $dim0 over the quotient space Y.
The {$
}-foliation of 0, or rather its foliation into H-orbits", of dim = g i m 0 ,
"Notice that subgroup H does foliate 0 into disjoint orbits: for any pair ad(H)t, = ad(H)t2, or ad(H ) t l ad(H ) t P = 0.
n
t1;t2E 0, either
292
56.3. The Kirillov orbit method
allows to introduce local canonically-conjugate coordinates, like the position and momenta {qi;pi}-variables of the previous section. Formula (3.4) then defines a 1-D representation of an admissible Lie algebra 8,or a local representation x of group H. We don't know, however, whether x could be extended through the global representation of H , and how many different extensions can be defined by x. The answer turns out to involve some topological constraints on the orbit, and its quotientspace SJ = O / H (fig.3).
Theorem 1 (Kirillov): (i) Local representation ,y(ezpY) = e'(' I '), Y E s, extends through the global representation of an admissible subgroup H , iff orbit 0 is integral, i.e. the fundamental form (closed 2-D surface) I:c 0),
RB takes on integral values on any 2-cycle
f
0, = m. (3.5) I: In the standard terminology, one says that cocycle RB h m integral cohomology, H2(O;Z).
(ii) Different extensions of x are parametrized by characters of the fundamental group r =r,(SJ), i.e. b y elements of the 1-st cohomology group, H*(SJ;T)= H'(O;B). Once the character x is constructed we define TO, as an induced representation of G, T = ind(a H ; G ) . More precisely, irreducible representations of group G are labeled by 2 parameters: an integral (quantized) orbit 0 c @I*, and an elements p of the first cohomology group H'(SJ;T); so T = To@. Representation T is naturally realized on functions (sections) over the quotient-space A = O/"foliation d i m A = $dime, and one can show T to be irreducible.
!&",
The proof of irreducibility exploits specifics of the Lie group structure of G. For nilpotent Lie groups one applies the dimensional reduction, and the natural 'functorial properties" of the correspondence "orbits
+
representation" with respect to operations of induction
and restriction: S I H+T = ind(S I H;G), and T+T
I H,
[Kirl,2]. Typically, both
operations yield decomposable representations { i n d S , or TI H}, and one asks for the direct integral decomposition of both. It turns out [Kir], that i) an irreducible T = To restricted on H, To
I =
fs
- direct
integral over the set of
H-orbits w c p ( O ) , where p denotes the natural projection 8*- 8*; ii) an
induced representation
T = ind(Sw I H ; G ) is expanded into the direct
To over the set of G-orbits 0, intersecting p-'(w) c 8*.
66.3. The Kirillov orbit method
293
The latter shows that irreducibility of ind(x; ...) is equivalent to the inverse image of point x E $*, p - ’ ( x ) being contained in a single G-orbit. This explains irreducibility of induced T for admissible subgroups H . We refer to [Kirl] for further details.
A modification of the above construction involves compleified Lie algebras (5 and 5, and “holomorphically induced representations” (56.1) in spaces of holomorphic functions/sections over 0. In such context certain integrability conditions on 0 arise, so orbits are “quantized”, in the sense that certain parameters describing 0 become discretized. The induced (irreducible) representations on 0, that arise in this way, are parametrized by characters of the fundamental group ~ ‘ ( 0 The ) . foremost example will be the Borel-Wed-Bott Theorem for compact Lie groups, which gives irreducible { T} realized in sections of “holomorphic vector bundles over 0”. Let us illustrate the foregoing with 3 examples.
3.2. Heisenberg groups W, and W, are conveniently represented by ( n + l ) x (n+l)matrices
where row 2 and column y E Wn, t E R. Their Lie algebras @ have the same (uppertriangular) form with zeros on the main diagonal. So the dual space @* can be identified with the lower-triangular matrices
via pairing of elements {X = X(z,y;t) E @} to functionals {$ = $(t;v;a)E @I},
($1 X ) = tr($X) = [-z+v.y+ at. The co-adjoint action consists of conjugation of 1c, by elements {g}: +g-’+g, and subsequent truncation of the lower-triangular part of the product, ad;($) = [! -I l$gl-.
(3.6)
The reader easily computes (3.6) to be ( [ - a b ; q + a a ; a ) ;for g = g(a;b;t). So the co-orbits are hyperplanes 0 = 0, = {(t,9;a):a = const} N
points {([;9;0)}
W2”, and all
in the hyperplane a = 0 (see fig.4). No orbit-quantization(3.5) required
$6.3. The Kirillov orbit method
294
here, as all (0,) (and all their cohomologies) are trivial. Each orbit has the standard polarization, and corresponds to an irreducible representation T A(A # 0) induced by an admissible subgroup H = {g = g(0;b; t ) } of G.
Fig.4. Co-orbiis of Heisenberg group H, consist of two families: 2n-D hyperplanes: z = A # 0, that correspond to irreducible represeniaiions {TA} of f6.2. The second family is made of all points (0-D orbits) of the plane z = 0.
3.3. Euclidian motion group En = R" D S q n ) is also realized by (n+1) x (n+l)matrices:
I; column [ Hence its Lie algebra consists of matrices, g = ( u ; a )=
a E R"; u E S q n ) .
~1
with the bracket,
and the dual space O* could be represented by lower-triangular matrices
{$=[:
1
o ] ~ E 4 n )r ;o w t E R n ,
with the pairing
(4 I X ) = tr($X) = t r ( P K )+ t - 2. The adjoint action of G on 0 then becomes
while the co-adjoint action
Here (u denotes the left multiplication of row-vector ( by orthogonal matrix u. Symbol a A ( stands for the antisymmetric part of rank-one matrix = ( u i t j ) , so ( a A <)ij = a i t j - &aj.
66.3. The Kirillov orbit method
295
Clearly, co-orbit 0 c 8*is fibered over its projection onto the abelian component
W" of 8*, a sphere S, = (1= ( u : u E Sqn)} of radius r = I ( I. We leave it to the reader to verify that fibers { T r l :E~S,} are (n-1)-dimensional linear subspaces of the antisymmetric so(n)-component of 8*. Figd. Co-orbits of the motion group E, are and a cylinders {(<;s): 1 € 1 = r > 0}, discrete (quantized) set of points ( 0 - D orbits) {(O;m)} on the s-azis. The former give a family of co-D represeniations {T'}, the laiier correspond t o characters {x,} of the compact quotient M2/R2N So(2).
We shall elaborate the rest of construction in the simplest case of plane motions
IE,.
Here co-adjoint elements are identified with pairs {(s;()}, where s w P s = [-s labels the anti-symmetric component. There are 2 classes of co-orbits: cylinders
1
O = O(() = { ( s - u A ( ; f u ) : u E R2; = "0 E SO(2)}, > 0, fibered over the circle S, = { I ( I = r } , and l-parameter family of degenerate (O-D) orbits { (s;0): s E W} (fig.5). The canonical 2-form on cylinders 0, is
of radius r
0, = rdd A ds. No quantization condition (3.5) arises here, as $Qr = 0, for all 2-cycles y c 0 7 (there are no nontrivial 2-cycles!). The reduced space (base) 9J-T has fundamental 1 )Z, whose characters (dual group) form a torus r = T. group 'I = ~ ~ ( 9 = We fix a point ( = ( 0 ; t )€ 0 , and find an admissible subalgebra $ = W 2 of (. According to the general theory, character x t ( u ) = eY ' a (u E$j), extends through a representation of the subgroup H = ezp$,
in f-different ways, labeled by angle
q5 E [0;27r]. So we get a 2-parameter family of characters { ~ ~ ; >~ O;q5 : r E [ 0 ; 2 ~ ] }of the admissible subgroup H , that induce irreducible representations: T r ; 4 = i n d ( ~ , . ;I ~H ;G ) . The appearance of a second parameter
4 seems to contradict our previous description of
the dual object of M, in J6.1. There we have shown M, to consist of a l-parameter family
{T'}, plus a discrete (quantized) set of characters IT"' = eims} of the quotient So(2). The contradiction is resolved by remembering that the general results (Theorem 1) apply N
to a simply connected cover M , which represents a central extension of M, via group I . The condition Tr;$I I = Z (trivial) selects a single member T' of the family {TrT9:4}.
56.3. The Kirillov orbit method
296
The quantization of degenerate (one-point) orbits {(s;O):s = 2x772) results from compactness of the admissible quotient H / [ H ; H ]N Sq2),H = MI,. Let us remark that all higher-D motion groups En (n 2 3) are themselves simply connected, so the quantization rules of Theorem 1 apply directly to them (problem 2). Co-orbits of examples 3.1-3.2 had real polarizations { H } , so the resulting irreducible representations {To} were obtained by the usual induction procedure. Our next example, compact group SU(2), lies at the opposite extreme. Its irreducible representations are all finite-dimensional (chapters 4-5), so they can not be induced in the usual sense. However, we shall see all of them realized as "holomorphically induced representations", according to the Borel-Weil-Bott prescription of 56.1. 3.4. Quantization of SU(2) and the Borel-Weil-Bott Theorem. Here
(5
= @*
N
R3
and co-adjoint orbits are spheres 0, = {tr(X2)= -?}
N
SU(Z)/U(1).
We identify each orbit with a complex projective space CP' via the map
Iu wl-+
G 3 -~i7 where matrices
{[' *]:
(u;w)/{(e"u; e"w)}
N
C x C/mod(C);
X = e"} make up the diagonal subgroup U(1) c SU(2).
The projective space ONCP' has standard homogeneous coordinate z in two neighborhoods 0 *, that cover it: z=$inO+={(u;w):w#O}, and z = $ i n O - =
{(u;w):u#O}.
Furthermore, space 0 is equipped with a family of symplectic structures which come from different orbits { O r } ,
One can show that R, is the only G-invariant 2-form on CP' (problem 3). The argument exploits the G-action on CP' by fractional-linear transformations:
Parameter (Y is simply related to the radius r of 0. The quantization condition of Theorem 1 requires (Y to be an integer m. The corresponding representation space is formed by sections of the holomorphic line bundle L, whose l-st Chern class is given by 0,. Following the general prescription of geometric quantization we express form R (3.9) as the exterior derivative of a l-form12 w = w (on each hemisphere 0 ). So we
297
56.3. The Kirillov orbit method write R = dw+ = dw-. The difference w+
- w- = d4 = 1 'dg, 2ri ,
where g = g(z) is a holomorphic transition function on the overlap 0+n0- N C\{O}. But any holomorphic nonvanishing function g ( z ) in C\{O} can be written as f+(z)z"f-(!), for some integer m, with holomorphic functions f* on C. This shows that any holomorphic line-bundle over CP' is equivalent to the one with transition function g = z". Taking such g(z,) we calculate 2-form R, on the 2-cycle 0 N S*, and find dw- =
-
4
g - ' d g = m.
equator
So the requirement that L have a proper line-bundle structure sets the integral over 0 (cohomology class) of R to an integral value rn! Holomorphic sections of L consist of functions {f,} transition function g: f+(Z+)
on 0, related by the
= g(z+)f-(z-) = Z"f-(Z-).
So if one asks both functions be analytic, both must be polynomials in z of degree m. Thus we get (mt1)-dimensional space of polynomials. To establish the Borel-WeilBott Theorem for SU(2) we only need to compute the SU(2)-action on sections of L. Remembering the fractional-linear action of G on CP', we get the familiar form irreducible representations of chapter 4 (§4.2),
Remark The above construction of irreducible representations, based on co-orbits in 0*, is a special case of the more general Geomefric quantization procedure on an arbitrary symplectic manifold (classical-mechanical phase-space) 9.The latter is typically equipped with a canonical 2-form, and a Poisson-Lie algebra
(9
of so-called primary obseruables,
functions of 9. In some cases, e.g. Lie group G = e z p ( 0 ) acting transitively on 9,the phase-space can be identified with a co-orbit 9 N 0 c 0*, via the momentum-map (see 8.1), so the above theory applies here. We refer to [Kirl]; [Kos]; [Woo]; [Hurl; [Sni] for
further details.
w can be interpreted as a connection form of a line-bundle L over CP'. An easy -Form computation shows, m Zdz * z t = 2*i(l+ 27 ); =
*
56.3. The Kirillov orbit method
298
3.5. The character formula and the Plancherel measure. We shall conclude this chapter with a universal character- and Plancherel-formula on Lie group G in terms of the orbit structure of 6*,due to A. Kirillov. These results apply to large classes of groups and representations, including compact Lie groups; SL,(W); exponential groups'' G (whose adX-map has no purely imaginary eigenvalues, so ezp:O-+Gbecomes a diffeomorphism); principal series representations of noncompact semisimple groups, and many more. Characters {xT = tr(T)} of irreducible representations of G are given by certain conjugate-invariant distributions on G , x ( g - ' s g ) = ~ ( z for ) , all z , g E G. We pick a pair of neighborhoods U c G and V c (5, related by the exponential map, ezp:V-U, and define an invariant distribution associated to an orbit 0 c 6*, ($0
If ) = J
J
0 u
f ( e z p X ) e i ( X I 4 I X dP&;
(3.10)
for any test-function f on 6. Here d X denotes the usual Lebesgue measure on (5, while /?= Po is a 2n-form (volume element) on 0,obtained by taking n = g i m 0 wedgefactors of 0 = nu,
p=$
(3.11)
flA...Afl *-
So distribution q50 can be thought of as the Fourier transform '3t+x of an adGinvariant measure d p on 0, pulled back to the group by the ezp-map. Since exp respects adjoint actionJconjugation on G and 6 , g-'(ezpX)g = ezp(adgX), the resulting distribution do is clearly conjugate-invariant. Distribution related to character x of representation T 0.
do is closely
Proposition: There ezists a conjugate-invariant function p = p0 on G, equal 1 at the identity {e}, and different from zero o n U c G, so that 1x0 = &4ul
(3.12)
Clearly (3.12) is equivalent to (3.13) The correspondence between characters {xu}, given by (3.13), and representations {To}, based on co-orbits, has not been proven yet in a complete generality, although the result is believed to be true. "This class includes fairly many nilpotent and solvable groups, affine, Heisenberg, etc.
56.3.The Kirillov orbit
299
method
Let us remark that formula (3.12) reveals the nature of distribution xo, that is closely connected with the geometry of orbit 0. Thus compactness of 0 (e.g. compact G) implies that xo is a regular function (hence To- finite-dimensional!). If 0 is a cylinder (example 2), so 0 is made of subspaces {to L}, then distribution xo contains &type factors. A particular case: L = 8 - annihilator of a subalgebra 8 c (5, implies that xo is supported on a subgroup H = e x p 8 c G.
+
For orbits of maximal dimension density pa in the denominator (3.13)proves to be a universal function: sinh(tl2) -
p ( e z p X ) = detF(adX);where F ( t ) = --
tl2
7O0
(t/2)2k
(3.14)
Next we proceed to the Plancherel formula for Lie groups. We remind that the Plancherel measure d p on any unimodular14 group has the form
(3.15) for a suitable class of test-functions {f}on G, e.g. f E L' nL2(G). One could understand (3.15) as a decomposition of &function on G into the the sum/integral of irreducible characters,
f ( e ) = /;r(Tt)dP(T);
or 6 ( e ) =
J G_XTdP(T).
(3.16)
In case when the character-formula (3.12) holds the meaning of (3.16) becomes transparent: it gives a decomposition of the Lebesgue measure on 0' into canonical measures on orbits. Indeed, let us assume that there exists a 1-1 correspondence between orbits and representations, and that the orbits of maximal dimension have the universal density function po of (3.14). The Lebesgue measure d( on @* is clearly adkinvariant, hence it can be decomposed (resolved) in the sum/integral of the canonical measures
{Po} (3.11)
on orbits,
(3.17)
Fourier-transforming (3.17) we get
G is called unimodular, if the right- and left-invariant Haar measures on G are equal. -Group Most examples considered in the book are unimodular: all compact groups; simple and semisimple noncompact groups; nilpotent (Heisenberg) groups; motion groups, etc.
56.3. The Kirillov orbit method
300
Now we apply the character-formula (3.12), remembering that p g ( e ) = 1, and find 6(z) =
J
x&)
dP(%
orbit-space which yields the requisite decomposition (3.16).
To compute dp explicitly we need a set of adG-invariant coordinates on 6*.Let us assume that maximal-D orbits are joint level-sets of the family of functions Xl;...Xk (k = codima), so {Aj} parametrize the orbit-space. Let us also introduce coordinates on each orbit 0 (2n=dimO). The orbit-space 4 = 6 * / a d f = can then be identified with the level-set: cpl = ... = cpZn = 0. To proceed further we need a notion of Pfaffian of a skew-symmetric matrix A = (akj). Any such A corresponds in a 1-1 way to an exterior %form: w A = z a k j e kA ej. The n-th exterior power of wA is a constant multiple of the only highest rank 2n-form, W AA . . . A W A= Ce, A...At+,,.
Constant C, represents a polynomial of degree n in variables {akj}, called the Pfaffian of A, and denoted by PfA. Pfaffian has many interesting properties, for instance, (PfA)’ = detA (problem 4). Returning to the Plancherel measure we can state the following general result. Let ((A) be a point on an orbit 0 with coordinates {Al;...Ak}. We denote by A the matrix of Poisson brackets at ((A), aij = {pi;pj}(t(’))*
Theorem 3: The Plancherel measure on group G is given by the formula Idp(A) = J(A;O)PfA(X)dA,...dXh1
(3.18)
where J(A;cp)is the Jacobian of the coordinate change: (-i(X;cp),
Formula (3.18) takes on a particularly simple form, when coordinate functions {Xi;cpj} are linear, i.e. elements of algebra 6 . Then
CC~’,;
where
{cc}
aij(X) = (((A) I [ c ~ i ; ’ ~= jl) are structure constants of the algebra 0.Now the Plancherel memure turns
$6.3, The Kirillov orbit method
301
Idp = P(A)dA,..A,
(3.19)
into
with a homogeneous polynomial P(A) of degree n = gimO(A) in {Aj}, with coefficients depending on the structure constants of 0. It is worth to remark that formal application (3.19) yields the correct Plancherel measure for complex semisimple groups (e.g. SL,(C)), and compact Lie groups. In the latter case integration in variables A’s must be replaced by summation over the discrete set of the “properly (Borel-Weil-Bott) quantized” orbits (problem 5)! In the real semisimple case (e.g. SL,(R)) it predicts the correct answer only for the so called discrete-series representations, as we shall demonstrate in the next chapter.
56.3.The Kirillov orbit method
302
Problems and Exercises. 1. Show that closedness of the canonical 2-form Sag (3.3) on a co-adjoint orbit 0 C (9' follows from the Jacobi identity for the Lie bracket on (9. i) Use the general formula for differential of a 2-form Sa on a manifold A given any triple of (tangent) vector fields X ;Y ;Z on A,
+
+
dR(X;Y ;Z ) = X [ R ( Y ;Z ) ] - Y [ R ( X Y ; ) ] Z[Sa(X;Y ) ]
+ R ( [ X Y ] ; Z ) - R ( [ X ; Z ] ; +YR ) ([Y;Z];X); Here X[. .I means (Lie) derivative of function (e.g. R ( Y ; Z ) )along vector field X , and [ X ;Y ] , etc. - Lie commutators of vector fields. ii) Consider functions { f(<)} on a*, restricted on orbit 0,pick an element X E (9 (view it as a vector field on U), and show that the Lie derivative,
X[fl(t) = (€ I I X ; V f ( O l ) Here V f ( 0 denotes the gradient of f 1 0 ' (an element of O ) , and [...]is the Lie-algebra bracket of X and V f. iii) Apply part (ii) functions {f = R ( Y ; Z )= (C I [ Y ; Z ] ) ;etc.} on a', and fields { X ; Y ; ZE (9}, and show the differential of R, is expressed in terms of the Jacobi-term, dR = 2(( IJac(X;Y;Z));where J a c = [X;[Y;Z]]+[Y;[Z;X]]+[Z;[X;Y]]. 2. Verify Theorem for Euclidian motion groups E (n 2 3) (Hint: reduced co-orbit space 91 2: Sn-', have 2-nd cohomology group H2(S"-') N Z).
3. Prove that the only G-invariant 2-form on CP' N SU(Z)/U(l) is Ra (3.8) (Use fractional-linear action of G = SU(2) on CP', take R = f ( z ; i ) d z A dz , and compute 1 f =const(l+zS
).2.
4. Check the identity: Pf(A)2 = detA, for any skew-symmetric A (Hint: bring A to a
canonical form, where W A =
ajej A en+ j).
5. Verify the Plancherel formula (3.19) for (i) S0(2), Syn)and other compact Lie groups; (ii) the Heisenberg group; (iii) Euclidian motion groups.
$6.3. The Kirillov orbit method Additional comments: The imprimitivity systems and representation theory of group extensions (56.1) were developed by G. Mackey [Mac]. Our approach to induction, holomorphic induction and Borel-Weil-Bott Theorem follows [Kirl] and [Ta2]. There are many excellent expositions of the ubiquitous Heisenberg group and the oscillator representation (see for instance [How];[Ta2];[Kirl]).
The orbit method appeared first in the work of Kirillov [Kir2] on representation theory of nilpotent groups. It was further developed by Kostant, Auslander, Pukanszky et al ([Kirl];[Pu];[Kos]). Later an important connection between the orbit method and classical mechanics was discovered by B. Kostant, and further developed by Souriau [Sou], Kirillov, Kostant [Kos] and others. Since its inception the orbit method came to play the ever increasing role in many parts of geometry, mechanics, geometric quantization ([How]; [Hurl; [Sn]; [Be]; [How]; [Kos]; [Kirl]; [Dir]), and more recently in connection to the string theory. Further details and examples of quantization will be given in $8.5.
303
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Chapter 7. Representations of SL2. Lie group SL, consists of 2 x 2 matrices of determinant 1. It is the first (and most prominent) member in the large family of noncompact simple/semisimpZe Lie groups. It appears in many different disguises: conformal Sq1;l); Lorentz S0(1;2); symmetries (fractional-linear maps) of the hyperbolic (Poincare-Lobachevski) geometry in
H ($1.l ) ,
and other. The analysis and representation theory of SL, reveals many fascinating features, that combine both "compact" (chapters 2;6) and "noncompact" theories (chapters 4-5). They find deep connections to the number theory, automorphic forms, Riemann surfaces, spectral geometry, some of which will be explored in $37.5-7.6. Most of the chapter ($57.1-7.6) deals with the real unimodular group SL,(R). T h e last section
(07.7) indicates some extensions of the theory to the complex (Lorentz) group SL,(C), and its higher-dimensional cousins
Sql;n).
57.1. Principal, complementary and discrete series. Our goal in this section is to construct and analyze unitary irreducible representations of SL,(R). Those turned out to consists of 3 series: principal, complementary and discrete. In all 3 cases the induction procedure of $1.2; $3.2 and $6.1 plays an important role.
1.1. Principal series. Group G acts by fractional-linear transformations g:z-og
= =bx+d' * on R (or C). We shall show that R is the homogeneous space of G, as
well as the upperflower half-planes C+ = {Sz > 0); C- = {Sz < 0). Namely,
Proposition: i) Real line R is isomorphic to a quotient-space of G, modulo the
Bore1 subgroup of the upper-triangular matrices,
B= =h=[ ii) The Poincare upper half-plane compact subgroup of G.
a
]aeRX;heR.
W N G ~ K where , K
= Sq2) is the maximal
To demonstrate (i) we use a unique factorization of (almost) any g E G into its upperflower-triangular parts,
where h E B; o E X = =
[ ;] So .
family {z) provides coset representatives for the
quotient GIB. The G-action on the quotient-space G I B E R is computed by multiplying a coset representative z by g E G, and factoring out the upper-triangular term,
306
57.1. Principal, discrete and complementary series. (14
xg = h(x;g)x9.
Here h ( z ; g ) denotes a B-valued cocycle ( h : X x G - + B ) , and xg marks the image of a coset {z} under the action of . Then we et' .U=[ 1 i]=[a:+c b:+J=[ (bz+d)-' b : + d ] [ A (1*3)
a ][ 1
]
bz+d
The second statement (ii) was explained back in chapter 1 (§1.1), where we realized the Poincare-Lobachevski half-plane W as SL2(R)/S0(2). It can also be demonstrated by the so-called Iwasawa decomposition (problem 1). We denote by
(a)
E
two special characters on the multiplicative group of reals
R X =W+UR-NW+XZ2, € + ( a )= 1; € - ( a ) = sgn a,
and write any character
x on R x as a pair ( s ; c * ) , or ( s f , for brevity), with s E R,
The principal series representations of SL2(W)act on space L2(R),by the formula
Another realization of the principal series representations is obtain from the natural action of SL2(R) on space L2(R2)= 36+ @ 36-, sum of even and odd functions,
: T f(z; y) = f(az+cy; bz+dy); f E 36
*.
Each of two spaces can be decomposed into the direct integral of spaces of homogeneous functions, X f s = {f E 36*:
f(lz;ty) = tsf(z;y)},
by writing f = f ( r ; 0 ) in polar
coordinates and Mellin-transforming the radial variable,
f ( r ;0 ) + T f ( r ; 0) r-'-'dr, s = ip. 0 The latter can be identified either with functions on R with the L2-product, f(z;y)= I Y
I Sf(zlY;f l)+f(z;
or with L2-functions on the circle, f = rsf(l;O)+f(l;O). restricted operators T
* 136 *
* I), It is easy t o see that the
are equivalent t o the principal series
*
TS , for purely
'The factorization formula (1.1) as well as coset-transformations (1.4) are particular cases of a general setup, namely a subgroup H C G, and a subset X c G of coset representatives X n H = {e), with the unique factorization g = hz, for almost any g E G. Given such pair ( H ; X ) one can identify X with the homogeneous space H\G, and produce an H-valued cocycle h ( z , g ) by factoring product zg = h ( z ;g ) 2 .
57.1. Principal, discrete and complementary series. imaginary s. Indeed, taking the R-realization of 36
I y I - S f ( a ~ + ~ yb;z + d y )
= I bz+d
307
*’, we get I ‘C
f
( b z + d ) az+c f(m).
Theorem 1: Formula ( 1 .d) defines a unitary irreducible representation T S of SL,(W) on Lz(W), for any s # 0. Representations T8;+and T-“;-are equivalent for all s E W, while all other pairs are inequivalent. Proof: Let us observe that operators (1.4) form a representation of G, due to the cocycle property of h ( z ; g )of (1.2). Indeed,
(T,f)(z) = 4Z7L7)f ( Z 8 ) , where scalar cocycle o(z;g) consists of a E-character
x
composed with h. Unitarity of
T S follows easily from the change of variable formula. Indeed, operator T gtransforms f(z)-mf o 4(z), where 4:z
:I;.
:
So the L2-norm of T g f is preserved,
= Ilf 1IZ7all f, IITgf112= I (a 0 4-9f I 14‘ I Here 4‘ denotes the Jacobian of the coordinate change
,/m.
4. 4’for the fractional-linear #(z) = sg, we find 4’ = -whence comes the (bz+d)” attached to a unitary character, x( ...) = I...Jisc (...) in normalizing factor I bz+d I
iff a(.)
=
Calculating
-’
*
(1.4). Irreducibility: The reader has probably noticed that representations {T’
* } are induced
by one-D characters of the Bore1 subgroup B. There exist a convenient irreducibility test for induced representations based on the notion of Mackey ’s imprimilivily syslem: group
G, homogeneous space X u G / B , a (continuous) family of vector spaces (vector bundle) {Yz:z E X},and a family of operators (cocycle) o(z;g):Y,+Yz,.
In chapter 3 (53.2) we
described a “finite/discrete version” of Mackey’s results. The continuous version goes along the same lines, but some technical modifications are needed due to topological setup.
Namely,
with
each
point
ycX
we
associate
its
stabilizer,
By = {g: yg = y} = y-’By, and an inner automorphism AdV:h‘+h = yh’y-l, that takes
By onto E . Automorphism Ad, pulls back any character (representation)
x
from B to
By: x+x~(h’$’x(yh’y-l),
for h’ c B,
.
The Mackey’s irreducible test for T Crequires 2 steps, 1) to check irreducibility of
x
(obvious for characters);
2) to show that two characters (representations)
intersection E
x
and
xy
restricted on the joint
nBy are different (non equivalent).
More generally, one shows that the commutator algebra of T = I n d ( x I E ; C) is
57.1. Principal, discrete and complementary series.
308
decomposed into the direct sum (discrete or continuous), of the intertwining subspaces, Com(T) = C o 4 x I B )
2 W x I B flBy;xu I B fl By),
the summation extends over all non-diagonal G-orbits w C labeled by a point integral
0...
In the continuous case direct sum
(e, y).
dw. To apply the Mackey's test in our case,
x = {y = [:
X x X, each of them being
g ... is replacedpy the,direct
G
SL,(R), B = {[a
we compute
y-'By={
&-a)-y OtbY
b
The low off-diagonal entry must be 0 on the intersection-subgroup B
b = $(a+;),
and
nBy,which implies
i.e.
y-'hy
=[
I/o
for all h E B n B , .
So the "pull-back" character XY(a) = x(;), whenever the off-diagonal entry of h b # 0. It follows then that two characters other, so
x2 = 1. But
x and xy, restricted
on B n B , , are inverse one to the
*
the latter is impossible for x(a) = cis% (a), unless s = 0. This
proves irreducibility of T'
* for all s # 0.
More elementary proof of irreducibility utilizing a simple Fourier analysis, rather than the Mackey's test, is outlined in problem 2.
Equivalence: The above argument, based on Mackey's test, also yields an intertwining space Int(TX1;TXZ),for a pair of induced representations. Ignoring the diagonal of X x X (of measure 0): there ezists a nontrivial intertwining Q E Int(TX1;TX2),iff two characlers
are relaled b y
x1 = xi,
on the intersection B n B , f o r (almost) a n y y E X . In our case
xl(a) = x,($), for all (Cartan) diagonal matrices a, hence
xlx2 = 1.
Rewriting the latter
in terms of parameters (sf) yields: TBt T-';-, while all other pairs are different (nonN
equivalent), QED.
1.2. Discrete series representations are realized in holomorphic functions on the Poincare-Lobachevski upper-half plane, P = {z = x iy, y>O}. Group G act on P by fractional-linear transformations: z In fact, P is a homogeneous space K\G, where subgroup
+
.;: :
coincides with the stabilizer of point i E P, and represents a maximal compact subgroup dzdy in G. There exists an invariant (Haar) measure on P, computed in 31.1, dp = 2 ' We shall consider vector-spaces
%ts,f
Y of holomorphic/antiholomorphic functions
$7.1. Principal, discrete and complementary series.
f(z) in P with norm
llfl12=
309
,...,
If(r)I2y"-ldydz,n=1,2 P . and define discrete-series representatrow in %,fby
The reader can recognize (1.5) as a holomorphically induced representation of
36.1, space P has clearly an analytic structure, and function a ( z ; g ) = b z t d , hence (bz+d) m, becomes a holomorphic in variable z cocycle a: P x G-+C*.
*
Density y"-' of operators 2
in the definition of %,-norm follows from the unitarity requirement
Tgf = a(z;g)f(zg), in L2(P;w(z)dA). Another realization of discrete series is obtained in the Poincare-disk
D = { z : 1 z I < 1) - a homogeneous space of the conformal twin SU(1;l) of SL,(R) (see $1.1, problem 5). We recall that the latter means 2 x 2 complex matrices {g}, that preserve an indefinite hermitian (1,l)-form in C2, (2;;) = zlwl - z2m2, so
1
I ~ I 2 - I P l Z =. ~
{g=k
;[
Lie algebra su(1,l)consists of all complex matrices { X = -$s E R;c E C}, and the correspondence between two groups (subgroups of SL(2;C)!)is given by the Mijbius element
;q
E SL(2;C).
.=L[ J z -i1
Namely, conjugation, u:g+o-'ga, maps .%(2;R)+sU(l; 1).Conversely,
["p
o-l:P I ] -
sends SU(l,l)-+SL(2;R).
H
The Mobius element u:z+z0 =
pa is
PI
-Im(a
Zm(a+P)
a)
].
Re(a+P)
= w, takes P into D, and conjugates
*
fractional-linear actions of both groups. Clearly, the SL,(W)-invariant volume-element on P is taken into SU(1; 1)-invariant element dwdw on D. (1-
Y2
In the unit disk-version space %,
I w I 2)z
consists of holomorphic/antiholomorphic
functions in D with norm 'Indeed, fixing point zo = i, and a map gZ:i-+z, given by matrix gz=[2$T/1J;]
*
we can easily compute the density, that makes T unitary, W(Z) = I a(zo;gz) I I (Jacobian gz)(zo) I = (J;j)2(n+i)y-2
= y"-'.
57.1. Principal, discrete and complementary series.
310
As above one can easily check unitarity of operators Tn of (1.6).
Theorem 2: Representation T f n of (1.5)-(1.6) are irreducible and distinct f o r all f n. To prove irreducibility we shall use the infinitesimal method of 51.4 (see [Lan]), and
analyze the generators of Lie algebra st(1,l) acting in space h,. Lie algebra st(1,l) has a Cartan basis of 3 elements
and the corresponding one-parameter subgroups are
We compute the corresponding one-parameter groups of operators, TeZp(th)f
= ei(n+')tf(ei2tz); ~ ~ , ~= (zsh ( ~ t+ch~ t ))- nf - *zch f ( t+sh m ) ;t T,,p(ty)f = (z i sh t
+ ch t)-"-'f(-);
and differentiate the latter in t at t=O to get the generators. This yields
+ 2zs,]; T~ = -(n+l)z + (1-z2)sz; T, = -i[(n+l)z + (1-r2)sz].
T , = i[(n+l)
Let us observe that monomials {zk}L form an orthogonal basis of eigenfunctions of operator T hin h,,
and Th(Zk)
= i(n + 1 + 2k)Zk.
Applying operators Tx and T, to the basis {zk} we find
T x(zk1 - kzk-'- (n+l+k)zk+';
~
~
(
=2-i[kzk-' ~ )
+ (n+l+k)z'+'l.
So linear combinations T X + i y ,T X - i y represent the raising/lowering operators in the basis {#}, and 1 = zo is the lowest-weight vector of T,,a situation reminiscent of $4.4 (raising/lowering for Legendre functions and and spherical harmonics). Notice, that operators T x o i y are not in the Lie algebra st(l;l), but in its complexification SL(2;C)! Now we can establish irreducibility in two steps:
$7.1. Principal, discrete and complementary series.
31 1
(i) first we observe that vector fo = 1 is cyclic in 36,,, since powers { T $ + i y ( f o ) }span the entire space; (ii) if a function f has (f I fo) = f(0) # 0, then the projection
takes f into a nonzeros multiple of fo, hence Span{T,(f)} 3 Span{T,(f,)} = 36, so f is also cyclic. Finally, any vector f E 36, can be shifted by a n element g E G, so that
(T,f)(O)= (T,f I fo) # 0. This shows that any vector f is cyclic, hence
1.3. Complementary
T" is irreducible, QED.
series representations are labeled by the real parameter
X E [ - 1;1], and look similar to the principal series. Namely,
for a suitable class of functions on R. The class of functions {f} is defined by the norm
'-',
where R, denotes the Riesz potential r:') I x-y I a fractional power of the Laplacian (-A)-,/' on R ( A = 8:) ( s e e chapter 2). We leave as an exercise to the reader (problem 4) to show that (2'') are, indeed, unitary irreducible representations of SL, with respect to the product (1.8). The details are similar to the previous analysis of the principal and complementary series, so we shall skip them (see problem 4). Finally, we shall state without proof the general classification Theorem of Bargman [Bar], about irreducible representations of SL,(R). Theorem 3: T h e principal, discrete and complementary series comprise all unitary irreducible representations of SL,(w). The proof (see [Lan],[TaB]) involves an infinitesimal analysis of the associated representations of Lie algebra SL,. The latter are shown to possess the highest (lowest) weights vector, labeled by a complex parameter s. Writing the unitarity condition for the resulting (group) representations T Sand analyzing them one comes up with 3 basic series.
312
57.1. Principal, discrete and complementary series.
+
. -4 -3 -2
-1
S-
Fig. 1: Space G of irreducible represeniaiions of SL,(R). Here vertical axis represents the principal series; interval (-1;l) on the horizontal axis - complemenfary series, while integers { f 7n} correspond to the discreie series.
Problems and Exercises: 1. Show the Iwasawa decomposition: any matrix g E SL, is uniquely factored into the product UN, of a unitary/orthogonal U ,and an upper-triangular N with positive entries on the main diagonal (use Gram-Schmidt orthogonalization). 2. Show irreducibility of the principal series representation T A (A = s f) by taking any Q E Com(T),and analyzing 3 families of operators that commute with it: (i)
(Tb:b = [: ,] b E R } ( i i ) (Ta:a = [
Step 1 . Show: Tb& = &Tb 3 K ( z ) on R.
aER
}
(iii)
(T,:
Q = K(x-y) is a convolution with
2 . T,Q = QT, 3 distribution K ( z ) = co6(z) Heaviside functions on [O;+co) and (-03;Ol. 3 . T,Q = QT, 3 K = cb(z), i.e.
1.
u=
a distributional kernel
+ c l H + ( z ) + czH-(z),
where H
are
Q = cZ - scalar!
3. Formulate and prove the continuous version of Mackey’s imprimitivity criteria (Do it first for spaces of continuous functions (or cross-sections), e ( X ) , then use density arguments of chapters 1;2;3).
4. Complementary series representations: show i) inner product (8) is positive-definite, i.e. XJ = {f:(RAf I f) < m} forms a Hilbert space (use Fourier transform to show positive- efiniteness of the convolution operator
RA);
’’:
ii) representation TAis unitary in XA(note that operators T Ahave the form At1 where q5 denotes the fractional-linear map ” I++
A-1
- *and check the identity bz+d’
R A ( ~ - Y ) [ ~ ’ ( ~ ) ~ ’=( RA(z‘-Y‘)~ Y)I~ for Riesz potentials; iii) establish irreducibility of representations { T A }by analyzing infinitesimal generators, along the lines of Theorem 1.
5 7.2. Characters of irreducible representations.
313
$7.2. Characters of irreduubk representatbns. Character of a representation T is given by its trace: xT(g)= t r T In chapters 3-5 we studied characters on compact (Lie$ groups, and found them to form a family of nice (differentiable) functions, orthogonal in S2(G).
.
For co-D unitary representations t r T does not strictly speaking make sense, 80 characters shoufd be understood as distributions on G, defined via pairing to suitable teat-functions { f(z)}, (xT I f) = tr(T,). Precisely, functions {f} that produce a trace-class operator T , (see Appendix B). In this section we demonstrate that the principal, discrete and complementary series representations have sufficiently many Then we compute the trace-class operators {T,} (all f E corresponding distributions { x T } on an open dense set of elliptic and hyperbolic conjugacy classes, and find both of them to be given by nice (analytic) densities on G.
Cr).
2.1. Characters, as distributions on G.All 3 series (principal, complementary and discrete) were shown to be induced T = ind(x I B;G). So the corresponding groupf(g)Tidg, are given by integral kernels algebra operators Tf = IG Y) = x ( 1 ;b7y)f(bry)db, (2.1) Bz where B, denotes a stabilizer of point z E X N B\G, and coset representatives {yy} map 2-y. Picking a suitable class of functions {f} on G (typically smooth, compactly supported), one can show integral kernels, {Kf(z,y)} to be also smooth and rapidly decaying. Hence, by the standard functional analysis (Appendix B) operators {T;:f E CF} belong to the trace-class (or Hilbert-Schmidt class) on L * ( X ) , or
q”;J
holomorphic functions, like %,-spaces. Furthermore, for representations in L2-spaces3 , trT; = K;(z, z)dz.
So character xT is defined as a distribution on e F ( G ) ,
However, such distributions have often continuous/smooth densities with respect to the natural coordinates on G. Our goal here is to compute these densities for the principal and discrete series representations of G = SL2@3). We denote by A,, A’;
the eigenvalues of matrix g E SL,(R), *A-l=
9’ 9
trg
*d-
2
3The situation is somewhat trickier for operators in holomorphic function-spaces, like 36,, discrete series representations.
for
5 7.2. Characters of irreducible representations.
314
The eigenvalues are r e d for hyperbolic elements g ( t r g > 2), and form a conjugate pair of unit complex numbers { e *id} for elliptic elements g ( t r g < 2). Since characters are conjugate-invariant functions on G, one expects them to depend on
{A$ '} only. Indeed,
Theorem 1: (i) T h e character
xs
of the principal series representation TSf is
equal t o e (Ag); for hyperbolic g for elliptic g
(ii) T h e character of the discrete series representation T
is equal t o
(iii) The complementary series character x p looks like the principal, with the real parameter p E [-1;1]in place of purely imaginary is, so
(Ag); for hyperbolic g
e
Xp(d =
for elliptic g
Proof: (i) The principal series operator T;* can be thought of as a distributional kernel on W of the form,
K g ( z ; y )= I bx+d
I
f
( . . . )az+c b ( m- 9).
So its trace is formally given by trT;* where a(t)denotes
I t I , e+(t)
= I(aiS-'ef)(bx+d)s(s9-
x)dx.
= 1, e - ( t ) = s g n t.
We apply the general change of variable formula for the &function to integral (24,
In our case, @(x) = sg - 2, so @ has real zeros (fixed points of g:z+sg) only for hyperbolic 9,and these are -d f Xg
51,z
=b .
Evaluating integrand ais-'e = I bx+d
I -,
and derivative
@' = (bx+d)-2
- 1 at
5 7.2.
we get ( b z t d )= *Ag,
z=
315
Characters of irreducible representations.
and @'(z)=
As2- 1. Substituting
the latter in (2.2)
yields the first character formula. (ii) The proof is more involved for the discrete series, due to the fact that the latter were realized in spaces of holomorphic functions on P or D (rather than L2(R)), so trace formulae of type (2.2) are not available here. The relevant formulae involve more complicated reproducing kernels. However, for elliptic g we can easily compute using the conformal SU(l,1 version of operators T t n , h =
[i
Sh(W),acting on %,(D), and
x*,
the fact that the
are diagonalized in the basis { z k } p = ;, with eigenvalues
{ f (2k+n+l)}. It follows immediately then that
To compute the hyperbolic part of the trace formulae we shall use yet another realization of T*" in L2-spaces on the half-line, L2(W+;t-"dt). Observe that the Fourier/Laplace transform 00
~ : f ( t ) - /f(t)eitzdt = ~ ( z ) , 0
takes unitarily L2(R+; P d t ) into holomorphic functions in the upper half-plane
P = {Zmz> 0}, square-integrable with weight y"-*dydz. Indeed, F(z+iy) = '3(e-tYf), so by the W"-Plancherel formula, 00
I f ( t ) I 2e-2Ytdt = 27r
6
J
I F(z+iy) I 2dz.
-00
Multiplying both sides with yn-ldy and integrating from 0 to
L H S = -r(n) p-11fIk2(t-ndt)
00,
we get
I F ( z )I 2yn-1dydz = 2?rIIFIf36,'
= RHS = 2.1
We shall transform the representation operators T * , from spaces 36,
to
L2(W+;...) by conjugating with 4. Since the inverse transform 9-1: F-
k
I
am2
e-iztF(z)dz,
= yo
we find that the transformed operators are given by integral kernels
Kn(t;v)=
i(r)zg-€z)
dz
(2.3) (bz+d)"+" It is convenient to change variables: z-t = b z t d . Then we compute: z = dz = zg - ( z = d() - (t r) ( t ) ] , and after substitution in (2.3), kernel K , takes
a; b v
i[(q+
the form,
Kn(t;v)= &e
y;
+
i(v)l
Smt = to
-i(r)lt + €t)/b&
p+l'
316
3 7.2.
Characters of irreducible representations.
Another change of variable, t+t-', brings the integral over the line Imt = ... into the integral over the circle C = { I t+i I = i} Kn(t
;rl
- h i ( - ) / b ,f ,-i(t~+C/t)lbtn-ldt,
) -2nb
C
+
Remembering, that exponential {t' - (a+d)t 1) in characteristic polynomial of g, the inner integral in (2.4) yields
(2.4) represents
a
b bt i [ ( t + l / t ) - (a+d)l= (t-Ag)(t-A;l).
Thus we derive the following trace-formula
The contour integral (2.5) is easily evaluated: for hyperbolic g the smallest of two eigenvalues (AS' < 1) is inside the circle. Taking the appropriate residue in (2.5) we finally get
and similar derivation applies for negative n, QED.
57.3.The Plancherel formula for SL,(W).
317
$7.3.The Plancherd formula for SL,(R). In this section we shall derive the Plancherel/inversion formula for group G = SL,(R), and get the direct-integral decomposition (see 56.1) for the regular representation of G on L*(G), into the sum of primary components. An interesting feature of SL, come8 from the fact that its Plancherel formula combines both the continuous (principal series), and discrete (discrete series) contributions, 80 SL, behaves like compact and noncompact group at once. We find the explicit Plancherel measure on SL,, including the density of the continuous branch, and multiplicities of discrete components.
3.1. The general noncommutative Plancherel formula was discussed in 56.1. It extends the classical Fourier-inversion formula (§2.1),
jG
f(0)= _?(E)dE,
(3.1)
d[ - properly normalized Haax measure on G,and yields a similar representation for any function f E eF(G), in terms of irreducible characters on G.Namely,
where (xT I f ) = trf^(r),and
I(*) =
L
f(z)r;'dz.
Integration in (3.2) extends over the (unitary) dual object 6 of G, and d p ( n ) denotes the Plancherel meaure of G. From the inversion formula (3.2) one easily derives the Plancherel Theorem for L2-functions on G. Indeed, applying (3.2) to a convolution f*f*,where f*(z) = f(z-'), the integrand becomes
)I l H S
tr[f^(r)3(ir)*] = ?(r)
2
(Hilbert-Schmidt norm)
so (3.2) turns into
(3.3) Let us also remark that inversion formula (3.2) at a particular group element go = e yields values f(g) at all other points of G
1-
(3.4)
Finally, from (3.2) and (3.3) we get a decomposition of the regular representation on L2(G)into the direct integral of irreducibles,
s7.3.The Plancherel formula for SL,(R).
318
(3.5) with multiplicity (finite or infinite) equal to the degree d ( r ) . Our goal here is to prove formula (3.2), and to compute the Plancherel measure dp for SL,(R).
*
*
Plancherel Theorem: Let {x, }, denote the principal and and {x =:}" discrete series irreducible characters of SL2yW). Then for any function f E CF(G) one has
Formula (3.6) shows that the Plancherel measure d p is supported on the union of the principle parts G,+UG,(G,* N R), and the discrete part G, N Z\{O},of the dual A
object
h
A
A
6, and is equal to
In the rest of this section we shall outline the proof of the Plancherel formula for
SL,(R).
3.2. Decompositions of G, Conjugacy classes, and the Haar measure. There different ways to decompose G, i.e. choose coordinates in G. Typically they are defined in terms of three special subgroups of G: maximal compact group
{
I{= u = u diagonal (Cartan) group
A={.=
.,I
cossin* 8 - [-sin
1
0<6<2*;
*at=[
tER}
along with its positive part, semigroup
A+ = {at: t > O}; and upper/lower unipotent subgroups: N = N , ={n=[';];and[;
l]n~R}.
We shall use two decompositions of SL,, called Iwasawa: G = K N - A , or AN+IC,
$7.3.The Plancherel formula for SL,(R).
319
and Cartan: G = KA+K. This means that any (almost any) matrix g E G can be uniquely factored into the product: (i) g = una, or anu, where u E K , n E N a E A+ (Iwasawa) (ii) g = uav = u4atutD,u,u E K , a E A+ (Cartan).
*,
To demonstrate the first decomposition4 we observe that K\G N P- the Poincare . . half-plane, and K fixes point {i}. So any g : z 4 = z = s+iy, can be decomposed as
But the conformal Syl,1)-set-
The second up. Then
The quotient-space K\G N D (Poincare disk), subgroup K stabilizes {0}, so each g = ugz, where gr: O+z = reie, is given by
m:
Thus the NAThe latter obviously factors into atueI2 with cosht = factorization in the Iwasawa's K N A can be thought of as real-imaginary axis in the half-plane-realization of K\G, while A+K of Cartan's I
(ii)
Cartan: g = u a u
4 t $
d4 dtD + d g = sinh 2t dt (2a)Z'
Let us remark that the Cartan's KA+K-decomposition for SU(1,l) represents a hyperbolic analog of the Euler-angle decomposition in SU(2), with hyperbolic rotations in place of ordinary rotations .
[::z:]
["fn",.'b",
1
The similarity extends also to the Haar measure on both groups, [dg = sin 0 d0 d4 d$I for SU(2) vs. Idg = sinh t dt d 4 d$I for SU(1,l). *For another derivation of Iwasawa decomposition see problem 1 of 57.1.
$7.3.The Plancherel formula for SL,(R).
320
The proof of Proposition 1 follows from the general factorization formula of the Haar measure for any pair G 3 K (compact subgroup),
deb) = d K ( 4 d K \ G W I for 9 = w,, and an explicit form of the invariant measure on the quotient K\G, both in the halfdzdy on D u A + K . plane-realization: -on N A u P, and the disk-realization: y2 (l-r2)2
tdrdb
Indeed, for K\G
rz P, g
= ugz with
Hence follows,
d n do
IQI
=dzdU ZYZ'
which proves (i). Similarly one can verify (ii) for K\G cz D: given any g = udg,, where 9, = Writing polar radius r in terms of the hyperbolic radius t , r = tanh t , we get
d ( d )=rdrdJl = sinh2t dtd$, Q E D . (1-r2)2
Corollary: The discrete series representations {T "} are subrepresentations of the
regular representation5 R in L2(G). An irreducible representation T can be embedded in the regular R, if its matrixentries, t ( g ) = (Tg$ I $), are L2-functions on G . Due to irreducibility of T it suffices to compute the L2-norm of a single matrix-entry, f ( g ) = (Tgl I l), where 1 means a constant function in the disk-realization of space 36,.
Using the Cartan decomposition
g = udatu4, and the transformation properties of 1 E 36 under T (1) = ein41, we see that f depends on variable t only, and ud
f(t)=
jD
( 2 sinht
+ cosht)-"-l(
1-
I 2 I 2)n-ldzdT
= cosh-(n+l)t
I
the
I<-action,
=
/ ( I + re+tanh t)+-lr(l-r2)drd+.
But the later integral is evaluated in a straightforward way, so one gets
f ( t )= ;cosh-("+')t, and
3.3. Conjugacy classes and the Harish-Chandra transform. One of the interesting features of the SL,(RB)-representation theory is the presence of different types of conjugacy classes in G = G+UG-UG,,, (all three open subsets in G ) . These are 5S0 they could be thought of as "discrete components" of the regular representation, whence the terminology!
57.3. The Plancherel formula for SL,(W).
321
= U { g - l ( f at)g: g E G}. We shall parametrize them
(i) hyperbolic classes G by variable t E W
t>O
C*t,
A\G = N + K .
{ g - l t i e g : g E G } , labeled by 8 (0 5 8 5 x ) ,
(ii) elliptic classes Go = U
o<e
C,
2:
K\G = N-A (or A + K ) .
The Haar measure on G can be disintegrated into invariant measures on classes C, (or Co),and certain density on the sets of classes: two half-lines f(0;co) and a semicircle ( 0 ; ~ ) .These are given by the formulae of the type we derived in 55.6 for compact Lie groups:
dg I G * = (et - e-')'dt dCt(nu) = 4sinh'tdt d N + K ( . . . ) dg I
I
= eie - e-i8 fds dCg(z) = 4 sin28 d8 d p ( z )
(3.8)
Now we are ready to proceed to the Plancherel Theorem. The proof will be given in 3 steps. In step 1" we shall introduce and compute the so-called Harish-Chandra (conjugacy-class), or I f -transforms, of CF-functions f on G , which essentially amount to averaging f over the elliptic, or hyperbolic conjugacy class. Then (step 2") we shall link 2 H-transforms (hyperbolic and elliptic) at points, where two sets of conjugacy classes meet. Finally, in step 3" we shall express irreducible characters in terms of Htransforms, which would eventually lead us to the Plancherel formula on G . Step 1": Given any function f on G , we shall define its elliptic and hyperbolic Htransforms, by averaging f over the corresponding (elliptic/hyperbolic) conjugacy classes. Precisely, (elliptic) f + F f = F K ( 8 ) = (eio - e-ie)
(hyperbolic)
/
f(g-'uog)dg, on Go; K\G
*
f + F f = F A ( t )= FA( f a t ) = I et - e-t
(3.9)
I
f(g-'a,g)dg,
on
A\G
G**
In other words, "averaged function" f is multiplied by the "square-root" of the Weyl-type density (3.8). From the disintegration formula (3.8) it follows,
T
Densities that appear in the above integrals represent the omitted "half-powers" of Weyl-factors (3.8). We can also express any character, or more generally any classfunction (distribution)
x on G in terms of its H-transforms,
$7.3.The Plancherel formula for SL,(R).
322
/
/
K+
A+
( x I f) = X ( e ) ~ ~ ( e ) 2 i s i n e+d e X ( a r ) ~ A ( a 1I sinh ) 2 t Id t +
j x(-at)...
(3.10)
A+
In the proof of the Plancherel Theorem we shall need the H-transforms of Kinvariant functions on the group,
f (u-574
= f ( g ) , u E K.
Let us remark that the Plancherel formula (3.6)is sufficient to establish for Kinvariant functions {f}. Indeed, any function-space P. on G, (like er), can be splits into the direct sum of m-isotropic components with respect to the K-action on P. by the conjugation. Precisely, 00
f=Cfm; where each term, fm(g) =
-00
I
e-imOf( "0-1 suO)de,
K
satisfies the equation f ( u i 1 g u 4 )= eim4 f ( g ) . Since averaging over G-conjugacy classes includes integration over K, it follows immediately, that all nonzero Fourier-components {fm: rn # 0}, will be H-transformed to 0. Consequently, any character (x I fm) = 0, unless m = 0, so (x I f) = ( x I fo), for all characters x and any function f on G. Proposition 2 Two H-transforms of a K-invariant function: f(u-'gu) = f(g), for all u E K, are given by
(ii) F A ( t ) = 2a7'
f(["t
(3.11)
ft]dr.
-00
The derivation of (3.11) is fairly straightforward. For the elliptic F K we evaluate integral (3.9), using the Cartan KA+K-decomposition, g = uarv. This yields the first relation (3.11-i), 00
F K ( 6 )= 2a(eie - e-") ~ f ( a ; ' u O a rsinh ) 2rdr, 0
To get hyperbolic F A we evaluate second equation (3.9) using the Iwasawa AN+K-decomposition, i.e. g = m u , with n = Then
F A ( t )= I et - e-t
I
/
[' f]
f(u-'n-'qnu)dn du. NK
The inner conjugation, n-'atn, in the variable of f yields matrix
57.3.The Plancherel formula for SL,(R).
323
while the outer conjugation, u-'( ...) u , leaves f invariant, and thus only contributes factor 27r to the integral. So
I (kt yy])t,
W
F A ( t )= 47r sinh t
f
-00
whence combining 2tsinht into a single variable
2,
we get the second relation (3.11-ii).
Next we shall recast the FK-transform in a more convenient form by making the change of variables: s = sind, = e2'sid, = e-2'sin4, and introducing a new function
t
f 4 € d = f[-q Jr-7;it defined in the region
(3.12)
{ ( t , ~I )t q: I 5 1). Then (3.11) becomes,
Step 2. Our goal now will be to relate two types of H-transforms at the points of G , where elliptic and hyperbolic conjugacy classes meet, namely elements { f I}E G, or points 4 = ( 0 ; ~ )in K , respectively {a,,; -ao} in i A (fig.2). To accomplish this objective we need some properties of functions F K and F A defined by (3.13). -A
At
Fig.2. gives a schematic view of the two conjugacy classes: elliptic types of [represented by the circle) and hyperbolic [infinite lines A * ) . Two classes meet at the points { r t e } . The Plancherel formula results from matching elliptic and hyperbolic Harish-Chandra transforms at
Lemma 3: The elliptic H-transform, FK(4),has a jump-discontinuity at 8 = 0, that is equal to 2iFA(ao)-the value of the hyperbolic F A at a o = I , while the right and left derivatives of F K at 4 = 0 are equal. Similarly at 4 = 7r, the jump-discontinuity of F K is equal to 2iFA(-a,) - the value of the hyperbolic F A at - a o = - I , while derivatives are equal. Precisely,
(b) -F d K (O+) = -F d K (0-) = -2nif(0;0) = -7rif(e), at 8 = 0, ds
ds
(3.14)
87.3. The Plancherel formula for SL,(W).
324
and similar relations hold at 8 = a. The Lemma’s conclusions follow immediately from equations (3.14). The first of them (3.14a) is obtained by the representing F K in (3.12), and the obvious relation j(z;O) = negative values of s, we observe that the integrand to K-invariance of the original function f on G, which implies in particular,
So changing variables from s to -s in (3.13) results in the change of sign of two
-
integrals
F K (--9)
=-2airf(z;$)dz-
j ( z ; O ) d z = -iFA(uo); as s+O.
S
Next we differentiate function F K of (3.13) in variable s, and evaluate at s = 0 (respectively 8 = 0),
+.
ddsF K = 2 ~ { - 2 ? ( s ; s ) 2 s r ...-
I.]-
-4ai?(O;O) = -4aif(e).
As a corollary we get the requisite relations between F“ and F A at two limitpoints f I, as well as the representation of f( i e ) in terms of derivatives $ F K ,
12aiFA(uo)= FK(O+) - FK(O-); -4aif(e) = $FK(O
*)I
at 6 = 0
(3.15) 12aiFA(-uo) = F K ( a + ) - F K ( a - ) ; -4af(-e) = $ F K ( a f ) I a t 8 = a. Step 3. Now we shall evaluate the principal series characters for Ii-invariant functions f utilizing their conjugacy-class H-transforms. We recall the character
formulae, obtained in 57.2, eist+ e--ist
x s & (9) =
I el-e-t
I
Then we can write by (3.10),
I
00
( x , I f) = xS f
0
6%;
on
G f ;and x s f ( u ) = 0,
on Go.
I
00
f
(ut)FA(u,)I et-e-‘ I d t + [same integrand at ( - u t ) ] .
Both integrals reduce to
I
(eist + e-ist)FA( * a t ) & .
Remembering that characters are even functions of t on & A , the latter become simple “Fourier transforms” of functions FA( f u t ) . Let us also remember that
formula for SL,(R).
57.3.The Plancherel
325
xs+(a-t) = xs+(-at), while xs- changes sign, when +-‘-at. So their sum xs++xs- cancels out on - A and equals to 2xs on A , while their difference, xs+ - xs-, cancels out on A and is equal to 2xs on - A . Thus we get
Next we evaluate the discrete series characters,
(x
f
1 f),where
For our purpose it will be more convenient to evaluate the sum of two We find
I
00
(Xn I f) = X,(t)FA(t) I et - e--t I d t
+
I
xn+ x - ~ .
2n
0
xn(t9)P(t9)(e”- e-”)dt9 = (3.17)
0
=
%
2n
00
F A ( t )d t J
,
+ [ FK(t9)ein”BdB. J
0
0
The latter equation yields Fourier coefficients {bn} of function F K (3.16) in terms of the discrete series characters and the hyperbolic F A ,
b, = (xnI f)- T e - I I tFA(t)dt. 0
Differentiating the Fourier series F K = ein’
Next we rewrite the (-n
I
bneine, we get
t ( F A ( o + ) - ~ A ( o - ) ) s ,+
(...)sn.
of the n-th coefficient of ( F K ) ‘ ,as
- e-tngFA(
iat),
0
and taking into account the parity (even/odd) of function FA( f a t ) , and that of F K ( 6 ) , we represent -J$FK by the following expansions 0000 0 0 0 0 i E n ( X nI j ) + f e~ - I t I n s g n t & F A ( a i ) d t + f ~ ( - l ) n J e - l - t l n s g n t adF A (-at)dt(3.18)
J
1
1
1 -00
-00
The rest of argument involves some basic Fourier analysis on R. First we shall apply the Plancherel formula on W for all integrals of (3.18),
J
I
f ( t ) g ( t ) d t= 2r T ( s ) ~ ( s ) d s . Then use a simple Fourier-transform relation
57.3. The Plancherel formula for SL2(R).
326
and find by (3.16) the 9-transformed derivatives of FAin terms of characters,
.d A 9 . d t F ( a t ) -& [(Xs+ I f )+(Xs-
(*..)I. These formulae allow us to rearrange terms in (3.18) into those of xs+, and those of xs-; namely, xs-
I f )I; and
d A zF
-.)
is [(-) -
c (A)+c (A) '
n=even
8
xs+
+n
n=odd
+n
Two series are easily summable by the Poisson summation formula (s2.1 of chapter 2),
c(+) 2
even
8
+n
=$ C e - R S I
I =icoth %
2
EL-+) odd 2+n2
= ... = gtanh y.
Thus we get the final result, Z~ i f ( e ) = ;di eFK
00
00
-00
-00
(O)=iCI n l ( x n lf ) + f / (xs-l f ) s c o t h y d s t f / (xs+lf ) s t a n h y d d s , n#O
which completes the proof. Corollary: The regular representation of SL2(W) on L2(G) is decomposed into the
s u m of the continuow and discrete parts,
Rg={
.
+
+
continuous t
continuous -
discrete
The L2-Plancherel formula,
Here
7 denotes the usual noncommutative Fourier transform of f , i.e. ](a)
for the principal and discrete series irreducible representations { T}.
= nf,
57.3.The Plancherel formula
for
SL,(W).
327
Problems and Exercises: 1. Consider the natural action of G = SL2(R) on L2(R2), ( T , f ) ( z ,Y ) = f(az+cy;bz+dyh and show that representation T is the direct integral of the principal series representations, T = $T'+ds$ $T'-ds. Hint: i) observe that R2\{O) can be ide tified with th homogeneous space N\G, where
: ] b E g
N=[['
consists of upper-triangular unipotent matrices, on L~(N\c).
80
is the regular representation of G
ii) Note, that representation spaces :63 can be identified with even/odd function f ( z , y ) on R2 homogeneous of degree (is - l),
= {f(tz;ty)= t i a - l f ( z , y ) } .
:63
Any such f is uniquely determined by its values on the line {(ql)},and can set
II f It = II f(z;1) pd.. iii) Show that T :I63 subspaces X!).
is equivalent to the principal series T'
'
(but 36, are not
iv) Establish the direct-integral decomposition by explicitly writing map '3: L2(R2)+
636,ds,
with properties (i-ii), as Mellin transform o f f (52.1) in the radial directions, '3: f-f(z, y; s) =
00
J' f ( t z ,ty) t-i'dt.
(3.19)
v) Use the Mellin inversion/Plancherel formu?a: OD
00
u(t)
=-!-I ir(s)t"-lds; 27L,
and
00
I I u ( t ) I 'tdt = 2 x J I G(s) I 'ds. 0
-00
to resolve any function f on R2 into the integral f(z,y) = & J f ( z , y ; s ) d s = j f ( s ) d s , 80 that Tgf = J ' T i [ f ( s ) ] d s ,
and to show
IIf I t 2 = 111f(s) If%$. Change variables z = tn, y = 1 ( t
> 0; K E R), and write the Lz-norm as
IIf 1IZ= J' JI f(t6t ) Ptdt dn; Then apply Mellin-Plancherel formula, to integral
ZTJ' to get the latter in the form
1 f (...;s)
I f(n,l;s) I 'dnds, , QED. 6
57.4. Infinitesimal representations of SL,
328
57.4. Infinitesimal representations of Sb; spheriul functions and characters. Infinitesimal method of 51.4 reduces repreeentations of Lie group to those of its Lie algebra, and greatly facilitates their study. Further reduction for SL, is achieved by passing to a single generator L of the center of its enveloping algebra 8 (called Casimir). Any irreducible T' must be scalar on L, thus we get an infinitesimal character ~ ( s ) .Character x uniquely identifies all three irreducible series, and determines the eigenvalues of the Laplacian on G and the hyperbolic plane H = K\G. Another reduction arises from the compact subgroup K of G. The latter gives rise to the algebra of spherical functions and spherical transform. Once again irreducible { T'} are completely determined by "spherical reduction", and spherical functions give generalized eigenfunctions of the hyperbolic Laplacian.
We shall keep the above convention for the principal, complementary series: Ti'
(-co < s < co); TS(-1
discrete
and
< s < 1); and T n (n = f 1; f2; ...).
4.1. K-invariants and spherical functions. We shall show that the even part of the principal series Ti'+; as well as the complementary series T S possess a unique I(invariant vector q50 E 36,:
Tud0= do; for all u E I(. Such vector is easy to exhibit in any of realizations of the principal and complementary T , constructed 57.1. In the %,-realization (a subspace of L2(R2)),we find is-1
do = (zz+ yz)T(for
s- 1 principal T ) , q50 = (zz y')T(for complementary T ) . (4.1)
+
A I(-invariant allows to embed an irreducible representation T into the regular representation in bounded functions on homogeneous space A = K\G, by assigning each 4 E 36 - a representation space of T , a function on G equal to its matrix entry, 4 4 4 = f ( s )= v g 4 140).
(44
Clearly, f depends only on the class of g (mod K ) , f = f(z), z E K\G. Thus we get a map W:X-+L"(A), that intertwines T and the regular representation R in
Loo(A). A particular matrix entry f o ( g ) = (Tg& I 40) is double-invariant (under both left and right multiplication with K), fo(ugu) = f o ( g ) ; for all u;u E K.
Such functions {f} on G are called zonal spherical funcfions, by analogy with spherical
87.4. Infinitesimal representations of SL,
329
harmonica on S2 (chapter 4). One can show that spherical functions {f E L'(G)} form a commutative (convolution) subalgebra of L'(G), which could be reduced to a functionalgebra on the half-line [O;co),via the Cartan decomposition (problem 1).
Let us remark that embedding (4.2) does not make T a subrepresentations (discrete component) of the regular representation R on A, which was the case with compact groups. Indeed, W embeds I16 into Loo(&), rather than a subspace of L 2 ( A ) . Irreducible decomposition of R for non-compact groups and homogeneous spaces are typically of the continuous (direct integral) type,
RN
Tis+dp(s).
In the next section we shall compute the corresponding Plancherel measure' d p for the regular representation of SL, on the Poincare half-plane W N G / K . 4.2. Representations T I K . In chapter 3 (53.1) we have shown that any representation T of a compact group K is decomposed into the direct sum of primary
components (multiples of irreducibles), T N @ _T @ d(n);d ( a ) - multiplicities (finite d = 0;l;... or m). T E K
In particular, an irreducible representation TS of G = SL, restricted on K = S0(2), admits such decomposition, cos8 sin8 T'IK= m e e i m 8 @ d m ; f o r u = u 8 = -sin8 cos8
(s = is f ; s; or n),
(4.3)
direct sum of characters. We shall establish the exact form of decomposition (4.3) for all
3 series.
Theorem 1: For any irreducible representation T6 of SL, all multiplicities {d,} in (4.3) are 1, the set of characters {xm= eim8} (spectrum of 18 I K ) , coincides with an arithmetic sequence of the difference 2: m = m,;m, f2; m, f4; ...; (or halfsequence, positive or negative). Precisely, (i) principal series:
~~
'Complementary series {Ts}do not enter regular representation R on A, as they don't enter a larger regular representation R on G , of which R A can be shown to form a subrepresentation,
R A C RG.
57.4. Infinitesimal representations of SL2
330
(iii) complementary series has the same expansion as an even-principal series part.
C1
]
h a r k Theorem 1 rephrased in terms of generator H = 1 of rotations states: operator T , is diagonalazed with a discrete spectrum of eigenvalues, {m, f2k}. The principal-series spectra contain all even/odd integers, while the discrete-series has either the lowest weight vector (m, = n, for positive discrete series), or the highest weight vector (m, = -n, for the negative discrete series). The reader may compare this result to weight structure of irreducible representations of SU(2), and other classical compact Lie groups in chapters 4-5. Proof: The discreteseries case waa already established in S7.1. For the principal series we shall use yet another realization of T i s f , in even/odd functions on the circle K. This form of Ti'
* results either from Iwasawa decomposition G = BK (remembering that
principal series T are induced by characters of B), or from the X,form homogeneous
functions
in
R2,
and
restricting
+
T = {z2 y2 = 1) N K. In either case, space X,
those
on
a
realized as unit
circle
* is identified with even/odd parts of
L 2 ( K ) , where K acts by translations. So T sI K turns into the regular representation on
K, and the Theorem easily follows.
4.3. Infiniteaimal generators and Laplacians on G and H. We consider a basis of 3 elements in the Lie algebra 0 = sl,(W):
(4.4) which generate one-parameter subgroups: K = {exp tH:O 5 t 5 2n) - rotations; M = {exp t V -co < t < co} - hyperbolic rotations; A = {exp tW:--03 < t < co} - diagonal (Cartan) subgroup Here W denotes the Cartan element, while V , H are combinations of raising/lowering elements (chapter 4),
V=X+Y; H=X-Y. Any element X E (5 gives rise to a left-invariant vector field (or 1-st order differential operator) 8,. Vector fields generate the algebra of all left-invariant differential operators on G:
{ax}
m
$7.4. Infinitesimal representations of SL,
Algebra
331
a(@)contains a central (Casimir) element
L = AG = t(W2+ V 2- H2)= $W* - Z ( X Y + Y X ) ] , (problem 2). In fact, one can show that L generates the center of ?I,
(4.5) so any central
element is a “polynomial of L”. The proof however is more involved. The corresponding differential operator (still written as L, or AG, to indicate manifold G) commutes with all left and right translations,
LR,f = R,Lf, for all g E G, and smooth functions f. So we get an invariant Laplacian on G. The reader should not confuse A with the Laplace-Beltrami operator on G, with respect to some left (or right) invariant Riemannian metric. The natural bi-invariant (left and right) metric on tangent spaces {T,(G)} is given by the Killing form (55.1) at the point {e}: ( X I Y )= tr (adXady), for X , Y E (5 = T,(6), shifted then to other tangent spaces {T,:z E G} by left/right translations (see problem 3). For noncompact semi-simple Lie groups the Killing metric has typically indefinite type7, so the corresponding Laplacians (Casimir elements, 55.7) represent nonon UP+’. commutative generalizations of the wave operator: 0 = 8; The situation is different on symmetric spaces A = K\G, as we know from ch.5 (55.7), like the Poincare-Lobachevski half-plane W. Let us remark that the G-action, {g:z-$,(z)} on any homogeneous space A = H\G, gives rise to a Lie algebra of vector fields (1-st order differential operators), representing 0,
(a,f)(4
d
= 3 I i=Of(4,(i](z));g ( t ) = exp t X *
The latter extends to the associative hull ?I(@)3 p ( X ) + p ( d x ) , whose elements turn into higher order differential operators on A. The reduced Killing form on the tangent space {T,} (hence, all other {T,}) becomes positive-definite (as will be explained in the next chapter), so A becomes the standard positive Laplacian on A (see 85.7). Let us compute the invariant Laplacian for the Poincare-Lobachevski half-plane W. It is convenient to write functions on W, as f ( z ; f ) . Given a one-parameter group t+(t;z) of holomorphic transformations of W, generated by a vector field B, the corresponding 1-st order operator,
(4.6) 7The Killing form is positivedefinite (Riemannian) iff group G is compact (see 95.7)!
$7.4. Infinitesimal representations of SL,
332
We apply (4.6) to one-parameter subgroups generated by 3 basic elements (4.4),
+
t. z cos - cos sin t1; exp tV z :zzcshh t + sh exp t H :z--)~ sin t + ch t , exp tW:z-te2tz; and find
a,
= - [(z"l)az
+ (z2+1)&]; a,
=(l-zy,
+ (1-z2)a,]; a,
=2
4
+Ed,.
Substituting in (4.5) we get the familiar expression of invariant Laplacians on M and D,
A , = -y2((ai
+ a;) = ( v ) 2 a : =; A, = (1- I z I ')(ai+aE).
(4.7)
Let us also remark that Laplacians (4.7) represent the Laplace-Beltrami operators A, on Riemannian manifolds W and D, with respect to the natural hyperbolic metrics: ds2 = +(dx2+dy2) on 04, and A ( d x 2 + d y 2 )on D. Indeed, given a metric g = g. .dz'dzj Y 1-r ZJ on manifold A, the standard Laplacian is defined by
and (4.7) follows by a straightforward computation. Since the invariant Laplacian commutes with the group action (belongs to the center) its restriction on any irreducible subspace' {X,}, embedded in Loo(W), must be scalar, TS(A= ) x(s)I. Number x = x ( s ) is called the infinitesimal character of T S , (since for higher rank groups, where dimZ('U) > l), x turns into a character on 1;. For SL, infinitesimal characters could be computed explicitly. Proposition 3 Infinitesimal characters { ~ ( s ) } of the principal, complementary and discrete series are given by x ( s )= 4
(4.8)
where parameter s takes on imaginary values (SEiW) for the principal series; [ - 1;11, for complementary series; and s - integer n = f 1; f2; ...;fo r discrete series. Thus sE
$ (discrete) 8For a cc+D representation T in space 36, all generators { T X : X E @},hence T ( A ) are unbounded operators, which should be considered on a dense core 36, of smooth vectors ($1.4).
57.4. Infinitesimal representations of SL,
333
The derivation of (4.8)is fairly straightforward. It suffices to compute x ( s ) on any fixed vector doE J6,. For the principal and complementary series {T'} we can choose K-invariants {q50} of (4.1), for the discrete series {T*"} the highest/lowest vectors {&} (problem 5 ) .
Problems and Exercises: 1. (a) Show directly that spherical functions {f(ugu) =f(g); all U , U E K} on groups G = SL,; K = S0(2), or compact G = So(2); K = U(1)1: T', form commutative subalgebras of L'(G), by using Cartan decomposition G = K A + K to reduce spherical functions {f(g)} on G to singlevariable functions {f(t):t > 0) for SL,, or (f(0): 0 5 0 < r}, for SY2), then writing the convolution formulae for reduced functions (The general argument based on Cartan automorphism 0 for pairs G,K was outlined in problem 1 of f5.7). 2. Show that element A (4.5) commutes with all B E 8, hence all of N(0) (check it for the Cartan basis {W;X;Y}). Furthermore, the center of 9l consists of polynomials in A: f(A) = C a k A k . (The proof is somewhat involved: each element A E can be uniquely represented by a polynomial p = xakmnWkXmY"in 3 (noncommuting!) variables {w;x;Y} (in the given order). One has to write the commutation relations: ad&) = 0; adx(P)= O;ady(P)= 0 3. i) Show that a bi-invariant metric ( ( 1 ~ ) ~ on Lie group G , restricted on the tangent space a t {e}, T,(G) N 8, is invariant under the adjoint action,
(Ad,(
I Ad,9)
= (( I v), for all
LV E 0,g E G.
Conversely, any Adg-invariant metric on 8 2: T,(G) extends to a bi-invariant metric on
G. ii) All Ad-invariant products can be easily described for semisimple Lie algebras. Killing form, (X I Y)o, is one of them; for simple 8 Killing is the only one; if 0 = @ 8,- direct sum of simple ideals, and ( I ) denotes the Killing form on a, then any Ad-invariant form on 8, is given by fa,( I), (Hint: any bilinear form is obtained from a nondegenerate Killing form by an operator B, (X 1 Y)= ( B ( X )1 Y);form ( I ) is Adinvariant iff operator B commutes with Ad,!). 4. Verify directly that Laplacian (4.7) on H is invariant under all fractional-linear transformations g : z s , in SL,. What happens to A under the GL2-action?
5. Compute the infinitesimal characters of Proposition 3.
87.5. Selberg trace formula.
334
$7.5. Selberg trace fbrmuh. In this section we shall study the regular and induced representations R of group G = SL, on compact quotient-spaces Ab = G / r , where r is a discrete subgroup of G. We shall show that representation R breaks into the diserete sum of irreducibles, each entering R with a finite multiplicity. We prove a reciprocity Theorem (similar to Frobenius reciprocity of chapter 3), and establish the Trace-formula for operators R, ( f E €$') - a noncommutative analog of the Poisson summation. Then we shall proceed to evaluate the contribution of various uparts of f" to the traceformula. This eventually lead us to the celebrated Selberg trace formula on compact quotient-spaces of SL,. After proving the general result we shall outline its ramifications and special e88e8, and find interesting connections to spectral theory of Laplacians on hyperbolic Riemann surfaces H/r, Poincare half-plane modulo a discrete (Kleinian) subgroup r c SL,. We also give an application of the representation theory of SL, to geodesic flows on negatively curved Riemann surfaces.
Let
r
5.1. Induced representations on G / E reciprocity and the general trace-formula. be a discrete subgroup of G , a be a representation of We denote by
r.
R = Ra =ind(aI r ; G ) - an induced representation of G . It is an interesting and challenging problem to find a decomposition of Ra into irreducible components of G . A simple commutative prototype of such setup consists of a lattice I' c Wn (rN Z"), and a representation/character 4 7 ) = cia*-', y E r. The induced representation Ra acts on scalar/vector functions {f(z) on W"}, that satisfy the a-Floquet condition (a generalization of the periodic condition), f(z t 7)= eZa 'Yf(z); for d l z E
w"; 7 E r.
Representation R" is easily seen to decompose into the direct sum of characters,
(5.1) sum over the dual lattice
I"', in a special periodic case R, 2 @ , i m * z,. 2 E W".
rn E r' Both results are simple corollaries of the Poisson summation formula of $2.1. In
particular we get spectrum of the Laplacian A, differential operator) on
(or any other constant-coefficient
W"/rN T", with periodic/Floquet boundary conditions, Spec(A,) = {A, = (rntcr),: m E r'}.
(5.2)
Our main goal in this section is to derive a noncommutative version of (5.1) for quotients SL,/r, and then to link such decomposition to spectral theory of Laplacians
27.5. Selberx trace formula. on spaces
W/r. Throughout this section quotient-spaces
335
J b = G / F will be assumed
r
compact. The corresponding are called co-compact (or uniform lattices), and SL, can be shown to have a plenty of those. Uniform lattices are characterized by the following properties (see [GGP], chapter 1): i) r has finitely many generators {71;...7m},and a finite number of relations among them; ii) the G-conjugacy class {g-l7g:g E G} of any 7 E r is closed in G; iii) l' contains only elliptic and hyperbolic elements.
A hyperbolic element 7 is called primitive, if y # y?, for some yoE T,and k > 1. Each elliptic element 7 has a finite order ym = e. Let us remark that (iii) easily follows from (ii), since a conjugacy class C ( 7 )= {g-lyg} of any parabolic element 7 contains limit points fI, not in C(y). So parabolic C can not be a closed subset.
r
First we shall state the general trace formula for an arbitrary pair c G, of a (locally compact) group G, and discrete subgroup T,with compact quotient G / r . Let Ra be induced by a finite-D representation a! of r. Operators {RF}are given by integral kernels,
q"; Y) = c f(gz17gy)47); 7Ef
where gz;gy E G are coset representatives of points { z ; ~ }in A. Clearly, compactly supported (or rapidly convergent) functions {f} give rise to compact operators {Rf}.In fact, operators {Rf}belong in the trace class (see Appendix B), and (5.3) X X On the other hand, any representation T , which contains a compact operator { T f }has "discrete spectrum", in the sense that T is decomposed into the direct sum (vs. direct integral) of irreducible components:
T N TTs@m(s), (5.4) each one entering T with a finite multiplicity {m(s)}(problem 1). If xs denotes the character of T S ,then combining (5.4) with (5.3) we get the general trace-formula,
57.5. Selberg trace formula.
336
for a suitable class functions f on G (e.g. e$). In what follows, however, it would be more convenient to write the RHS of (5.5) as
where T,;G, are stabilizers (centralizers) of an element 7 in r and G, respectively, and d g denotes the invariant (Haar)measure on the quotient space G,\G. 5.2. Tram-formula on SL,. Our main goal is to compute (5.6) explicitly for group
SL,(W), then to deduce all possible information about the LHS of (5.5): irreducible components (2') of Ra,as well as their multiplicities. To state the general result we remind the reader 3 character formulae for irreducible representations of SL,, obtained in '$7.3. All three X > 1, for hyperbolic g
-
terms of the eigenvalue {A = X ( g ) } ; and complex, X = eie, for elliptic g
-
complementary: x s ( g ) =
I i s t I I -3. Ix-x-ll
-.A-m
COSS sin0
(5.7)
'
hyperbolic g
.
m
L. hyperbolic g
discrete: x S m ( g ) =
-.
elliptic g
elliptic g
Given a function f E CT(G),we introduced its generalized Fourier transform/ coefficients {T(s f ); T(s); fn)}, by integration of f against irreducible characters,
T(
j(s)
-
= / f ( g ) x , ( g ) d g = tl.(Tj);s = 2sf;s; f n ,
so ?(is+> = f ( g ) x i s + ( g ) d g , etc*
Following [GGP]we have to compute the contribution of different conjugacy classes of to tr (RY).Those are conveniently divided into 4 groups: hyperbolic classes, elliptic classes, and 2 special elements { e } and { - e } . Contribution of hyperbolic elements. We shall split all hyperbolic conjugacy classes y = {g-lyg} c T,into primitive classes Pr, = {S :7 # yi}, and their iterates (9 : y = 7;; 7,-primitive}. We have to evaluate each term of (5.6) for a hyperbolic 0
$7.5. Selberg trace formula.
337
-[
element y X-l]. The RHS of (5.6) involves the conjugacy-class H-transform of f , which was essentially evaluated in $7.4.Namely,
I
W
JD\G
f(s--’rs)ds= FA(r)= 4r I 1 - A-l I -oo { T(is+)+ &-)
sgn A}
I A I %s,
(5.9)
in other words the hyperbolic H-transform of f is equal to the inverse Mellin transform (52.3) of the even and odd principal series generalized Fourier transfom of f. Next we compute volume of the quotient-space A
wol(ry\Gy) =
Jq= In A,
(5.10)
1
since G, I IW (multiplicative group of reals), while r, = {ym: m E Z} forms a discrete lattice in G,. Combining (5.10) and (5.9) we come up with the total hyperbolic contribution to the trace,
Here the outer summation extends over all hyperbolic primitive classes {y }, while the inner consists of their iterates {y ’}); symbol 1; if a(-.) = I € a = { -1; if a(-.) = -I‘ Note that hyperbolic terms involve only the principal-series characters.
contribution of elliptic elements. We consider the set of primitive elliptic elements, and notice that each of them has a finite even orde2: y2m= e. We need to evaluate an elliptic H-transform of f in the RHS of (5.6). For any elliptic element u = u * - [ case sine -sine
case
].
do
I&+
-do
ch (e -
T)-?(is-)
Chy
sh (e - y )
shy
The first half of formula (5.12) (sum) consists of the discrete series “generalized Fourier coefficients” of f,while the second half (integral) is made of two principal series “generalized Fourier transforms”. The derivation essentially follows arguments of 57.4, where (5.12) was established at 6 = 0 (problem 2 provides further details). ’Indeed, if y had an odd order p , then -7 would have an even order 2 p , so y = (-y)p+’ not be primitive. Hence each elliptic y is equivalent to a rotation by angle 0 =
5.
could
$7.5. Selberg trace formula.
338
The volume of r7\G7 is easily found to be &, where y 2 m = e (7-primitive). Indeed, G , = Sq2)N [0;2r],while ry makes a finite subgroup of order 2m in Gy. Combining contribution of all elliptic classes, we get
(5.13) The outer summation extends over all conjugacy classes of primitive elliptic elements {y }, the inner over all iterates (7': 1 5 k 5 m-1}, from k=l through $ x order of y.
contribution of elements {e} and {-e}. Clearly, terms of (5.5), which correspond to { fe} are
4 a )v o V \ G ) { f ( e ) + d ( - e ) } , where symbol
1; if a(-.) = I c = -1;if a(-e) = -I*
{
It remains to express f( fe) (or &functions at { fe } ) , through irreducible characters (generalized 9-transforms) on G. But those are precisely the Plancherel formulae obtained in the previous section
A similar relation holds for class {-e} f(-e) =
-$jj=1 (-1)n-'n(T(tn)
t ?(-.mI)
+;I
t (l+c))(is+)sth
-m
We shall now summarize all 4 contributions into the final result. Selberg trace formula: Let r be a uniform lattice in G = SL,, a ( y ) - a finite-D representation of r of degree d(cr), and R = R" - the induced representation ind(a1 r ; G ) of G. Representation R is decomposed into the direct sum of irreducibles (principal, complementary and discrete series), each entering R with a finite multiplicity, ~ N e ~ ' ~ k * gT '~ ~@ &Ne ,.C B T * ~ B N $ . (5.14) k
-
j
d
principal
complementary
discrete
Operators { R f :f E eF(G)} belong to the trace-class, and trace of
Rf can be
67.5. Selbern trace formula.
339
ezpanded an two different ways, according to (5.5), the LHS being
c
f
k
+c j
?(sj)N8i
+ c ?( f
(5.15)
;
while the RHS is made of contributions of variow conjugacy-classes of G,
00
+ i/ (l+c)J(is+)
s th
y + (l-~))(is-)
scth?
-00
The first line of (5.16) contains the hyperbolic contributions (7 E Conjh(G)), lines 2-3 elliptic part (7 E Conj,,,(G)), lines 4-5 come from the classes { fe}. Looking at a cumbersome 5-line expression one can't help wondering about its meaning and utility? Here we shall give some partial answers to this daunting question, derive a few Corollaries of (5.16), and then mention some interesting connections of the Selberg-trace formula to spectral theory of Laplacians on hyperbolic surfaces.
5.3. Discrete series multiplicities. The first simple consequence of Selberg is the formula for multiplicities of irreducible components of Ra. For discrete series {T "} those are precisely the coefficients N , f of {)( fn ) } in (5.15). Hence, N;t = ('i(""l€)(dla)~~~(r\Crn - m-1 c7e*n= tra(-,') . r k ], (5.17)
*
k=14ktSanm
summation over elliptic classes
{7 },
and { fe}, as only those carry the discrete-series
components. A similar relation holds for N;. In special cases, when discrete subgroup has no elliptic elements, formula (5.17) simplifies to d (a)vol(r\c) Nn+ = N , = (l+(-l)n-b) "* n. The reason for the splitting of the discrete-series components in both sides of the trace formula (5.15)-(5.16) is fairly general and simple. It has to do with orthogonality
I'
57.5. Selberg trace formula.
340
properties of the discreteseries matrix entries and characters. Roughly speaking they behave like the characters on compact group (see. chapter 3). The latter, as we already know, have a unique decomposition for any conjugate-invariant function q4 on compact G, q5
=
asxs =
b
s
X
s for~ all irreducible ~ ~ {s),
due to their completeness and orthogonality in L2(G). For non-compact G , like SL,, characters (even discrete-series) are no more L2-functions. However, discrete-series matrixentries f ( g ) = ( T i $ I + ) do belong in L2, {T”} being embedded in the regular representation R I L2(G). Moreover, one can show that entry f can be chosen in space
L1(G), e.g. the highest/lowest entry fo = (T:+o I $o) (problem 3). Such f can then be paired with any (principal/complementary) matrix entry fl(g), the result being
(f I f J = 0, for all fl E SP4(T811 111)). Hence, operators T i = 0, for all principal/complementary series {Ts), as well as their traces,
I
tr T ; = f ( s ) x s o d s = 0. The latter provides the requisite orthogonality relation between the discrete and other series matrix entries and characters. It remains now to insert such f in both sides (5.15)-
(5.16) of the Selberg trace-formula, and observe that all non-discrete terms vanish. Along with splitting into the “discrete” and “continuous” parts trace-formula allows yet another splitting into the positive (even) and negative (odd) components”. “Odd and even” functions, representations, etc., arise on SL, due to the element {-e}, f(-z) = ff(z);T-,= fI; etc. Clearly, principal {is+}, as well as complementary {s}-series, hence their characters, are even, while the principal {+}-series is odd. Separating the even and odd terms in both sides of the trace formula we get a complete splitting, Discrete
m, C I(fn)N$
n=O
=
d(a)uol(r\G)
t ?(-n))t
=,
“No simple orthogonality relation would help this time, 84 principal/complementary characters form an “over-determined system” on G. Indeed, the trace-formula underpin the overdeterminancy of { x , } , as its the LHS (5.15) exhibits a discrete sum of characters, N: x i , N,X, while the other side (5.16) is represented by various combinations of “discrete sums” (over {y )) and “continuous integrals” of that same set of characters,
c
* +c
Notice, that the RHS of Selberg formula (5.16) contains only the principal series characters!
57.5. Selberg trace formula.
341
5.4. Recipricity Theorem. Next we turn to the principal/complementary part of
spectrum of Ra. Our goal is to find all irreducibles { i s k f } and { s j } , as well as their multiplicities {N} in (5.15). There are two possible approaches to the problem. One of them, based on a Frobenius-type reciprocity principle, allows to reduce a decomposition
of induced R = ind(a I F;G), to the study of certain r-invariants of irreducible representations {T'} of G. We remind the reader the classical Frobenius reciprocity Theorem for compact G (33.2): irreducible n
I 'I
if
E
enfers an induced representation R = ind(a I GG), iff the resfriction
contains a copy of a, n I f 3 a @ N,. Furthermore multiplicity N , ( R ) (of n in R )
is equal t o multiplicity N , ( r ) of a in
T I r. For trivial a = I ,
it says: n C R , iff R I f
confains r-invariants, and multiplicity N,(R) is equal to dimension of r-invariants in n.
The main difficulty with non-compact G has to do with the fact, that unitary representations in Hilbert (L2-type) spaces, like h,
c L2(H),
have typically no -'l
invariants. The latter appear in larger (Loo-type) spaces, associated to
{P}, which have
a dense (core) intersection with L2.Although topologically, representations in L2- and Lco-type spaces may look different, their joint core makes them essentially equivalent (they are also equivalent in the sense of characters, and/or infinitesimal characters!).
For group SL, such "expanded" spaces {h,} could be realized" in L 2 ( H ) .In 57.5
57.5. Selberg trace formula.
342
we described them, as eigenspaces of the invariant Laplacian A on M, u
X,={f(z): Af
=qf}.
We look for (finite-D) subspaces A, c f,, which "transform according to a" under the action of In other words, there exists a basis of functions { fl(z); ...fd(z)} in A , ( d = d(a)),so that
r.
(5.19)
3,
where zg means the usual fractional-linear action, of SL, on W. In the case of trivial a = I , we look for r-invariant eigenfunctions of the Laplacian,
Af =& f ; and f(zy) = f(z); all -y E r, the so called automorphic f o m on W . Functions (5.19) could be thought of as generalized (vector-valued) automorphic forms of weight a. The main reciprocity result for pairs (SL,;r) can now be stated as follows. Reciprocity Theorem: Irreducible representation T of the principle/ complementary series enters an induced representation Ra = ind(a I r;G)with multiplicity N,, iff the ezpanded space %, contains a subspace A, of automorphic forms of weight a. Furthermore, multiplicity of a an
T I r).
dimA, N, is equal to (i.e. multiplicity 4,)
For 1-D representations a (characters of r),reciprocity Theorem states that N , is equal to the number (dimension) of automorphic form of weight a. The proof of reciprocity can be obtained by a careful analysis of automorphic forms (eigenfunctions of the Laplacian), for all 3 series of SL,, as in [GGP] (chapter 1). But there is also a fairly general argument (due to Piatetski-Shapiro), valid for all semi-simple Lie groups G . We shall skip further details and refer the reader to [GGR]. Reciprocity Theorem reduces our "spectral problem" for Ra to the study of automorphic forms, a difficult and largely unresolved question by itself. So its utility in this regard is somewhat limited (as compared to, say, Frobenius reciprocity for Laplacians on spheres and other compact symmetric spaces). Another approach to the Spectral decomposition problem for Ra is closely related (in fact, includes as a special case) Spectral theory of Laplacians on Riemannian surfaces "We have done it in $7.1 for the discrete series, and for principal/complementary series it follows from the results of $7.5: existence of K-invariants in {T'}.
57.5. Selberg trace formula.
N(A) N g4") v o l ( r \ G ) X 2 , as X+w
343
(5.20)
Then the LHS of (5.18) becomes exactly N(A), while the leading asymptotic in the RHS is given by the last integral-tFrm of (5.18),
0 Strictly speaking functions { f} with the Heaviside-type "Fourier transform" are not allowed in the Selberg-trace formula, they can not be C r , and the corresponding operators { T ; } do not belong to the trace-class. However, any such {f} could be approximated by a family of nice (regular) functions {fc}, which suffices for the proof of (5.20) (see [GGP]).
Formula (5.20) means, in particular, that the number of complementary series constituents of Ra is always finite, all of them being located on interval [0;1], whereas no accumulation of Isj+} and {sk} is allowed at any finite point A, of [O;m). We shall indicate the connection of representation R" to Laplacians on Riemann surfaces F\H. To simplify matters let us take trivial cr = I. The representation space L2(F\G) contains a subspace 'V of K-invariants (made of all K-invariants of the evenprincipal and complementary components of R"). Space 'V can be identified with L2(r\H) = L2(r\G/K). Furthermore, the Casimir (central) element
57.5. Selberg trace formula.
344
A, = a{Wz- 2 ( X Y + Y X ) } E Z(!!l) turns into an invariant Laplacian on H. Taking quotient Al, = r \ H reduces A, to the natural Laplace-Beltrami operator on the Riemann surface AL (group plays the role of "periodic boundary conditions" for A,, the same way as discrete lattice Znc W" yields the "periodic" torus Laplacian A,, from the 03"-Laplacian).
r
We observe that the Casimir becomes scalar on each primary subspace L, c L2(r\G) (L,N 36,@ N , - a multiple of irreducible T', contained in R ) ,
Furthermore, each primary (spectral) subspace L, contains exactly N , "Kinvariants" (since each irreducible M, has a unique K-invariant, described in 57.5). We denote by T, a subspace of K-invariants in L,. Then the entire space Y of K-invariants is decomposed into the direct sum of eigenspaces of R, I T CY A,,
y,
T = $T,;
the eigenvalue: A, = on Ts has multipicity N , = dimT,. Here {s} varies over the spectrum (principal; complementary; discrete) of the induced representation R. Hence, solution to the spectral problem for R yields a complete solution of the eigenvalue problem for the Laplacians A , on Riemann surfaces r\H, and vice versa. Both problems, however, are equally difficult to solve exactly. In the next section we shall explore some aspects of asymptotic spectral analysis of A, and establish interesting links to geometry of A, particularly the relation between eigenvalues of A , and the length of closed (periodic) geodesics on A. 5.6. Geodesic flows on negatively curved Riemann surfaces. Finally, we shall give a simple application of spectral decomposition L 2 ( A )= $IT,, to a geodesic flow on A = r \ H . The geodesic flow (Appendix C) is a 1-parameter family of transformations on the unit cotangent bundle
S * ( A ) = { ( z ; ( ) : xE A;l/(ll= 1) = 0, generated by the vector field E = 8,a. 8, - a,a. a,, where a(x;() = root of the metric tensor form on covectors (.
/=
- square
Theorem 3: Geodesic jlow o n any compact R i e m a n n surface A = r\G i s ergodic, in the s e w e that a n y function f E LZ(R), invariant under the flow, is constant. Indeed, vector field S on J b is the image of geodesic generator on the PoincareLobachevski plane (or disk) D. But S*(D) is naturally identified with the group G = SU(1;l) itself, and since field E on S*(D) commutes with the G-action (by
$7.5.Selberg trace formula. isometries) it must be a left-invariant field, given by a Lie algebra element X E 4 1 ; 1).
To find X we just take a geodesics through {0} in the direction of z-axis, and find the 1parameter group of X to be,
hence X = an operator
[ ].
Any (geodesically) invariant function f on 0 must be a null-vector of
{T$}, for
some u in the spectrum of RA. But we know all generators {rx}
for (principal, discrete, complimentary series
T),
neither one has a nontrivial kernel for
s # 0. Hence, kernel(Rx) = 0, and the flow exp(t2) is ergodic.
345
5 7.5. Selberg trace formula.
346
Problems and Exercises: 1. Consider a unitary group representation T with sufficiently many compact operators {T ), (for a suitable class of test-functions {j)in the groupalgebra, and show that suci T has always a discrete (direct sum) decomposition into irreducible components: T 2: @ TP@m(p), of finite multiplicity, m(p) < m. Steps: P
i) take a symmetric element f = 'f on G , and the corresponding self-adjoint compact operator T * show that a non-zero eigensubspace 6 (A # 0) of T intersects G-invariant f subspaces of T; choose minimal among such% (always exists, due to fin of Bx!); and show that T 136 is irreducible.
(k}
ii) derive the direct sum decomposition of T, and show that all multiplicities must be finite. 2. Derive formula (5.12) for the H-transform of an elliptic element u = u8. Steps: i) Show:
7
--*
FK(~e)q5(~e)d8 = IG,lf(g)d(g)
I eit-e-it I- dg,
integration over all elliptic conjugacy classes in G ,where {e it} denote eigenvalues of matrix g E Gel;q5 - an arbitrary test-function, constant on conjugacy classes. ii) Apply (i) with q5 = e-int(e-i'
7
--*
FK(sle)e--ine(e-i6-eie) d8 =
- e i t ) to get
-inl
Jcelf(g) eif -e-ild
9;
aEd compare it with formulae (5.16) for characters and the ensuing transforms of f, {f(s)= (f Ixb): s = +is; s; or i n}, to show
iii) Deduce from (5.21) the formula for the elliptic Harish-Chandra transform o f f
-
,
Fh
iv) It remains to express the_ last sum Fh in (5.22) through the continuous series characters and transforms { f ( i sz t )}. Use the Fourier expansion of the conjugate Poisson kernel in D, A(,i8 - ,-i8 A-n(eine - , - i d *toget = 1 2Acos 8 A'' 1
-
+
v) Use the formulae for the hyperbolic H-transform (cf 57.4),
57.5. Selberg trace formula.
347
.
vi) Formula (v) simplifies, using the relations f(-is f ) = j ( i s f ), and the well known Fourier integrals:
Xis
71 -2XcosB+ 0
m
d
Xis 1 - 2Xcos 0
+
X2
r s i n h s(0-r) dX = sin 0 sinhrs ; O < O < r ;
dX = -
r s i n h s(O+r). -r sin 0 sinhrs '
< 0 < 0.
vii) Complete the derivation.
3. Take a discrete-series representation T", pick its highest/lowest weight-vector $ J ~ (T:$, = erne; for u = u E K), and show i) matrix entry fo(g) = $T;$,I $o) is in L'(G) (use Iwasawa decomposition); ii) show that operators: Ti = 0, for all principal and complementary-series representations {P} (Pair fo tooa spherical function 40(g) = (TdqoI qo) (Tlqo = qo; for u = u0 E K); check that (fo I q50) = 0, by comparing K-actions on fo;do;then use irreducibility of T'!).
348
57.6.Laplacians on hyperbolic surfaces W/r $7.6. Laplacians on hyperbolic surfaces H/r. Section 97.7 develops the Spectral theory of hyperbolic Laplacians A A on Riemann surfacea A = H/r, from a different prospective. It doea not directly involve the representation theory of the preceding sections (f7.1-7.6),but exploits the heat-kernel method, following McKean's approach [Mcl.
The hyperbolic space can be realized either as Poincare-Lobachevski half-plane W,or disk D. The distance between two points z = x1 ix, and y = y1 iy, in the half plane W is given by
+
+
while in the disk realization (problem 4) it corresponxi to
The geodesics in D are all circles perpendicular to the boundary, while in also include all lines parallel to the y-axis (see fig.3 and Appndix C).
W they
Fig.): illustrates geodesics in the hyperbolic geomeiry of the half-plane (left) and the disc (righi). We have shown 2 families of geodesics. One consists of concentric circles, ceniered at {0} in H, which are MSbius transformed into the family of vertical circles, centered on the horizonial axis of D (the left and righi poles in D correspond t o poinis (0) and {co} in M). The dashed circles connecting two poles of D represeni radial (dashed) rays in H. The second family of geodesics is made of parallel lines in H, ihat get iransformed into a family of circles converging io the right pole (00) in D. The hyperbolic geomeiry clearly violates the Euclid parallel lines aziom.
6.1. Fundamental regions in H. A discrete subgroup 'I of SL, divides space W into a union of fundamental regions: U A,,, non-overlapping geodesic polygons {A,,= 7 ( h ) :-y E each bounded by a finite number of geodesic arcs {Cj}.Such tessellations of W are similar to partition of the Euclidian plane (or Wn-space) into the union of fundamental rectangles by a discrete lattice I' c W2. Each fundamental region B c W2 is then mapped onto the quotient space, B+RZ/r N Y2 (2-torus).
r},
Throughout this section we shall assume that group
I'has no fixed points inside
$7.6. Laplacians on hyperbolic surfaces W/r.
349
W,i.e. no 7 E r fixes a point zo of Sz, > 0. This means l‘ has no elliptic elements (fixed points in W turn into “sharp vertices” of the quotient-space, where A loses its smooth structure, a situation reminiscent to quotients of sphere S2, modulo any of discrete “Platonid’ symmetries, chapter 1). Compactness of W / r (hence, of G / r ) also implies the absence of parabolic elements. Thus is made entirely of hyperbolic elements.
r
To construct r-fundamental polygons in H we pick any point zo E W, 7 E
r, and
consider a subset A = { z E W: d(z;z,)
5 d(z;7j(zo));j
= 1;2; ...}.
Such A forms a geodesic 4N-gone in M, the boundary arcs transformed one into the other by elements ( 7 ) . Boundary arcs
{Cj} being
{Cj}are
naturally
divided into opposite pairs, and under proper identification we get a smooth (analytic) Riemann surface A = W/r of genus N . Figures 5 below illustrates a deformation of octagon ( N = 2), into a surface of genus 2.
r then turns into the fundamental group12 of A, in other words each r corresponds to a class of equivalent closed loops { w y}. Furthermore, fundamental class { w } of r N x 1 ( A ) , contains a minimal length path yo, which Group
element 7 E any
N
coincides with the closed geodesics of the given homotopy type. We shall illustrate the foregoing with a few examples of fundamental regions. Examples of fundamental regions H/E 1. The modular group
r = SL,(Z)
is generated by 2 elements: translation by 1 in the
horizontal direction h: z+z+l, and reflection u: z-+l/z,given
Two types of fundamental regions of
by matrices
r are shown on fig.4:
12Two path yl; y2 on a manifold Jb are called (homotopy) equivalent, if y1 can be continuously deformed into y2 inside Jb. Any pair {yl;y2} of closed path (loops) passing through a fixed point zo E Jb can be formally mulfiplied by combining them into a single path y l q 2 (=yln followed by “y2”). Such multiplication is easily verified to respect the (homotopy) equivalence on space Q(z,) of closed path, through {z,}.
So the set of equivalence classes of {y E Q(zo)} acquires the group structure: the identity consisting of all path contractible to {z,}, while the inverse {y-’}, being given by y, traversed in the opposite direction). The resulting group is called the fundamental (or 1-32 homotopy) group of Jb, and denoted by nl(Jb). It carries some important topological information about Jb. For instance, the fundamental group = nl(Jb) of a 2-D surface A, when quotient modulo its commutator = yields the so-called homology group H l ( A ) = r/r’. The latter forms a commutative group, isomorphic to Z N x C (direct sum of the “free and torsion components”: ZNand C), and dimension N of the free component is precisely the genus of Jb!
r
r’ [r;r],
350
57.6. Laplacians on hyperbolic surfaces W/l"
3;
(I) vertical strips (light shading) are shifts of the basic strip {-$I %r 5 I z
I > 1);
(11) Fundamental triangles (dark shading), if fact, quadrilateral, since point i E C3 should be considered a vertex, obtained by shifting the basic triangle {O; f i ~ ]The . latter
3+ &
is bounded by geodesic arcs: C,,,=(I 63 1 =l}; and C3 = { I z 1 = 1). The first type regions are obtained by fixing point { 2 i } , shifting it by h:2i-2i f 1, and inverting by 0:2i+$, then writing the corresponding geodesic bisectors: d ( z ; 2 i )= d(r; f 1+2i) (vertical lines); and d ( z ; 2 i ) = d ( z ; q i ) (circle I z I =l).
Obviously, the fundamental triangle is obtained by a-inverting the fundamental strip. Let us remark that the fundamental region Jb of the modular group SL2(Z) is not compact, it has cusp (0) a t {co)(i.e. at the "infinite boundary" of H, aH = R
U (m}). But Ab has
finite volume (problem 2 ) . Topologically manifold Jb looks like a sphere with an (infinitely) long spike (Fig.5). Fig.4: Two types of fundamental regions of ihe modular group
r =sr,~).
Figd. Riemann surface H/SL,(Z)
with a
cusp (spike) at {m}.
2. Our next example (Fig.6) demonstrates a typical fundamental polygon in H, whose vertices
are (finite order) fixed points of certain elliptic elements (yj E Z'}.
$7.6.Laplacians on hyperbolic surfaces W/r.
351
Fig.6 A typical fundamental polygon in H.
Topologically quotients A = D/r, drawn in fig.6, turn into Riemann surfaces of higher genus g. Fig. 7 below demonstrates a transformation of an octagon with properly identified sides into a genus 2 Riemann surface. Let us notice that images of the fundamental region J b under all elements
{r E r } form a regular
tessellation of D into
4g-gons, an impossible fit in the Euclidian flat-land, for any g 2 1. A remarkable feature of the hyperbolic plane is the existence of regular tessellations of any order, hence of higher genus hyperbolic surfaces Jb/r of constant negative curvature!
Fig.7. (i) A Riemann surface of genus 2 (A) is cut into 2 truncated tori (B);
(ii) The tori are then unwrapped into squares (with holes in the middle, which correspond to the original cut) via 2 siandard fundamental cuts;
a
b
d
(iii) The inner circle is stretched through a vertex to turn each square into a pentagon C
a
(D)?
s7.6.Laplacians on hyperbolic surfaces H/I'
352
b
(iv) Finally two pentagons a r e glued together along the diagonal (the original cut) to form a n octagon (E). finally two pentagons a r e glued together along the diagonal (the original cut) to form a n of octagon (E). The fundamental group genus-2 surface has 4 generators {a; b;c;d}, a n d a single relation: aba-'b-'& - ' d - = e, which corresponds to traversing the octagon clockwise.
r
b
(v) The resulting fundamental group r of genus-2 surface has 4 generators {a; b; c; d}, and a single relation: = e, which aba-'b-'cdc-'d-' corresponds to traversing the octagon clockwise.
d"
d
6.2. Kernels, traces and heat-invariants.We are interested in eigenvalues {A,}
of the Laplacian A on hyperbolic surfaces A = W/F.These are usually impossible to compute exactly (in any closed form), so one would like to get some approximate or asymptotic expansions of {A,}. The latter often involves the study of certain "means" of {A's}, (cf. chapter 2), and the related transform (Laplace, Fourier, Stieltjes, etc.) of the counting (spectral) function N(A) = #{k:Ak _< A}. We shall mention a few of them Heat-kernel (Laplace):
m
o(t)= x e - t A k = je-t+iN(A) Z(s) =
Zeta-function:
cm-
Resolvent (Cauchy-Stieltjes): R(C)= - J7
= tr(etA);
0
CXks= JA-SdN(A) = tr(A-"); 0
c&
dN(A)
o(C-4
Wave-trace (Fourier):
m
mdN(X)
= J-
0C-A
- tr(C - A)-'; -
or
= tr(C - A)-"';
X ( t ) = C eW
G = JeiMdN(X) = t r ( e w ) .
The first two are called theta and zeta-functions of A, by analogy with the classical (Jacobi, Riemann) theta and zeta functions, which correspond to an "integral spectrum" { A, = k } . All transforms represent traces (regular or generalized) of various Green-functions of A: theta gives the trace of the heat-kernel, tr(eVtA), (the
$7.6. Laplacians on hyperbolic surfaces
W/r.
353
fundamental solution of the heat problem: ut = Au), Cauchy-Stieltjes transforms R corresponds to resolvent of A (and its powers), while di~tribution'~ X ( t ) on R yields the wave-trace, t r ( e i t d ) (Green's function of the wave equation: utt - Au = 0) (see $2.4).
A11 transforms are related one to the other, for instance, zeta is obtained from theta via T-integral, 00
Z(s) = &j! a consequence of the obvious relation
@(Ws-lat,
This often allows to link large-) asymptotics of spectrum { A k ] , to asymptotics of zeta, theta, etc. functions, via the so called AbelIan/Tauberian Theorems. If for instance, sequence Ak
-
kp, as k+m, for some p > 0, then its theta-function @(t) =
is easily seen to obey
-
e(t)
e-"k
-+re-tAApdA,
r(p+l)t-p-';
a t small t ,
and similar relations could be derived for other "kernels of A". The converse, however, is not true, typically means, like 8,Z, etc., behave much better, than irregularly distributed sequence of eigenvalues. For instance, the heat-trace admits an asymptotic expansion, due to Munakshisundaram-Plejel, tr etA
-
t-"/'{b,
+ b l t + b,t2 + ...},
(6.2) whose coefficients { b k } , called heat-invariants, carry some important geometrictopological information about manifold m. For instance, b, is proportional to vol(rn), with Const, depending on n; 6 , =
I
Kdz
-
integral of scalar curvature14, K ( x ) , over
the natural (Riemannian) volume elEment on m. Higher invariants involve certain universal polynomial expressions in curvature and its covariant derivatives, integrated over m. All of them could be computed in principle, but the formulae quickly become unmanageable [Gill. So to get a better insight into the structure of spec(A) one need to is no difficult to show that heat-semigroup and resolvent (( - A)-', of Laplacians tIon compact Riemannian manifolds m, are given by nice integral kernels: K t ( z ; y ) and R ( z ; y ) (cf. chapter 2 ) . Hence both are compact, and belong to the trace-class. But the wave-kernef (made of unitary operators { W ( t )= e z p ( i t 6 ) ) ) could be "traced" only in a generalized sense, as a distribution on R (see chapters 1-2).
14Scalar (Gauss) curvature K(z,,) at a point xo on a surface e measures the deviation of surface area of the geodesic sphere: S, = {d(z;z,) < 0}, from the Euclidian (flat) area: I S, I = ?TC2Kc4 + For an n-D manifold we take all geodesic 2-planes {e,} through 2, (images under the e x p map of all planes in the tangent space T20) ' and average the Gauss curvature of e, at zo over all {e,}. The formal definition of K involves the trace of the Ricci tensor.
... .
57.6.Laplacians on hyperbolic surfaces H / r
354 *
look for a different set of "geometric spectral data". One such class consists of the length of all c l d path (geodesies) in A.
In this section we shall compute the heat-trace (theta-function) for Laplacians on A, and will link it to closed geodesics in AL. A simple prototype of the main result and the basic approach below is the flat (torus) case ? N R"/r, discussed in chapter 2. Any lattice (discrete subgroup) r CR" can identified with the image of the standard (unit-cell) lattice Z" under a linear map A:R"+W". So the Laplacian on W/r, pulled back to the standard n-cell To= [0;1] x... x [0;1], turns into a 2-nd order constant-coefficient elliptic operator: L = Bd .d = C bijd:j, where matrix B = (tAA)-
'.
The eigenvalues of L in To (respectively, A in T) are readily computed while closed path in
= 4T'Bk. k; k = (k1;...;k,) E Zn. T,labeled by n-tuples m = (ml;...mn),have length em = I Am 1 = B-'m*m.
To get the heat-kernel Lon To, we take the free-space Gaussian, e w t z ,
averaged it over the lattice I',
Gk)+
C
G(z
+ m),
m E r
then applies the Poisson summation formula. This yields ~ ~ - t ( 4 a ' B k . k--) (4rt)"" m k
7
whose LHS gives the theta-function, x e - t A k , while the length of all closed path,
t-+
c
2
e-A@m
(6.3)
RHS involves exponentials of
.
In the hyperbolic setup the role of lattice r is played by a discrete subgroup of SL,, and "tori" become fundamental regions W/r. We shall take as above the free space heat-kernel K t on W, which depends only on the distance d(x;y), between {x} and {y}15, K = K(d(x;y)), average it via the r-action, and analyze all terms of the resulting 15The latter follows from the symmetry properties of the Laplacian A on H Laplacian is invariant under the SL2-action, hence any "function of A", e.g. K , = e-tA, is also C-invariant. But a G-invariant integral kernel F(z;y) must depend on the distance d(z;y) only, due to a double-transitive action of Sb on H: any pair (2;y) is taken to any other equidistant pair ( 2 ' ; ~ ' ) by an element g E SL, (problem 3). This is a general feature of so called rank-one symmetric spaces (see J5.7).
57.6.Laplacians on hyperbolic surfaces H/r.
355
series. The analysis could be carried out for any radial function K ( r ) , r = d(x;y), decaying sufficiently fast at {m}, to assure convergence of the r-series below. Any such K defines an SL.+variant integral kernel K(z;y) = K(d(z;y)) on 04, which can then be reduced (quotient) to the manifold A. The resulting kernel
The trace formula below gives trKA in terms of length of primitive inconjugate elements { p E r}.A hyperbolic element 7 is called primitive, if 7 is not an iterate of another element, i.e. 7 # for m 2 2,and r0 E r. We shall introduce the length of qEG, as
7r,
One can check (problem4) that e does, indeed, represent the length of the smallest closed path (geodesics) in the fundamental class of q E r . For hyperbolic q - one finds
- I"
m2dy
e(q) = J
= 1og(m*),hence e(qn) = 210g(mn).
1
Now we can state the main result of the section.
6.3. Trace formula: Given a radial function K ( r ) , r = chd(x;y),on W,the reduced kernel K A (6.4) on a compact quotient space AL = W/r belongs to the trace-class. Its trace, t r K A = K,(x;x)dx, can be expanded in a double series, over integers n = 1;2; ... a n A v e r all inconjugate, primitive elements { p E r},
Here pn denotes the n-th iterate of the primitive path p , and l(...) - the length of the path. The proof involves several steps, some of them similar to the derivation of the trace formulae in the preceding section. Namely, 1) Group
r
is split into the union of conjugacy classes of powers of inconjugate primitive
elements:
r = {e}U{y-'p"r:
yE
r rP},
57.6.Laplacians on hyperbolic surfaces M/r.
356
where r pdenotes the centralizer (commutator) of element p in f. 2) We observe that integrals J A K ( z g ; z ) d z , over a fundamental region A, are equal, for all
conjugate elements {g = y-'pny:y E r}.SO
where [Rfp] denotes the index of subgroup clasa {e},
r pc r. The first integral corresponds to the trivial
while the remaining terms give nontrivial classes {p"}.
[r:r,] = # {cosets in r
Notice that index
fp)is always finite
3) Let also remark that K(zg;z)dz,
is identified with an integral of K over yet larger fundamental region A,
N
H/fp,of subgroup
r pc r, A, is made of [I? rp]non overlapping copies of the original A! 4) Next we note that f,=
{pk:k
E 2 ) coincides with the subgroup generated by {p}. A
hyperbolic primitive element p E r is conjugate equivalent to a diagonal matrix q = Such q acts on H by dilations with m', strips { 1 5 I Sz I
60
a fundamental region of q can chosen to consist of two
5 mz} (fig.8). By the distance formula (6.1) the argument of K becomes
(I - mZn)' I z I 2mZnxZ2 Then integration over the fundamental region Aqyields, chd(x9"; z) = 1 +
/%r
'
m2
K= AAb,
dzlK(l+~(m"-m-n)'(1+~~)2~)
122-w
Changing variables: z 4 #z2;t =
2),we get
l o g m 2 7 K ( 1 + f ( m " - m - " 2 ) (1 -00
+ t2))dt.
Remembering that e(q) = logm', e(q") = logm'", the argument of (6.8) turns into
=
+ 2 s hne(9) 2 7
n+ 1
t2).
We call the new variable u, and make yet another change, t-u, to brings (6.8) to the form
which completes the proof of (6.7).
7.6. Laplacians on hyperbolic surfaces W/r.
357
Fig.8. Fundamental region of a hyperbolic element h can be chosen either as a strip {A 5 1922 I 5 m} ( A ) , or annulus {f 5 I z I 5 m} ( B ) .
The heat-trace. Now it remains to specify (6.8) to the heat-kernel Kt(z;y) on A. But first let us find the heat-kernel on W. We have already observed that K depends only on the hyperbolic distance r = d(z;y) between points z,y E W. There is a wellknown formula for K , in terms of ch r (see problem 51,
Next we shall compute the contributions'of various conjugacy classes in (6.7),
while the pn-term contributes
The inner integral gives the standard Beta-factor B(i;i)= a, while the outer integration results in e-e2/4t (4*t)'/2
Substitution in (6.9) yields the following version of the Selberg truce f o m d a for the heat-kernel @(t)= t r ( e t A ) = e-tAk, of the hyperbolic Laplacian A = A ,
with coefficient
358
57.6.Laplacians on
hyperbolic surfaces W/r.
Remarks: Formula (6.10) has many i:teresting
interpretations and applications in
Spectral theory of differential operators, and in quantum mechanics. It links directly the “Laplace-transformed” eigenvalues {A,}
of Ah on the one hand, and the length (period)
spectrum of manifold A {l(y): over all closed geodesics y
c A}. In
the picturesque
language of Mark Kac [Ka]: One can hear ihe length specirum of A. This allows to address the Inverse spectral problem: Can one hear (uniquely deiermine) the geometry
(metric) of Ab from specAA? The length spectrum
reduces the Inverse spectral question to a Problem
in
geometry/dynamics on manifolds: given iwo metrics on A, whose geodesic flows have
identical period/length spectra can one conclude thai metrics are isometrically equivalent? In the classical case of hyperbolic surfaces A = H/r of constant negative curvature, McKean [Mc] showed that there are at most finitely many isospectral surfaces { A } .He utilized the Selberg trace formula (6.10), and the Klein-Fricke double and triple traces {tr(ykY,,,); tr(yjykym): y k E
r}.The latter
served to identify elements {yk} up to
r-
conjugacy, hence the length of closed geodesics {e(y)}, and their orientations. The work of McKean [Mc] still left a possibility of a unique solution of the Inverse spectral Problem, until Vigneras [Vi] found in 1980 finite (but arbitrary large!) families of distinct isospectral surfaces of any genus. These examples, however turned out to be rather exceptional. Wolpert [Wo] showed that a generic hyperbolic surface is uniquely determined by spec(A)! For more general hyperbolic manifolds of non-constant curvature Guillemin and Kazhdan [GK] established “infinitesimal rigidity”. They also used a reduction to a dynamical Problem. In the study of dynamical systems it was found that uniqueness usually requires an additional ”discrete set of data”, for instance association of homotopy classes to the periods {L,,,}, [KB]. The latter was previously used by McKean [Mc] in the constant curvature case. Under such hypotheses, the geometric Problem was shown to have a unique solution [KB]. Though the length spectrum brings one tantalizingly close to settling the inverse spectral question (at least on hyperbolic surfaces), there remains the frustrating “labeling problem”, whose spectral content is unclear. In the quantum-mechanical context Laplacian A A represents the hamiltonian (energy-
operator), which describes the evolution of a quantum system (particle), constraint to move on a surface/manifold A. The underlying classical mechanical sysiem coincides
57.6.Laplacians on hyperbolic surfaces W/r.
359
with the geodesic flow on the phase-space p(A)- cotangent bundle of A, given by the corresponding classical hamiltonian/energy function: h ( z ;() =
gjj(z)(j(j
-
metric
tensor, as a function of phaoe-variables: z E A, ( E T,*. Formula (6.10) gives then a correspondence between “eigenualues of A ” (quantum energy levels), and the “length spectrum” (energies/actions) of classical trajectories (geodesics). Such relations exemplify the Correspondence principle of Quantum theory, which asserts close links between the classical and quantum systems. Usually, the correspondence holds only approximately/asymptotically at “large energies” (or for small Plank parameter), where the quantum dynamics becomes quasiclassicel. The remarkable feature of the flat (Tn) and hyperbolic Laplacians is that the quasiclassical expansions (6.3) and (6.10)
turns out to be exact! In the next chapter we shall explore in the greater detail the structure of classical hamiltonian systems, the quantization procedure and the role of groupsymmetries in the classical and quantum mechanics.
57.6.Laplacians on hyperbolic surfaces H/r.
360
Problems and Exercises: 1. Find the distance between two points: z = z1 +&and y = y 1 + iy2; in the Poincare half-plane H (6.1) (hint: it is easy to compute the distance between purely imaginary points d(ia;ib)= ln:. Use it and find a fractional linear map g:H+H, that takes 2 4 ; y-tib, and compute b). 2. Find the hyperbolic volume of the fundamental region Ab = H / C
r = SL,@) (fig.4).
3. Show that any 9,-invariant integral kernel F ( z ;y ) depends only on the distance d ( r ;y). Hint: F(zg;yg)= F ( r ;y ) , for all z , y E H, g E G; move 2 4 , and use the fact that stabilizer of { i } , K = S q 2 ) , acts transitively on all spheres S,(i) = { y : d ( i ; y )= r } , centered a t { i } , of radius r.
4. Show that the hyperbolic distance between points (6.6). Use the disk realization of H, and show that and d(O;Oq) = ln(
d ( 0 ; r ) =$In(*); l-lzl
(2)
and
{zq} (q
I I + I I ), where q = (I
E SL,) is given by
P [P
E SU(l;l)!
5. Derive the heat-kernel Kt(chr) (6.9) on the Poincare half-plane, in terms of hyperbolic radius r = d ( r ; y ) . Steps: (i) the Laplacian in hyperbolic polar coordinates (r;O) of D is given by A = 6’: + cth ra, + sh-2rag2. Changing r-m = ch r , bring it into the Legendretype differential operator,
Z = a(z2-1)8 +*02, 2
on the half-line [l;oo) X 1;
-1
ii) Laplace-transform the heat-kernel 00 K(1;...)-Q(r;A) = e e - ” K t ( r ) d t , 0
and check the resulting function &(.;A)
to solve the equation:
(Z+ AN91 = a(r),
(6.11)
i.e. Q(r;A) gives the resolvent-kernel of Z. But for radial function Q = Q ( r ) equation (6.11) reduces to the Legendre ODE: [(z2-1)Q‘]’ + A & = 0, with a suitable “source condition” a t z=O (cf. J2.3, chapter 2). Its solution is known to be a Legendre function of 2-nd kind and order v, Q = Qv(z), where A = v 2 f .
+
iii) The Legendre function Qv(r) has an integral representation ([Erd];[Leb]),
iv) Use this representation of Q, and compare it with the Laplace transform of the integrand of (6.9), ~[t-~/’ezp(-t/4-r~/4t}]. YThe later yields a modified Bessel (Kelvin) function K1/,(r A + 1/4), via an integral representation of Kelvin functions, 00 Y %ezp{-a(at 0 +:)}t-’-’dt = 2(;)”/’K A. ab). But in our
case
i,
order s = the corresponding Kelvig function becomes classical, ~
Complete the derivation!
~
= ~
~
(
= z
)$ e - z !
361
$7.7. SL,(C) and the Lorentzgroup $7.7. Sb(C) and Lorentzgmps. In the last section of ch.7 we shall briefly review the representation theory of the complex group SL,(C), the related Lorentz group of special relativity, and also higher-dimensional (pseudo-orthogonal) Lorentz group S q l ;n). Most results will be stated without proof.
7.1. The Lorentz group S0(1;3) of special relativity preserves the Minkowski (1;3)-form in [email protected] 2-fold cover is the complex unimodular group SL(2;C). To
show
the
correspondence
SL,-Sq1;3)
we
note
that
the
real
4-space
= {(zo;z1;zz;z3)}can be identified with the space of complex hermitian matrices, ~~~~
'p=
{x = [
c'E --2
'
ro;zl
]
= z2+iz3}1
where the determinant-form d e t ( X ) = (X I X) = z2-Czjz, defines the Minkowski product on
3, and conjugations, g:
x+g*xg,
preserve hermitian symmetry, as well as the Minkowski norm, det(g*Xg) = det(X). So we get a map (isomorphism) of
SL,/{ f I } onto the Lorentz group Sq1;3).
Irreducible representations of consist of 2 series: principal and complementary. Principal aeries representations are induced from the Bore1 subgroup B of all uppertriangular matrices,
here C* denotes the multiplicative group of complex numbers, C* N T x R*. The characters of B are labeled by pairs { (s;m):s E R; m E Z},
The quotient-space 96 = B\G can be identified with the complex plane C (via the Gauss decomposition g = n-hn+), and G acts on C by fractional-linear transformations, as in $7.1,
So principal series representations {Ts9m}can be realized in space LZ(C;d2r),by operators ,
-'
Here d2z denotes the standard area element &fz A dZ in C, factor I bztd I serves to unitarize Tslm, and the complex argument ~br+dl bz+d plays the role of "sgn(bz+d)" in the real case ($7.1). So representation TgVm is induced by the character
$7.7.SL,(C) and
362
the Lorentz group
of B. An alternative realization of TSfmon the 2-sphere will be explained below in the context of the general Lorentz group So(1;n). We remark that the Riemann sphere S2 gives a 1-point compactification of the complex plane, Sz= C U {co}. 1rs7m
Irreducibility of Ts9m can be now established either by the Mackey's test (Theorem 1 of $7.1), or directly (see problem 2 of 57.1). The principal series representations have characters, given by
xs,m(g) =
I I istm ~
~
+- I m A, I -(is+m)
I Ag - A g - l l
A
m 8
. 7
(7.3)
where {Ag&'} denotes the eigenvalue of matrix g E SL,(C). Representations T S * m and T-S,-m are equivalent, which can be verified either directly (by constructing intertwining operator W), or as a consequence of (7.3). Complementary series correspond to the imaginary s = ia, -2 < a < 2 , and m = 0 in (7.3). As in the real case 57.1 they can be realized in the Hilbert space,
determined by the Riesz potential in C,
1 f Ip =
1I I z
w I "-"(z)fod2zd2w.
Their characters are analytically continued in s functions more details we refer to [GGV], chapter 2).
{ x ~ ,of~ (7.3) } (for
Group SL,(C) has no discrete series as will be explained below. The discrete series of semisimple Lie groups were directly linked by Harish-Chandra [Har 5) to the existence of compact the maximal abelian (Cartan) subalgebras, and SL,(C), as any other complex G, has none. 7.2. The Plancherel-inversion formula for SL,(C) has the standard formulation: for any smooth rapidly decaying function f on G,
where { x ~ , denote ~ } the principal-series characters. So the Plancherel measure dp is supported on principal-series part (a countable union of half-lines), and has density, dp(s, m) = (s2
+ m2)ds.
Gelfand, Graev, Piatetski-Shapiro [GGP] derive a general Plancherel formula on SL,(F) over any locally compact field F (reds, complex, p-adics). Their result for complex C states,
57.7.SL,(C) and where the Plancherel density,
363
the Lorentzgroup
Gprine
and { T = rstrn}varies over all characters (7.2) of B . Singular integral (7.4) is understood in the regularized sense, whence follows,
+
p ( r ) = Const(s2 m2), with Const = 1 . 32r4
Selberg-trace formula for compact quotients r\G of Sb(C) looks similar to 57.5,
but many simplifications arise due to the absence of the discrete series and the “elliptic conjugacy classes” (any g in SL,(C) is conjugate to a diagonal matrix!). Chapter I of [GGP] provides the complete details.
7.3. Lorentz groups SO(1;n) preserves the Minkowski product,
“,yo-
e x i y j in 1
MIn+’. The preceding discussion of SL, has prepared the reader for the introduction to representations of more general semisimple groups. Here we shall briefly discuss one such class higher-dimensional Lorentz groups. The key to the analysis lies in two decompositions, Cartan and Iwasawa (see 555.7 and 7.1). Group G contains a maximal
compact subgroup K = S q n ) , and an abelian subgroup of hyperbolic rotations,
[ ].
generated by a (relativistic-boost) operator H = The commutator of H in G is he product A . M , where compact subgroup M = Sqn-1) c K gives the commutator (centralizer) of A in K ( M consists of orthogonal rotations in the span{z,; ...zntl}!). Element H (its adjoint map adH) splits Lie algebra (5 = so(1;n) into the direct sum of eigenspaces: 0-1@ 0,@ 0,
(7.5) of eigenvalues {-l;O; 1). Here subalgebra (5, = ‘u @ !Dl - (Lie algebras of A and M ) , while are made of matrices of the type, subspaces (5
*
where (n-1)-vectors u = (v2;. . . v ~ + ~u)= ; (uz;...untl) satisfy the relations u = u for g1, u = -u for (5-l. Indeed, element H acts by left multiplications on row-matrices [i].
57.7.SL,(C) and the Lorentz group
364
Decomposition (7.5) clearly obeys the Lie-bracket relations, for p,X = 0; f 1, [O,;OA] c so 0 form two commutative subalgebras R f of 0, and the sum I @ R+ becomes a 2-step solvable subalgebra of O (51.4). Now the entire space 0 can be split into the direct sum of subalgebras,
O = si @?I@ R. With some additional effort (cf. [Hell) one can show that group G can be similarly factored into the Iwasawa product, G =KAN. We define the Borel subalgebra of 0, as B = !Ill@ I @ 9,using the fact that the centralizer !Ill of '(I leaves subalgebra 9 invariant. The corresponding Borel subgroup is also factored in the product, B =MAN. The quotient-space B\G can be identified either with the n-sphere S" = M\h', or with a "closure" of N , using a different (Gauss) factorization G = B N ,
Principal series representations of S q l ; n) are induced by irreducible representations of the Borel subgroup B, so T = T S Y mare labeled by the pairs16 { ( s , m ) : sE W;mE M } . Let 'T = Ym denote the space of m, and 7r = R"'' = xs 8 am the corresponding (inducing) representation of B, A
7r;,m = &sa T m. ",
b=U.a.h, ( u E M , u E A , ~ E N ) .
Then T s ~ m = i n d ( a S ~B;G) m I acts in the usual way on the product-space L2(Sn)8 'Tm, and can be shown to be irreducible (problem 1). A different realization of TSjmcould be given in spaces L 2 ( N...) using a Gauss-type decomposition G = B.N . Let us compare these constructions with special cases of S0(1;2) (or Sf,@)) and
S0(1;3)= Sf,(C)/Z,. Examples. 0
Sf,(R).Here the abelian subgroup, A=
{[
A A - 1 1I #
> 0) (positive diagonal); K
=
-The dual object of the orthogonal group Sqn-1) was described in J5.3. It is made of all ordered p-tuples {ml 2 m 2 2 ... 2 f mp; p = [;I}.
57.7. SL,(C) and the Lorentzgroup
M = { f1);N =
{[ :}I 1
365
upper-triangular.
Bore1 B made of all upper-triangular matrices. The principal series {TS* * } were realized in L2(R),where W N B\G, or L2(T), another (projective) form of B\G.
SL,(C) has A - positive real diagonal matrices; K = SU(2), or S q 3 ) for the Lorentz S q 1 ; 3); 0
M - unitary diagonal
{Ieie 1 ,-ie
(maximal torus in K);
N , B consist of upper-triangular matrices; the quotient-space B\G = M\K N S2 (from the Iwasawa factorization), or C = R2 (from the Gauss G = Be N ) . In (7.2) we were able to write down explicitly operators using the fractional-linear (conformal) action of SL,(C) on the quotient B\G
N
{T2m},
C.
It is interesting to note that the action of higher Lorentz groups on the homogeneous space S” N B\G implemented
by
’
a
is also conformal. The conformal action could be stereographic
map
from
the
hyperboloid
Wn = {q,’- I z I = l} N K\G (where G acts isometrically), onto the sphere Sn= (202 + I z I = 1). So right translations of G on the quotient B\G are equivalent to its conformal action on S” (problem 2). As their low “SLz-cousins”the higher Lorentz groups have complementary series. But the discrete series appear in only half of them {Sq1;2n)} with even space-like part
(like SL,(R)), and are absent in the odd case {Sq1;2n+l), e.g. SL,(C)}. The Plancherel formula includes only the principal and discrete series (when present). The detailed analysis of SO(1;n) is contained in the last chapter of [Wal,73], and the original papers
[Hir].
$7.7. SL,(C) and the Lorentzgroup
366
Problems and Exercises: 1. Establish irreducibility of the principal series representations of S q 1 ; n ) by the Mackey's method.
+
'
2. We define the stereographic map fig 9 from the unit sphere S" = {zoz I t I = I} to It 1' il)}, parametrizing both surfaces by a hyperplane the hyperboloid H" = {z R" = { t= ( 0 ; z ) )in Rntf Here maps @:S"-R" and P:H"-.R" are given by
'
b:y = (yo; y)-z = L; and 3:z = '-Yo
r'
x-
where yo = f
1- I y I denotes a point in S",
Y X I 1
Fig.9. Two stereographic maps 8, P fake fhe unit sphere {y} and the hyperboloid { z } onto R", hence define a conformal map @ - l o # from H" into S".
(i) Show that both maps are conformal from S", H" into R". Hint: use the polar coordinates on all three surfaces: {(r;O)} in R", {(q5,0):yo = sind} in S", and { ( t ; O ) : z o = s i n h t } in H". Note that 8 and P transform only the radial variable r = cot$ = cothi.
(ii) Combine 8 and P we get a conformal map 8 y-x:
1
rd
Y
'
o
P from H" into S",
D = 1 r J%
(iii) Since G = S q 1 ; n ) acts by isometries on the symmetric space H" = K\G (K = stabilizer of the pole N = (l;O)), the stereographic map 8-' o P takes it into the conformal group of the sphere S". One can show ([Tay]} chapter lo), that S q l ; n ) comprises the entire conformal group of S". (iv) Compute the conformal factor p ( z ; g ) for any g E S q l ; n) on z E S".
Additional results and historical comments. A. Kirillov [Kirl] traces the onset of the representation theory to the late
XIX century,
and divides its history into 3 periods. First connected with the names of Frobenius, Schur, Burnside dealt mostly with algebraic aspects: finite group and algebras, characters, projective representations, as was outlined in chapters 1-3. Second period brought in compact groups with the contributions of Haar and von Neumann (invariant integration), Peter-Weyl (completeness of finite-D representations). At the same time E. Cartan and
H. Weyl [We31 (1939) unveiled the structure and built the representations theory of simple and semisimple Lie algebras (chapters 4-5). These results not only strike with their profound inner beauty, but find deep applications in a wide range of mathematical and
57.7. SL,(C) and the Lorentzgroup
367
physical subjects: geometry, differential equations, analysis on symmetric spaces (the orginal Cartan’s motivation), quantum mechanics and particle Physics. Soon the need came to study noncompact groups and their infinite-D representations. The first result along these lines, the celebrated Stonevon Neumann Theorem, essentially amounted to classification of unitary irreducible representations of the Heisenberg group. In 1939 E. Wigner [WigZ] made the first attempt to build the theory of elementary particles, based on the infinite-D representations. However, more systematic study (third period) began in late ~ O ’ S , when first classification Theorems for the classid complex Lie group: SL(n;C);
Sqn;C); Sp(n;C), were obtained by Gelfand-Naimark [GN], and Bargman [Bar]. Since then the development of theory went a t an ever increasing pace. For general surveys on the role of the harmonic analysis and group representations we recommend surveys [Mac2,5] and [Gr].
Semisimple gmupe: The basic features of the representation theory of complex semisimple groups were unveiled in the monograph [GN]. It was shown that the principal part here is played by the so-called parabolic subgroups: P C G. These are characterized by the property, that P contains the maximal solvable (Borel) subgroup (e.g. all uppertriangular matrices in SL,), equivalently, the quotient-space P\G
- compact.
Zelobenko
and Naimark [Ze] proved that all irreducible (even nonunitary) representations of G are elernenlaryl7, i.e. induced by 1-D representations of the parabolic subgroup. Monograph [GN] derived characters of elementary representations, and established the Plancherel formula for classical complex groups. It was also found that semisimple groups have complementary series, that do not “belong” to the regular representation RG. Then Harish-Chandra [Harl] extended the “classical results” to arbitrary complex semisimple groups. He also found the general Plancherel formula for semisimple groups in terms of his celebrated c-function ([Hara]), dp(X) = Const I c(X) I -,dX. This yields an explicit expression of d p on an arbitrary group G , or a symmetric space
X = G / K , in terms of the (principal series) weight A, and the root system C = {a} of G, or the restricted root system of space X (55.7). Namely ([Har2];[He12]),
n
dp(X) = Const aEC+
0I 4
where function q5a depends on multiplicity ma of root a,and is given by 17The study of elementary representations is often facilitated by the fact that the double conjugacy claases P,\G/P, are finite for any pair of parabolics P , ; P , (this result h known as Bruhat’s Lemma). In particular, for the Borel (maximal solvable) subgroup B, they coincide with the elements of the Weyl group W of G!
$7.7. SL,(C) and the Lorentz group
368
I
All 4 cases are determined by the relative values of multiplicities m, and m2,, according to the table.
m = 2,4,6,8
... ...
and m2, = 0 m = 1,3,5,7 and m2, = 0 m = 4,8,12, and m2, = 1,3,7 m = 2,6,10,14, ... and m2, = 1
...
The classification of all nonunitary (Banach-space) irreducible representations of semisimple groups was given by Berezin [Ber] (1962), based on his study of Laplacians (Casimirs operators) on G. But the problem of selecting unitary representations among all elementary ones turned out to be technically difficult, and has not been yet completely solved. Significant progreas has been made in the past decades, that has brought about a complete classification for a growing list of groups: S q 2 ; 3); Su(2; 2); Sp(1; n); SL,(F), for any F = RC;Q. For surveys of this important work we refer to [Kn];[KS];[Sp];[Vo]. The representation theory of real semisimple groups added further difficulties. Here elementary representations proved insufficient to build a complete system, even if one allows the holomorphic induction ($6.1). First example of non-elementary “strange series” appeared in the work of Gelfand-Graev for group S y l ; 2 ) . But only after the Langlands’ work [JL], it became clear that “strange series” can not be realized in “function-spaces” (0-forms), but require higher-rank differential forms. Langlands conjectured that all of them could be realized in higher “Z’-cohomologies”, by a combination of the regular and holomorphic induction (see $6.1). The most significant contribution to representations of real semisimple groups came from the works of Harish-Chandra. His papers [Har] gave, in particular, a complete classification of
80
called discrete series, i.e. {T} with square-
integrable matrix-entries, equivalently {T}, embedded in the regular representation RG. Harish-Chandra associated discrete series to compact Cartan subalgebras in 8. So certain semisimple algebras have them, like 4 l ; n ) , while others, e.g. s(n),n 2 3, or any complex
a),
do not. Harish-Chandra characterized discrete series by means of their
characters. This left an open problem to explicitly construct such {T}. The latter was accomplished by Parthasarathy [Par]; Schmid [Sch] and Atiyah-Schmid [AS]. Our main sources in 57.1-7.5; 7.7 were books [GGP];[GGV];[GMS];[Lan];[Ta].
Chapter 8. Lie groups and Hamiltonian mechanics. We shall outlines some interesting applicatione of Lie groups and symmetries to hamiltonian mechanics. Our main emphasis will be on integrable systems and systems that possess large symmetry groups. Although most of the discuasion does not directly involve the representation theory of chapters 1-7 (save for the last section, §8.5), the Lie structure theory of groups and algebras will enter many times. To make our presentation self-contained we included in the first section, 58.1, some basics of the hamiltonian mechanics: Lagrangian formulation, Minimal action principle; EulerLagrange equations; Canonical formalism; sympIectic/F‘oisson structure; conserved integrals, integrability and Darbeaux Theorem.
$8.1. Minimal action principle; Euler-Lagrange equation; Canonical formalism.
1.1. Hamilton’s Minimal Action principle. The state of a classical mechanical system of n degrees of freedom is described by its position vector: q = q ( t ) = ( q l ; ...qn), varying over W”, or more general Riemannian manifold At, called configuration space, and velocity (tangent) vector: q = (ql; ...qn). Its dynamical evolution is determined by the action functional,
s=
li
P.(q;q)dt,
t0
whose integrand .t(q;q), called Lagrangian, depends on position and velocity, and has physical dimensionality of energy. In many cases of interest Lagrangian represents the difference of Kinetic and Potential energies, P. = K - P , where h’ = 1 . 2 . 2Q
7
or more generally, is given by a Riemannian metric-tensor { g i j ( q ) }on At,
K = !jC g i , qiqj; while P = V(q)- a potential function. According to the Hamilton’s principle of Minimal Action: a trajectory (evolution) of the classical system must minimize (or give a stationary path) of the actionfunctional. So it satisfies the Euler-Lagrange equation:
a 6q(t) =P q -d(P..e=o. dt .P case of
(1.1)
Equation (1.1) represents a 2-nd order OD system in n variables. In the classical P. = “kinetic” - “potential”, (1.1) turns into the Newton’s equation,
9’’
-aV(q) = F- force.
The canonical formalism reduces the 2-nd order Euler-Lagrange equations to a 1 - st order system of size 2n. We introduce a new set of variables: pi = a. .P.(q;q)- conjugate momenta, ‘11
(1.2)
$8.1. Minimal action principle; Euler- Lagrange equation;
370 and
H ( q ;p ) = p
- L, hamiltonian/energy function.
Solving a system of equations (1.2) for q-variables’, we get q = Q(q;p),and these are substituted in the hamiltonian H ( q ; p ) .Then the Euler-Lagrange equations (1.1) are shown to be equivalent to a hamiltonian system
@ - a PH ;
Ap= - a p .
(1.3)
We shall first demonstrate the canonical formalism in the case of N-particle systems. The corresponding Lagrangians are N
L = 3’ - V ( q ) ,or 3Cmjq: - V, where ( m j } denotes masses of particles. Then the conjugate momenta: p j = r n j q j , and the hamiltonian,
+
+-
H
= c l 2mj p . J 2 V(q);“kinetic” “potential”, a familiar expression from elementary calculus/mechanics. For
more general (“kinetic - potential”) Lagrangians on manifolds A, the Euler-Lagrange (1.1) takes the form & ~ g i j q j ) - a q i v = 0,
while the canonical variables:
..
P; = E g i j Q j ; H = + C g ” P i P j
+ V(q),
( g i j ) denotes the inverse matrix (tensor) to ( g i j ) , and the hamiltonian system
becomes \pi = - a,v. So geometrically, momentum variables { p } can be identified with cotangent vectors on A, and the (position-momentum) phase space becomes a cotangent bundle I*(A).The change of variables (q;q ; L)+(q; p ; H ) , called the Legendre transform, can be interpreted as a map from the velocity phase space, tangent bundle T ( A )= { ( q ; q ) } , to the momentum phase space, cotangent bundle T * ( A )= { ( q ; p ) } , that takes solutions (trajectories) { q ( t ) ; q ( t ) }of the Euler-Lagrange equations (1.1) to those of Hamilton system (1.3) (problem 2). ‘provided it could be solved, i.e. the Hessian of & in q-variables is nonsingular, det(-) a2L # 0. aCr;aqj The latter is always the case with the classical Lagrangians, L = “kinetic” - “potential”, on A, whose Hessian turns into the metric tensor { g j k ( q ) } .
$8.1. Minimal action principle; Euler-Lagrange equation;
371
1.2. Symplectic structure and Poisaon bracket. Phase space 9 = T * ( A ) is equipped with the natural symplectic/Poisson structure, given by a differential (canonical) 2-f0rm2 on 9:
R = C d p i Adqi, (1.4) i in standard (dual) local coordinates { ql...qn;pl...pn}.In other words, we take a basis in the tangent space T q ( A )and the dual basis { d q l ;...dqn} in the cotangent space T z ( A ) , so each point x = ( q ; t ) E T * ( A ) , q E A, ( = C p j d x j E T i , can be represented by a 2n-tuple { q i ; p j } .
{aql;...aqn}
Symplectic structure on a general manifold 9,with local coordinates { x l ;...zm}, is given by a non-degenerate closed %form,
0 = X U j k d X j dXk, (1.5) with the usual proviso that R be independent of a particular choice of { x l ;...xm} (i.e. coefficients { a j k } transform as a tensor on under coordinate changes on 9).Clearly, non-degeneracy of R constraints dim9 to be even. In addition, one requires closedness of R, in the sense that its differential &?= E E i j k a i ( a j k ) d Z i A d x j I\ d X k = 0.
Here Eijk denotes a completely antisymmetric symbol (tensor) in 3 indices, normalized by elZ3 = 1. An alternative way to describe a symplectic structure on 9 is in terms of a skew-symmetric bilinear form 1, on tangent spaces {T,(T):xE T},
(1t17) = z b j k t j 9 k ; (77 E T Z . Two structures are related one to the other by R = j - ' d x A dx, in other words matrix (a.J k ) = ' ( a $ ) - ' . Clearly, the standard 2-form 0 yields the standard symplectic matrix, L
J
Examples of symplectic manifolds include: 1) The standard (flat) phase spaces: W2 = { ( x ; p ) }with R = dx A d p , and W2" with 0 = C d x j A dpi- Those are often convenient to write in the complex form: z = x -ti p E C", then R = A d?i.
3.z
2) Cotangent bundles "*(A) over manifolds A, with R = E d q j A d p j ; 'Let us remark that (1.6) is independent of the choice of local coordinates on A. Indeed, if ,denotes a coordinate change, then differentials { d q j ] are transformed by the Jacobian map f' = 8 9 while differentials of co-vectors by inverse transpose of f'. So the product C d p j pI d q j remain8 invariant. f:q+q'
(r),
58.1. Minimal action principle; Euler-Lagrange equation;
372
3) The 2-sphere S2= {(c$;O)} with R = sinc$dc$A do.
4) Co-adjoint orbits of Lie groups: O C ~ (chapter * 4). The tangent space T , (2 E 0 ) is identified with the quotient 0/6, - Lie algebra modulo stabilizer subalgebra of z OZ = { ( : a d i ( z ) = 0). As a bilinear form on tangent spaces,
I [(;171)
Q(f;rl)= .(
= (ad;(.)
Id; t77l E (5.
5 ) Finally, we shall mention the so called KiiMer manifolds, complex manifolds A with a hermitian metric-form: ds2 =
C b,,dz,dZv;
-
b,,,, = b,,,;
whose imaginary part is a closed 2-form7
R=
b,,dz,,
(1.7)
A dTv.
To check skew-symmetry of R one notes that in real coordinates {z,, = Sz,,; y,, = St,,},
R = Adz A d z + 2Bdz A dy + Cdy A dy, where A = C and B are real (antisymmetric and symmetric) matrices S(b,,,,) %(b,,,),
i.e. (b,,”)
= B + iA. Hence
and
defines a symplectic structure on A.
Symplectic structure on any phase-space 9 (cotangent bundle, co-adjoint orbit, etc.), allows to assign certain vector fields, called hamiltonian fields, to functions (observables) F on 9, F 4 Z F = j(aF).
In other words we take a gradient vector field of F and “twist” it by a skewsymmetric linear map 1 on tangent spaces {Tz(9’)}.The standard symplectic structure (1.7) on phase-space R” x R”, yields hamiltonian vector field, z, = a p F . - a q F .
aq
ap.
Each vector hamiltonian field generates a hamiltonian flow, {ezp(tE,)}-a fundamental solution of ODS,
5 = B(aF(z)), (1.8) which generalizes the canonical system (1.3). Hamiltonian flows possess many special features, for instance, all of them preserve the canonical (Liouville) phase-volume, d”q
-
d”p on 9,since all respect the canonical/symplectic form 1, or R (problem 3), and “Liouville volume” = R A ... A 0.
Symplectic structure also defines a Lie-Poisson bracket on the vector space of observables { F ( z ) }on 9, namely
{F;G) = (r(aF)I aG),
58.1. Minimal action principle; Euler-Lagrange equation;
373
which in special cases, W2" or T * ( h ) ,turns into,
{ F ; G }= a , F - d , G - a , F . a , F . The reader can verify directly all properties of Lie bracket (skew-symmetry and Jacobi identity) for {F;G}3. In fact, the Poisson bracket of any two observables corresponds to the standard Lie bracket of their hamiltonian vector fields,
-- --
{ & G } ~ E { C G= I [ZF;5,] = = F = G - Z G C F . Thus the space of observables { F ( z ) } on 9 acquires a structure of an m-D Lie algebra, a subalgebra of all vector fields bD(9).The corresponding Lie group consists of all canonical transformations on 9, i.e. diffeomorphisms {q5} that preserve the symplectic structure,
Tq5'%#) = 8.; equivalently coordinate changes, y = d(z), that preserve the canonical 2-form,
We shall list a few examples of canonical transformations: i) symplectic matrices { A E S p ( n ) } in R2" = { ( q ; p ) } ; they clearly preserve the standard 2-form: R = C d p j A dqj. ii) any coordinate change (diffeomorphism) y = @(z), from manifold .A to N, induces a canonical map,
YO);
q5:(x;o+(@(x);TAwhere A denotes the Jacobian matrix of @ at {z}, A = @: (problem 4). The reader has probably noticed some coincidence in terminology: symplectic Lie groups/algebras on the one hand, and symplectic structure/geornetn'es on the other. The relation between two becomes apparent now: Jacobian matrices of canonical (symplectic) maps are symplectic matrices4! 1.3. Conserved integrals; action-angle variables and the harmonic oscillator. The hamiltonian evolution (1.8) of the position and momenta variables { q ; p } gives rise to 3The Jacobi identity on general symplectic manifolds results from closeness of the canonical 2form 0. 41n this regard symplectic groups plays the same role in the symplectic geometry, as orthogonal groups in the Riemannian geometry. There exists, however, a striking difference between two kinds of geometries. The Riemannian geometry is fairly rigid in the sense that isometries of m (even in the best case of symmetric spaces) form only a finite-dimensional Lie group, whereas "symplectic isometries" (canonical maps) are always infinite-dimensional!
374
$8.1. Minimal action principle; Euler-Lagrange equation;
evolution of any other observable (function) F on T * ( A ) ,
P = {F;H}. Functions that remain constant along trajectories of the hamiltonian flow, { F , H } = 0, are called first integrals of (1.8). Indeed, the Legendre “back-transform” (q;p)+(q;q), takes such F into a function F ( q ; p ( q ; q ) ) ,constant along trajectories (solutions) of the E-L equation (1.1) i.e. gives a 1-st integral of (1.1). Hamiltonian H is itself an integral, as {H;H} = 0. In dimension n=l, it is the only integral. So 2-nd order E-L equation, Lq- $L4 = 0, is reduced to a 1-st order ODE: H(q;p( ...;q ) ) = E -const.
In the classical (Newton) case this yields I *+v,I.2 zq +V = E, ZP
whence we get an implicit solution in the form of integral
5 J;i&
= t - to.
(1.9)
More generally, each Poisson integral F of the hamiltonian system (1.3) reduces its total order 2n by 2. The proof is based on the following general result.
Dubeaux Theorem: Any set of functions (obseruables) {F,; ...Fk;G1;...G,} on a phase space (symplectic manifold) 0 that satisfy the canonical commutation (Poisson bracket) relations { F j ; G i }= Jij, can be locally eztended to a canonical coordinate system on 0,{pl;...p,;q,;...q,}, p; = pi (i=l, ...k) and qj=Gj (j=l;...m).
In particular, a Poisson integral F can be made the 1-st momentum variable p , of the new coordinates. Then hamiltonian H ( p ; q ) becomes independent on the corresponding canonically conjugate position ql, and p , is constant along the flow,
p1 =
Q1
= { p , ; H } = 0.
Setting pl= El-const we reduce H to a hamiltonian H ( E , ; p , ;...p,;q,;...q,) in ( n 1) degrees of freedom, so the system becomes:
p . = d H;
(4i‘=
‘li
-dPiH;
t
i=2; ...n; and p , = El; q1 = ql(0)
+ IH(...)ds. n
A hamiltbnian system (1.3) in n degrees of freedom is called completely integrable (in the Liouville sense), if it has n functionally independent, Poisson commuting integrals {Fl;Fz;...F,}, { H ; F i } = 0; and { F i ; F j }= 0, for all i , j . (1.10) Functional independence means gradient-vectors { VF,} are linearly independent
58.1. Minimal action principle; Euler-Lagrange equation;
375
at each point t = ( q ; p ) . In the standard terminology, commuting integrals (1.10) are said to be in involution. Let us remark that any set of n functionally independent integrals {Fl;...Fn} allows to reduce the total order of the system by n, i.e. to bring (1.3) to a first order system in n variables: Fl(G4)= El F,(q;q)'= En' Complete integrability yields, however, more than a mere reduction of order by n, as each subsequent integral F , respects the joint level sets and evolutions of all preceding variables in the canonical reduction. Hence, the system could (in principle) be reduced to order 0, i.e. solved completely! The procedure is best illustrated by the harmonic oscillator
{
H=izpf+wfqionW2".
As first integrals of H one could take all coordinate oscillators,
{ H i = ;(pi
+ wiqi): 1 5 i 5
72).
zHi.
Obviously, H= The joint level sets of { H i = E j } form a n-parameter family of invariant tori in WZn, and the hamiltonian dynamics consists in a uniform motion in some direction along the torus. Introducing polar coordinates (P; 8 ) in the i-th phase plane { q j ; p j } after , rescaling, qi+wiqj, the j-th hamiltonian becomes H 1. = L( ? + q?) = LP?. 2Pc I 2 1 In polar coordinates the flow of each
Hiis given by an OD system
!= O, i.e. P = E -con~t, 8 = 0, + Et. (4 = r The Poisson bracket { r ; e }= i,hence {$';fI} = 1, and the entire set of variables { H,; ...H,; 8,;...en} satisfies the canonical commutation relations: {H..B I' 3.}=6.. 1J' Returning to a general completely integrable hamiltonian system with n commuting integrals { F,; ...F,} (called actions), the Darbeaux Theorem implies the existence of a canonically conjugate set of angle variables: {el;...en}, that satisfy the canonical relations The hamiltonian H in new coordinates { F; 6) becomes a function of actions only, H = f(Fl;...F,J, since aej-- Fj = 0. The joint level sets { F j = Ei} form a foliation of the phase-space into invariant tori Tn (or products TkxWn-'),
and the dynamics
376
$8.1. Minimal action principle; Euler-Lagrange equation;
resembles the oscillator case. Precisely, if
4j
denotes the canonical coordinate change:
(P;q)-+(F;e>,and
+
F~ = E ~e,(q ; = ej(o) i g 8H F i ( ~... l ;E"), the hamiltonian flow in the action-angle coordinates, then solution in the original coordinates becomes (1.11) ( p ; q ) ( t=) @-I(... E ~...; e,(t) ...). Let us remark, that solution (l.ll), although written explicitly, may be of limited utility unless one is able to compute the canonical map 4j.
Our next goal is to establish integrability, find l-st integrals and, if possible, explicit solutions of different hamiltonian systems. We shall review a number of the classical models, and also discuss some newly discovered examples. Our principal tools, in the study of conserved integrals, will be symmetries of the problem, and the Noether Theorem, to which we turn in now.
$8.1. Minimal action principle; Euler-Lagrange equation;
Problems and Exercises. 1. Check the equivalence of the Euler-Lagrange equations (1.1), and the hamiltonian system (1.3). 2. (i) Demonstrate that the "squareroot-kinetic-energy" functional of a Riemannian metric {gij(z)), L[z] = ( cgij(z)2i2j)1/2, yields geodesics, as extremal curves. So the EulerLagrange equation for L is precisely the equation of geodesics (see Appendix C). The geometric meaning of this result is quite transparent: the Lagrangian density Ldt represents the arc-length element of the metric g! (ii) Show that the Legendre transform kinetic energy:,functional K = + c g i j ( z ) i i 2 j of metric g, becomes the hamiltonian H = i c g r 3 ( z ) p i p j , of the dual metric (gr3) = (gij)-l, on the cotangent space.
3. Check that the Jacobian map ];o :[f-
a hamiltonian vector field
2, = ( a ( p ; q ) ; b ( p ; q ) ) a; = apF; b = -8,F; has determinant 1, hence the hamiltonian flow of F preserves the Liouville volume d"p A d"q. 4. i) Any coordinate change, q+(q),
on configuration space Ab defines a canonical
transformation,
Ad: ( q ; P)+(d(d;=dC YPN; on the phase-space T*(Ab)(Check that A preserves the canonical 2-form dq A dp).
4
ii) Apply part (i) to transform the standard canonical (q; pkvariables in the phase-space R2x W2 to polar { ( r ;O ) } , and spherical {(r;d;fl)}-coordinates. Compute the hamiltonian H = p2 V ( q )on R2 in polar and spherical coordinates, and show
+
2
Hpolor= P,
1 2 + -P r2 + V ; Haher= p j +$p$
+ sin24b2)+ V ;
iii) Do the same for elliptical coordinates: where {rI;-P2) are distances fig.1). Show
of focal points: {( f a;O)} (see
Fig.1: Elliptical coordinates in R2 are made of confocal ellipsi: ( = r1 + -P2 = Const; and hyperbolae: q = -Pl - r2 = Const. Here z = a ch t c o d ; y = a s h t sine; and parameters ( = 2 a c h t ; q = 2acose. Elliptical coordinate change can also be viewed as a conformal (analytic) map, w =t + i b z =z+iy, given by z = $ e W + e-"').
377
378
58.2. Noether Theorem, conservation laws and Marsden- Weinstein reduction. 58.2. Noether Theorem, wnservation lam and Marsden- Weinstein reduction. Conserved integrals of hamiltonian systems are often derived from the oneparameter groups of symmetries via the celebrated Noether Theorem. In particular all basic conservation laws of Physics: energy, momentum, angular momentum, arise this way from the translational and rotational symmetries of the system. Going further in this direction, we consider systems with ‘‘large” Lie groups, or algebras of symmetries (not necessarily commuting). Such symmetries allow to reduce the system to fewer variables, via the Weinstein-Marsden reduction process.
Given a configuration space of a classical mechanical system with Lagrangian
L = L ( q ; q ) , we consider a one-parameter group of transformations
CC:(q;t)+(q*;t*) on
A x [O;co)(point transformations along with time reparametrizations),
L*
+
+ ...
= q e$(q;t) = 11,,(q;t) (2.1) = t + q q q ; t ) + = &(q;t)* In other words vector field $J = Sq and scalar field q5 = S t , represent infinitesimal 9*
...
generators of the family Cc. Obviously any group of transformations (2.1) of A generates transformations on the path-space of AL (trajectories of the system):
(40 t ) +uq*(t*); t*)= ( q + 6%t + S t ) = 11,€(Q A).
(2.2)
O
2.1. Noether Theorem: A n y one-parameter group of transformations (2.2) with
generators (11, = Sq;$ = St), that leaves invariant S = L(t;q;q)dt,gives rise to a conserved integral
I
J=p.bq-HSt=p.$J-Hq5
the
action-functional
Const
Here p = a. L, and H = p . q - P. are the canonical variables (conjugate momenta Q and the hamiltonian) of L. The proof follows from the general variational formula for a functional L (problem l), when we allow free motion of the end points as well as all reparametrizations of the timevariable: t+t* = t+6t, ‘1
6s = ( p a 6 9 - H a t ) [
to
‘1
+ J(Lq-$Li*469
- 4 61)dt.
(2.3)
to
The first factor in the integral represent the E L equation that vanishes along any critical path. If furthermore functional S is invariant under (2.2), then
6s = 0, and we get
J ( t l ) = ( p . 6 9 - H 6 t ) I t , = J ( t , ) - constant along any critical path! QED.
Remark: Conservation of Noether integral J = p a 11, - Hq5 can be recast in the hamiltonian formulation. Indeed, function J = J(q,p; t ) , considered on the phase space
58.2. Noether Theorem and the Marsden- Weinstein reduction.
379
of variables (q; p ) clearly satisfies the equation,
J,+(J;H}=O. Thus each symmetry of the Lagrangian L produces a Poisson commuting integral (symmetry) of the corresponding hamiltonian H. So any Lie group/algebra of symmetries of the Lagrangian gets represented by the Lie group/algebra of hamiltonians. 2.2. Conservation laws. As an application of Noether's Theorem we shall derive the basic conservation laws of classical mechanics.
Energy conservation. If Lagrangian is time independent, L L ( q ; q ) ,then time shifts: t-+t+, form a translational symmetry group of L, whose generators: II, 0, #=I. Hence the Noether integral becomes hamiltonian/energy function, 0
IJ
= H ( q ;;p ) = Const].
.Momentum conservation. We assume now that the space shifts q+q+cu in certain directions u leave Lagrangian invariant. Then the generators 4 = 0; II, = u yield
J, =p . ~ , the u-component of the momentum remains constant. The standard example is a N particle system in R3 with pair interactions,
e = ;xq; - V , where potential V = C v(qi - q j ) . Obviously, simultaneous shifts, do not change V , hence system,
(Q1 ... QN)+(Ql+";... nNt"), 21 E R3, L. Thus we get conservation of the total momentum of the
1-1
.Angular momentum conservation. The source of the angular momentum conservation are rotational symmetries of the Lagrangian, like the central potential (Kepler) problem: V = V ( I q I ) in L = ;q2 - V . Let us assume that P. is invariant under rotations in the ij-th coordinate plane. The corresponding symmetry generator is a linear vector field
+ and the Noether integral becomes
.ij=[
-1
]:[I' 0
J i j = Piq j - P jqi
380
58.2. Noether Theorem, conservation laws and Marsden- Weinstein reduction.
- the ij-component of the angular momentum. In the N-body system with potential V = x u ( I qi - q j I ), only simultaneous rotations of ( q1...qN) by u E Sq3) leave L invariant. So the total angular momentum is conserved,
Remark. A relativistic particle Lagrangian has the form
so the corresponding symmetry group consists of Lorentz transformations S0(1;3), whose
generators include rotations in spatial directions,
-
-
I
J i j =I:[ 1 5 i;j 5 3, i.e. Lie algebra R = 4 3 ) c so(l;3), as well as relafiuislic boosts:
generators of hyperbolic rotations,
C 4 1 ; 3 ) . The corresponding Noether integrals are JOi
= PoQi + P i 9 0
2.3. The momentum map and Weinstein-Marsden reduction. Let 8 = span{F,; ...Fm)be a Lie algebra of hamiltonians on the phase space A. Then Lie group G of O acts on 311 by canonical transformations { e z p ( t Z F ) F : E (5). Any such algebra O defines a map J from JL into the dual space 0*,by evaluating hamiltonians F E O at a point z E A (“point evaluations” are clearly linear functionals on C ( A ) ! ) ,
( J , I F ) = F ( z ) ,for all F E 0. The resulting momentum map, intertwines two actions of group G: canonical action on A, and the co-adjoint action on 6*:if g = g(t) = ezp(tF), denote denote a Lie group element (of generator F E 0), and #g = e z p ( t Z F ) - the corresponding canonical transformation, then
J(dg(z))= Ad;(J(z)),for all g E G; z E A.
(2.5)
Furthermore, map J preserves the Poisson structures on JL and O*. We remind the reader that the dual space of a Lie algebra has a natural Poisson bracket (chapter 6 and f8.l), defined in terms of the Lie bracket on 0.Namely, for any pair of functions Fl(z); F z ( z ) on @*, gradients { a F .} belong to the Liealgebra itself, so one can 3
Poisson-Lie algebras
(5
often arise as symmetries of hamiltonians
H on A. The
58.2. Noether Theorem and the Marsden- Weinstein reduction.
38 1
momentum and angular momentum operators are obvious examples of the momentum map,
P, = u . p (u-fixed direction), and J = {J. = ( p A z) . 3E 3)
k -pkzj}.
= p .z 3
The former corresponds to translational symmetries (in the u-th direction)
O 21 Wm, the latter to rotational symmetries, 6 N 4 m ) . We shall see that symmetries of hamiltonians { H } play a double role. On the one hand they apply to reduce the number of variables (degrees of freedom). On the other hand some, seemingly complicated, systems could be “lifted” to larger, but “simpler” systems, typically based on Lie groups and symmetric spaces. We shall analyze a few important examples of this sort in the next section. But here we shall concentrate on the “reduction part”. Given a Lie algebra O of symmetries of hamiltonian H , we consider a joint level set of all observables { F E O} (it suffices to pick a basis Fl;F2;...Fm in 0 ) ,at a “level toE O*”,
A. = J-*(to). Subset A, is invariant under the flow of e z p ( t H ) . Furthermore, A, is invariant under the stabilizer of to,subgroup Go = {g E G:Ad,(to) = 0}, an obvious consequence of (2.5). The action of group Go on A, splits it into the union of orbits (a fiber bundle), and the reduced space we are interested in is the quotient of A. modulo Go, the orbit space5 R.
In our setup space R, c R = Jb0/G,, also has a natural symplectic structure: the Poisson bracket on A. restricted to Go-invariant functions. This Poisson bracket turned out to be non-degenerate, hence yields a symplectic structure on Ro (a Theorem due to Weinstein and Marsden). We shall illustrate the reduction procedure by 3 examples. 2.4. Examples.
Radial (spherically symmetric) hamiltonian H = H ( I q I ; I p I ) in W3 x R3, a generalization of the classical Kepler central-force problem: H = i p z V ( I q 1 ). Any such H has an Sq3)(angular momentum) symmetry, the momentum map being,
+
J : (%P)+9 x P
E43).
Here we identify 3-vectors [ = ( a ; b ; c )with antisymmetric matrices,
51t should be mentioned here that orbit-spaces are typically non-smooth manifolds with corners and edges (take, for instance, R”, modulo Sqn)). But there is always a dense open set 0,of ‘generic (maximal-D) orbits” in Q, that possesses a smooth (differentiable) structure!
'-
M -
c -c
-L b -a
-b a
;
-
2
so q x p becomes a element of 4 3 ) . We fix a joint level set
4 = { ( % P ) : P x q = Jol, of the (constant) angular momentum (Noether’s Theorem). Without loss of generality vector J , could be taken, as (O;O;jo),where j , = I Jo I . Then vectors ( q ; p ) belong in the plane orthogonal to the (constant) angular momentum, which reflects the well-known property of central potential forces: planar motion!
A 3-D manifold 4 = { ( q ; p )E R2x W2; q A p = j,} is foliated into orbits of the stabilizer subgroup GoN Sq2)of J , E d 11 4 3 ) . Clearly, any u E Sq2)takes 21:
k;p)+(qU;pU),
Hence, the reduced dynamics in the phase-space N=W2 is given by the hamiltonian.
where j o = IJoIis the total (fixed) angular momentum. Thus we have reduced he number of variables to 1, which shows complete integrability of rotationally symmetric hamiltonians (problem 2).
Example: Integrable hamiltonians H. A commuting family of integrals { J,} maps phase-space 9 into the Lie algebra R". Here group G N U" (or R"' x Un-"') is generated by the flow {ezp(t,Z, + ...+t,Z,)} of hamiltonian fields {Z,} of {Jk};the coorbits { t } are points in w", stabilizers Gc = U", and inverse images J - ' ( f ) coincide with invariant tori in 9. So the reduced space J-'(()/T" is trivial. 0
0 Example: Left-invariant metrics on Lie groups. Tangent/cotangent spaces of Lie group G at each z E G can be obtained from a single space (Lie algebra), 6 21 T,, and its dual O* N T,*,by left translations,
q : t E T e - - + q - t E T q andpETB-+q*-'*pET;; , qEG. For the sake of presentation we shall consider the matrix group and algebra G
58.2. Noether
Theorem and the Marsden- Weinstein reduction.
383
and 0.Then q .( will be the matrix multiplication. The left-invariaot fields on G are of the form: ( ( q ) = q - 6 (( E O ) , and any left-invariant metric is uniquely determined by its restriction at {e}, a symmetric bilinear form B on 8,
(Bq-'
I q-' v); (;v E T,. L = $(B... I ...), and the *(
*
Thus we get a Lagrangian corresponding hamiltonian H = $(B-'... I ...) on phase-space T*(G), which possess a large symmetry group G of right translations {z+z.q; s;q E G } . Those clearly commute with left-invariant vector fields {( = ( ( q ) = q . ( } . The corresponding family of right-invariant hamiltonians (integrals of H ) consists of {J&; P') = (t * Q I I)'):
t E w' E q , on T*(G).
Identifying co-vector p' at point { q } with a co-vector p = q - * . p' E @* (via left shift with q - ' ) , we can write such functions J's on G x @* N T*(G)as, J&
P) =
I
I
- '& PI ) = (ad,(<) P ) = (t I Qd;;(P)).
So the momentum map becomes IJ(q;p) = ad,*(p)I
To get the reduced space we take the inverse image of p , E @*, J - ' ( P o ) = {(q;P):~dp*(p)= PO} = {(q;$-1(po):q E G}, (2.7) and note that J - ' ( p o ) projects onto a co-adjoint orbit 0 = O(po) (2-nd component in the RHS (2.7)). In fact, space J - ' ( p o ) is foliated over 0 with fibers, isomorphic to Go and its conjugates {G, = q-'Gq}. So the quotient-space J-'(po)/Go can be iden.tified with orbit 0, and the reduced hamiltonian becomes,
H = @?-'(I(),
restricted to 0.
In the next section we shall illustrate the foregoing discussion of symmetries, conserved integrals, integrability and reduction with a few classical examples, including Kepler one- and two-center gravity problem and the Euler rigid body motion.
384
58.2. Noether Theorem, conservation laws and Marsden- Weinstein reduction.
Problems and Exercises. 1. Prove the variational formula (2.7). Steps, i) Consider all possible variations of path q ( t ) , q-+q+6q, along with all reparametrizations, t+t* = t+6t. Write the variation of the functional, 6s = JL(t*;q*;$)dt* - J L ( t ; qdq ;z)dt, ii) Change new parameter t* back to the old t , and show that the 1-st integrand becomes
L(t+6t;q+6q; d(9+69))(1 + 6 i ) = L( ...;.,.;‘-)(l+ ...). d(t+6t) 1+6t The dot above any variable/function (q;6q; etc.) indicates its time derivative d. dt iii) Expand the latter in the Taylor series to the 1-st order in variations
-
6S= J L t 6 t + L q * 6 q + L 4 ( 6 4 - 4 6 i ) + L 6 i I
iv) Integrate by parts of all terms containing 64; 6t and get,
v) Observe that the off-integral terms above combine to p * 69 - Hbt, while the &factor inside the integral, after term-wise differentiation and cancellations d reduces to (Lq- ;iiL4). 4 6 4 and completes the proof.
2. Check directly that all 3 components of angular momentum J . .= q.p .- q .p.*
Poisson commute with H = !jp2
I3
8
3
3
8’
+ V, for any radial V, { J i j ; H ) = 0.
Show that { J i j l i< satisfy the so(n) of antisymmetric matrices,
relations of generators of the Lie algebra rotation in the ij-th coordinate plane.
68.3. Classical examdes
385
38.3. Classical wmples. This section will illustrates the concepts and methods of the preceding parts (§f8.1-8.2), particularly the role of symmetries, integrability and reduction, by a few classical examples: spherical pendulum, Kepler problem, Euler rigid body and 2center gravity problem.
3.1. Spherical pendulum. The configuration space here is the sphere
S2= {s2+y2+z2 = 1) in UP; and Lagrangian: L = $(k2+y2+i2)- V ( z ) .In spherical coordinates (q5$), p. =(;
$2
+ sin24 8’) - V (cos4),
and the corresponding hamiltonian:
H = $ ( p i t sin-’4p;) t V . The system possesses a rotational symmetry about z-axis, whose generator by Noether’s Theorem gives a conserved integral (z-component of the angular momentum) p g = 0 = g - const. One could verify directly the commutation relation system is reduced to a 1-st order ODE in +variable, $($2
{H;p e } = O! As the result
the
+ V(cosq3)= E ; 8 = g t7 g2 s m (p
which is then integrated as in 58.1,
I
2 (dE4-
)
=t
- to; e - eo = gt.
J -
3.2. Kepler 2-body Problem. The motion of a 2-body system about the joint center of mass is described by the Lagrangian, 1 = Tq2- V , with potential V = -2, 191 the corresponding hamiltonian H = &p2 V . In spherical coordinates (r;4; e),
L
+
+(i2 r2(@
+ + sin2482))- v ( T ) ,
and
H = L2(Pr2 t f . -2 (p(p 2 t sin-2p;))t V(r).
(3.1)
The S0(3)-rotational symmetry of central potential V yields conservation of the angular momentum J= q x p , by Noether’s Theorem. In particular, the direction of J remains constant, hence 2-body motion is always planar (in the plane orthogonal to 15). The Kepler problem is thus reduced to a plane motion, where
68.3. Classical examples.
386
e = $(i.' + r'b')
- V ( r ) ;H = f(p:
+ r-'p;) + V ( r ) .
But this L also possesses a rotational symmetry, L 8 + c . Hence the Noether integral p g = g - const, i.e. { p e ; H } = 0. Thus the 2-D system is completely (Liouville) integrable. Moreover, two integrals reduce it to a I-st order ODE, as above
+ $1 t V ( r )= E , =+ T=Ak 2(E2
:(i.'
= t; 8 = 8,
+ gt.
So we get a family of solutions r ( t ; E ' g ) depending on 2 parameters (constants of '(e t ; E , g S integration) E, g. To show complete [Liouville) integrability of the 3-D central potential problem does not require, however, the 2-D (plane) reduction. Indeed, 3 commuting integrals can be exhibited directly:
H = f p 2 iV ( r ) ;J , = J,, and J z = J i +
J i + J:.
The commutation relation: { J , ; J 2 } = 0, follows from the 4 3 ) Lie bracket relations. The reader could easily recognize J2, as Casimir (central) element of the enveloping algebra of 4 3 ) . 3.3. Eula rigid body problem. The rigid body in W3 has a density distribution {p(x)}. In the absence of external force its motion consists of the free (linear) motion of the center of mass, f = $zpdx, f = vt
+ b,
(momentum conservation), and rotations about f . We shall restrict our attention to nontrivial rotational components of the dynamics, i.e. study it in an inertial coordinate system6 with origin at the center of mass 0.
Fig.2: Rigid body with 2 coordinate frames, the fated frame (dashed) and the moving frame (solid).
Two coordinate frames can be assigned to a rotating body: the fixed rest frame (of the ambient space), and moving body frame, both centered at 0 (fig.2). We shall
'i.e. system whose axis move at a constant velocity u.
387
68.3.Classical exarndes
Q
posit ion:
q = utQ
velocity:
2,
angular velocity:
w = 6u-I = ut( R
V=nxQ
=q =w x q =Ut(V)
angular momentum j =
J
1
52 = u-19
vxqdp= /(wxg)xqdp
j =4J)
acceleration:
a =q
= Lj x q
J= / V x Q = /(RxV)xVdp J=BR
+w x (w x q )
A = u;
* ( a ) = j2 x Q
+ R x R x Q.
The angular velocities { w ; R } here could be understood in two possible ways, as antisymmetric matrices (elements of Lie algebra 4 3 ) ) , or 3-vectors: w,R E W3 CY 4 3 ) . So operation Q-4 x Q, coincides with the adjoint/co-adjoint action, ado(Q), under the standard identification of l@ and ' 43),
-2
B = /(ad$dp
1
-2
Similarly, multiplication with u E &3) ad,(Q) = u-'Qu. A symmetric matrix
corresponds to the adjoint action:
= /(&'I - ' Q Q ) d p
,['2;
represents the inertia temor of the body (in the fixed body-frame),
B=
-22
-2y
-2%
z2+r2
-yt
-2y
2'+y2
The inertia tensor relates angular momentum to angular velocity,
pzmi the same way as mass/metric tensor relates momentum to velocity in the standard,
Now the basic Newton's law: i p = F-force, takes on a form:
8.3.Classical examples.
388
We can write the latter as
$ j = ut(J
+ J x 0 )= ...
In the absence of external force the RHS (torque) is 0. So we get a system of equations for unknown (body frame) parameters: J(t);R(t), J -t J x R = 0;
J=BR or
IBb -t B R x R = 01
(3.2)
To simplify equations (3.2) and write them down explicitly we introduce a special inertial body frame, where matrix/tensor B is diagonalized,
.=[I'
1,
J
the eigenvalues of B are called principal moments of inertia. If R = (wl;w2;w3),then
(3.2) turns into a system of 1-st order equations for { w j } with quadratic nonlinearities,
I
I& -k (Iy- Iz)wzw3 = 0 I y L j 2 + (I,- I , ) w 1 ~ 3= 0 ; I,w3 + ( I , - Iy)w1w2= 0
whose solutions are given in terms of Jacobi cnoidal (elliptic) functions [Erd]. Let us remark that equations (3.2) arise in the usual way from the minimalaction principle with (kinetic) Lagrangian on the phase-space T(G) or T*(G), of Lie group7 G = S0(3),
JP. I
P. = K = i
'JI
w x q I 2dp = T
R x Q I 2dp = i(BR 10)=$(B-'JI J).
(3.3)
has 2 conserved integrals: hamiltonian, H = P. = ;(B-'J I J ) ; and total angular momentum: 3" = = I BR I '. Third integral can be chosen as any (e.g. 2) component of the angular momentum. But more systematic reduction-procedure should Lagrangian
exploit the momentum map. Let us notice that P. is a special case of left-invariant Lagrangians on group S0(3), considered in example 3, the metric being given by the 71t may be somewhat confusing to see Lagrangian (3.3) without velocity (time-derivatives) terms. The reader should keep in mind, however, that R itself represents the "velocity" (tangent) vector on G. So the 1-st order E L equations (3.2) are, in fact, 2-nd order in the position-variable { ~ ( t )E G}. However, (3.2) can solved directly for O(t), whence u(t) is recovered by integration of u - % = 0.
$8.3. Classical examples
389
inertia tensor B ,
e = $tr(tBt);f E (Jj = 4 3 ) ; H = ;tr(pB-'p);
p E @*.
Here Lie algebra 6 and its dual O* are identified via the ad-invariant product
(tld= tr(tv)* The momentum-map reduces H to a co-adjoint orbit 0 = 0 ( p , ) - a sphere in 0 N W3, with the natural (invariant) symplectic structure R = sinddd A dB, and the reduced hamiltonian becomes, $(B-'p 1 p), restricted on 0.So we see once again the integrability to result from the symmetry-reduction! 3.4. Twmxxker gravity problem. We place two centers at points {-a;a} on the
z-axis. The corresponding Lagrangian is given by
It possesses an obvious rotational symmetry about z-axis, so J , = p g is an integral. The existence of another commuting integral is not so obvious. It results from a hidden symmetry of the Coulomb potential: V = L, the so called Lagrange-Laplace191 Runge-Lenz vector,
where J = p x q is the angular momentum. One can verify the commutation relation { Li;H } = 0 , and show that a system of hamiltonians {Jz; J y ; J,; L,; L,; Ly; L,) satisfy the Lie algebra brackets of 4 4 ) or so(1;3) depending on sign of y, with the angular momentum part represented by ro(3)-matrices, while the Runge-Lenz occupying the 'J3-component,
0
0
0
0
O
a
b
e (3.4)
-p
J=
aJ,
-a
*c
0
+ p J y + y J , and L = aL, + bLy -t cL, (problem 2 ) .
In particular, hamiltonians J , and L, commute. Furthermore, the Coulomb potential is the only one (among other central potentials) to possess the Runge-Lenz type symmetry (problem 1). For the two-center gravity problem we modify the definition of the Runge-Lenz
$8.3. Classical examples.
390 vector to be
9-0
= P X J - 71IQ-al-
9+a
Yzlq+al.
The new L is no more constant under the hamiltonian flow, but one has Proposition: Hamiltonians H and L satisfy the Poisson bracket relation
t3
{ J ; H } = a x ((+1
2 )
x P},
where q1 = q - a; q2 = q t a; rl,z = Iql - q21. The proof involves standard calculations with Poisson brackets
{ L ; H } = { p X L - - 7191 r1
...;$+?+...}={p 2
xL.I?.+...}-{7+...;T}. 7191
' rl
1
PZ
(3.5)
1
The first bracket {p x L.r-'}= ' '
while the second
-Q'x L r:
+ p x ( 91T rl
x9);
2 91.p tFil 2
1- r1-3 (Pq: - P .919l),
and similar relations hold for qz; rz -terms. Combining the rl-terms of (3.5) we get b [ P x (91 x 9 ) - 91 x (P x 9 ) + 91 x (P x 9J.
4
(34
Remembering that q = ql+a, and interpreting x as adjoint action of 4 3 ) , a x b CI [a; b]
= oda(b),
we get in (3.6) adqlad,(p)-~d[gl;a](~)=adaadql(p) = a x ( q 1 XP), QED.
It follows from the Proposition, that z-component of L (in the direction of vector a ) , Poisson commutes with H , { L , ; H } = 0. Also J , and L, commute, as in the onegravity-center case (check!). Thus we get a complete set of commuting integrals: H;J,;L,. In cylindrical coordinates (r; B;z ) they become
Substituting p r = i.; p , = i ; p g = r2h = g in (3.7), it is reduced to a 1-st order
from which f and 2 can be found in terms of other constants and variables,
88.3. Classical examples
1
391
i. = f ( T ; e; E , F , g)
.
i = g(r;& ...)
S=
/r’
The latter could in principle be solvecfeexactly. We shall demonstrate it in the case of the planar motion using elliptical coordinates (see problem 4 of $&I), and the Liouville’s method of separating variables. The 2-center planar problem still possesses a Runge-Lenz symmetry, L , = ( J -~ a2p2,) az(T 71 - 5). 72
+
Here we use (z;y)-coordinates in the plane with centers placed along the y-axis (see fig.3). Fig.3. A two gravity-center system with masses at { * a } .
Vectors J is orthogonal to the plane, while L lies in the plane. Passing to elliptical <=+; r +r2 rl=- fl-r2
coordinates:
where {rl;rz} are distances the from the moving point to the gravity centers, we get
The hamiltonian in elliptical coordinates takes the form
The resulting H belongs to a general class of hamiltonian systems, studied by Liouville (1849), that allow separation of variables. Liouville hamiltonians H have the kinetic and potential-energy terms of the form,
where B = C Bj, and all functions {aj; Bj;Vj} depend on a single variable qi’ It is easy
to check that (3.8) has n Poisson-commuting integrals
F 3. = 4 2 3. p32 + V 3.- H b 3’.*
(3.9)
which also commute with H (problem 3). Integrals {Fi}are not independent, since their sum.
eFj=O. j=1
But H along with any (n-1) of {Fj} form a system of n commuting integrals, so (3.8) is completely integrable. Furthermore, fixing the energy-level: H = E , and the values of
$8.3. Classical examples.
392
conserved integrals: F j = a j , (3.9) is reduced to a system of uncoupled ODE’s, 4 2
2aj
3
+ V3. - Eb3. = a3.; j = 1;2;...
The latter could be solve explicitly (in quadratures) by writing (3.10)
One first introduces ”local time” (3.11)
then solves ODE’s (3.10) with the RHS = d r , to get solutions { q j = q j ( r ) } , and substitutes them in (3.11) to produce “physical time” 1 in terms of the “local time” r , t = j B ( q l ;...q,) ds.
So any Liouville system is integrable completely and ezactly (in quadratures)! Returning to the 2-center problem we find B = t2- q2; a l ( t ) = t2-a2; a2(q)= a2 -q2;
vl(t)= -yt;
~ ~ ( =7-y”). )
Hence system (3.10)-(3.11) becomes (3.12)
with quatric polynomials
+ + a);Q = 2(q2-a2)(Eq2- y’q -a).
R = 2(t2 - a2)(Et2 y t
So ( ( T ) , q(r)can be expressed through the Jacobi elliptical functions ([Erd]).
Problems and Exercises: 1. Show, if a “central-force” hamiltonian H = $p2+ V ( I q I ) Poisson commutes with
then f = y - const, and V =
L = p x J - f ( I q I )L, - Coulomb potentibf.’
IQI 2. Verify the Lie bracket relations (3.4) of 4 4 ) , or so(1;3) for a combined “angular momentum-Runge-Lenz algebra” { J ; L}. 3. Verify that Liouville integrals { F j } (3.9) are in involution and Poisson-commute with the hamiltonian H of (3.8).
58.4, Integrable systems related to classical Lie algebras
393
$8.4. Integrable systems related to d a h l Lie algebras. Many interesting examples of integrable hamiltonians arise as interacting n-particle systems on R, where H = $ C p: V, and potential V is made of all two body interactions: V = Cui,, u. . = u(qi - q,), or nearest neighbor interactions. The foremost those are CalogereMoser systems: u = l / z z (and other special u), and the Toda lattice: a chain of nonlinearly coupled mass-spring systems: u = ezp(qi - qi+l). Both types turned out to be closely related to some classical Lie algebrss, e.g. 4.).
+
fi:
In this section we shall develop the basic Lax-pair formalism for such systems, then outline the so called projection method following Olshanetski-Perelomov (see [Pel). The idea is to view such systems as reductions (projections) of the geodesic flow on “large” symmetry group, or symmetric spaces {X}. Although the number of variables increases, in passing from R” to 36, we gain in simplicity of the ensuing flows. While the Lax-pair formalism proves integrability of the Caloger+Moser and Toda hamiltonians, the projection method yields an effective computational procedure for conserved integrals.
N-body hamiltonians. We shall study hamiltonians of the form H = i C p ; i-V , where potential 4.1.
V=
C
v(qj-qj);
(4.1)
l
- sum of all 2-body interactions, or
V = Cv(qj- qj+1)
(44 i - nearest neighbor interactions. The former are often called Calogero-Moser systems (for special v), the latter are exemplified by the well known Toda lattice. It turns out that for special classes of 2-body potentials v(q) the resulting n-body hamiltonians (4.1) and (4.2) are completely integrable. For (4.1)these functions are
(I) v = 1 . “2’ .1
(11) v = 2. or -
sinh’aq’
2
L
o
cosh’aq’
2 (111) v = L! -or--2. *
sin2aq’
cos’aq’
(IV) v = a’p(aq) - Weierstrass p-function (V) 2, = 4-2 4- ,*q2 (VI) Toda lattice: V = i
bie
-a(qi-qj+l)
Systems (I-VI) are closely related to certain classical Lie groups and their homogeneous spaces G / K . As we shall see they represent “projections” (via symmetry reductions) of the geodesic flow (free motion) on such spaces.
58.4. Integrable systems related to classical Lie algebras
394
The latter can be illustrated by 1-body potentials of the type (I-VI). Indeed, free motion on the Euclidian 2-plane with hamiltonian
H = g P ; 4becomes upon the radial reduction with fixed angular momentum po = g, (I) H = ; p : t $ .
2
Similarly the W 2 harmonic oscillator H = :pz
+ w2(x2+y2),reduces to
g2 4- OZ?. (V) H = #ppT.4- 7
Hamiltonians of type (11)-(111) arise in the radial reduction of the free (geodesic) motion on the hyperboloid { x I x * x = x i - - x; = l}, embedded in the Minkowski 3-space with the indefinite metric: ds2 = dxf d x i - d x i , and the standard Euclidian 2-sphere
+
{ x i t X'1-i- x i = 1) respectively. Let us also remark that potentials (1-111) are special cases of the Weierstrass function (IV). 4.2. The Lax-pair formalism. Most known examples of integrable systems (both in finite and infinite-D) can be recast in the form of so called Laz pair formalism. Namely, each point in the phase-space z = ( q ; p ) is assigned a pair of linear operators (matrices):
L = L ( z ) (typically self-adjoint), and M = M ( z ) , in such a way that the Hamiltonian evolution is equivalent to the Laz equation
Iii = [L;MI = LM - MLI
(4.3)
We shall see that evolution equation (4.3) preserves eigenvalues of operator L(t), so we immediately get a family of conserved integrals: eigenvalues { &(q; p): 1 5 k 5 n}, or some functions of {Ak}, e.g. characteristic coefficients of L. Proposition 1: For any operator function M = M(t), the eigenvalues of L(t), that solves equation (4.3), remain constant {Xk(t)= X k ( 0 ) } . If M were constant, then L ( t ) = e-i'ML(0)eitM would be given by conjugating L ( 0 ) with one-parameter subgroup, generated by M , so the result would obviously hold. In general, the role of the group { e i t M } is played by the propagalor (fundamental solulion) U(l) of the ODs:
58.4. Integrable systems related to classical Lie algebras iUl= M(t ) u
{
U ( 0 )= z
395
*
Then L ( t ) = U(t)-'L(O)U(t) is still isospectral to L(0). Another argument is based on traces of powers of L. Indeed it suffices to check that all traces tr(Lm) = But derivative, -$Zm)
=
C XP- const, for m=l; ...n.
L"'-'-Jt
L j , 80
differentiating the trace of Lm, we get
t),
tr(lr"j = m t t ( ~ m - 1 and the latter is equal to
= 0,by (4.3), QED.
tr(L"'-'[L;M])
Our goal is to construct Lax pairs for the Calogero-Moser and Toda systems (IVI) and to show that the resulting integrals: eigenvdues {Xk(L)}, traces {xm= tr(Lm)}, or the characteristic coefficients {bm(L)} (coefficients of the characteristic polynomial p(X) = det(X - L), are in involution. 4.3. Construction of Lax pairs. We use the following Ansatz for matrices L and
M:
L = P -I-ix,
(4.4) where the diagonal part P = d i a g ( p l ;...p,) depends on the momentum variables, while the off-diagonal part X = (xiJ is determined by an odd function x(q) of positions only, 2 . . = X(Qi
'I
- qj).
Similarly, matrix
M=Z$Y, with the diagonal part 2 = diag(zi), made of functions " i = C'z(qi-qj), I
and the off-diagonal part Y = (yij), determined by an even function y(q), Yij = d q i - qj).
The Lax equation then takes the form
iP - x = [P;Y ]-I-i [ X ;21 + i [ X 21.
(4.5)
Writing down matrix entries of (4.5)yields a system of equations for diagonal entries ip'= 3 ix'(Xjkykj-Yjkxkj) = 2ix'xjkYjk; (we took into account the parity of functions x, y). The off-diagonal entries satisfy
Comparing the RHS of (4.6)and (4.7)with the H&il;onian
(4.6)
system of potential
58.4. Integrable systems related t o classical Lie algebras
396
(4.8)
Im:x j k ( z j - Z k ) = ~ x j m x ~ m - x m k x ~ m .
Introducing variables: [ = q j - qm; 9 = qm - q k , and remembering the form of matrix entries { z j } of 2, formulae (4.8) yield a functional-differential equation for an odd function x and and even z,
The analysis of (4.9) is outlined in problem 3 (see [Ca];[Pe]). One can show that
and function takes on one of the following expressions,
P; a cot h(a(); a sinh-'(at) a eot(a0; asin-'(at) dn a%(aO; am(.€);a sn(a€)
The last line gives x in terms of Jacobi elliptic functions: sn, cn, dn. The corresponding potentials v = x 2 + C are then found to be those of the list (I-IV). Potentials of type (V), u = w2q2
2
+c,require
a slight modification of the basic Lax
q2
construction. One takes the evolution equation
~ L = [ c ; M I ~ L~ =LL;* . Evolution (4.10) does not preserve eigenvalues of L, or traces latter can be shown to satisfy, x,(t)
(4.10)
xm = tr(Lm),
but the
= xm(0)ezp( f i w t ) . Now the integrals of motion
can be determined from a pair of auxiliary operators
N , = L+L-; N , = L-L+. T h e latter satisfy the usual Lax equation,
i N i = ";;MI, so their powers and eigenvalues provide conserved quantities. Specifically, we take a pair
of matrices
L* = L f i w Q ,
$8.4. Integrable systems related to classical Lie algebras with the above L and diagonal Q = diag(ql; with
2I.k
... 4,).
397
Then a simple identity
[Q;MI = X, =-q j l q along k , with the Lax equation: i L = [&MI, leads directly to the
modified equation (4.10).
4.4. Complete integrability of Hamiltonian (I-IV). Integrability of systems (I-VI) will be established in two different ways.
1-st argument is fairly general and simple, but it applies to the repulsive, shortrange potentials v(q). The former means that derivative v'(q) 0, so the Newton forces: F i , = -&(qi-qQj), between the i-th and j-th particles, have always the direction of
<
i.e. particles repel each other. "Short range" refers to the effective particle interaction at large distances: Ej potential v(q) decays sujjiciently fasf? at {oo}, then
qi-qj,
hamiltonian trajectories { q ( t ) ; p ( t ) } become asymptotic t o the f r e e motion (lines in the phase-space), qj(t)
-
qoj
+ pit, as t-mo.
So particles do not "effectively feel each other" at large distances.We observe that conserved integrals,
xm = tr(Lm) with
matrix
L of (4.4)represent
polynomials in
the momentum variables { p l ;. . . p n } , whose coefficients depending on { z ( q j - q j ) } i j . One can show that for any repulsive, short-range potential v , and any hamiltonian path { q ( t ) ; p ( t ) } , the distances between pairs increase, - qj(t)-,
as t+m,
hence their motion becomes asymptotically free. Therefore, large-time limits of { xm} turn asymptotically into functions of the momenta variables only. Precisely, if
dt:(Qo;Po)-(q(t);P(t)), denotes the hamiltonian evolution of H in R2n, then
cp7 - mth Newton symmetric polynomial, as b o o .
x m ( q ( t ) ; p ( t ) ) - + x m (= p)
Obviously, the infinite-time limits of
{xm]
Poisson commute. But the
Hamiltonian flow preserves Poisson brackets,
{ F 0 dt; G 0 4tl = { F ;GI 0 dt, for any pair of observables F,G on W2n. Since a commutator of two integrals
{ x j ; x k }is
also an integral, and,
lim { x3.;xk } o +t = 0,
t-w
it follows that
x j ; x m commute
for all time
t, including t=O, which proves complete
W
It is often sufficient to require 0
I u(q) I dq < 00.
58.4. Integrable systems related to classical Lie algebras
398
integrability (see problem 4)! 4.2. Remark: Let us briefly discuss the scattering process for type-I hamiltonians. Scattering refers to hamiltonian dynamics with certain asymptotic behavior at large time, typically linear (free) motion:
( 9 ( t ) ; p ( t ) ) - ( q f + t P f ; P f ) , as t-fw. So the =--oo"
asymptotic state (q-;p-) is transformed into "+w" state (qt;pt), by a
canonical (symplectic) map Y, called scattering map. Ordering particles according to their relative position on R (from left to right), we can write
pt
< ... < p;;
p;
> ...> p i .
Comparing conserved integrals at both extremes, xm(pt) = x,(p-), conclude that, p; = p i ; p ; = pi-,; vectors, q; =;:9
q;
= qi-,;
...
... A
we immediately
similar relation holds for relative position
So the scattering transform amounts to reordering
particles.
Fig.4 illustrates the scattering process f o r a train of particles traveling along R asymptotically free at { fa}.At t = - 00 the first particle q1 has the highest momentum, and the last q, the lowest. They interact via pair potentials { u ( q i - q j ) ] and ezchange momenta. A s the result the left particle (qJ becomes the slowest, while the right one (q,,) gets the highest momentum, with the rest of them occupying intermediate states.
The above argument does not apply, however, to the "long-runge", or attractive, potentials (11-VI). So we shall give another (direct) argument to verify vanishing of the Poisson-bracket for eigenvalues { X k ( q ; p ) }of L,which applies to all cases. 'Ind argument. Given a pair of eigenvalues {X;p} and eigenvectors 4 = (4,; ...I$,); $ = ($I;...$,,)
of L,
zf#J= ( P + X ) I $ = Ad; L$ = p$; we differentiate (4.11) in variables q and p ,
(4.11)
58.4. Integrable systems related to classical Lie algebras
399
that results from (4.11), and the functional equation (4.9), to bring the bracket { & p } into the form
p-Ak
#j
($k~jRkj+*kdjRkj)Zjk-~ (~kajRkj+dk*jRkj)zjk. k # j
Since the expressions inside the first and second sum are antisymmetric in (jk),each sum vanishes. Thus we establish integrability of systems (I-VI).
4.5. Integration of the equation of motion. The Projection Method. We have
shown complete integrability of hamiltonians (I-VI) by exhibiting a family of Poisson commuting integrals { x k ( L ) }or {xm(L) = ktr(Lm)}. Integrals {Ak}, however, do not yield an explicit solution of the hamiltonian system. The latter will be achieved via the so called Projection method due to Olshanetsky-Perelomov [OP]; [Per]. The idea is to lift the dynamics from n degrees of freedom ( q l ; ...qn) to a larger space of dimension N = n2-1, related to some Lie group, where the motion becomes free (geodesic). Namely, we take the space 36 of n x n hermitian (complex) matrices of trace 0, and consider a free motion: z ( t )= at b, i.e. H = ip'. The reduction (projection) from 36 to W n consists of diagonalizing matrix z,z-uQu-', where Q = diag(ql;... q,,), is made of eigenvalues { q j ( z ) } , and u is unitary.In case n=2,
+
1
!q and the projection consists of radial reduction on the free is 3-dimensional, Q = motion in W3 for a fixed value of the total angular momentum,
L' = I J
I ' = p i + sin-2dpi; J = p x q.
Indeed, in spherical coordinates hamiltonian
H = x p ; +T L' ) ,turns into the type-I
two-body problem upon reduction of the center of mass coordinLteg q = f(ql - q2). Let us return to the general case. To find the evolution of { q j } and of the conjugate momenta { p3. = q 3.} we differentiate the relation, UQU-' = at b,
nQ -
+
u
-
l
-Q ~ ~ - + ~ u-l W --a
~
and rewrite the latter as 'The center of mass coordinate T = & x m j q j ( M = CrnJ-total mass) moves rectilinearly (at a constant speed), due to the momentum conservation (58.2):
& - = J-p-const. dt M Hence, the dynamics of the n-body system can always be reduced (constrained) by the number of degrees of freedom of T
.
58.4. Integrable systems related to classical Lie algebras
400
5 = u(t)L(t)u-'(t) = a.
(4.12)
Introducing operators
P = Q, M = - k - ' U , and
L = P+i[M;Q];
(4.13)
and differentiating (4.12) once more yields the Lax equation for the pair L,M
Ji+ i[M;L]= 0. The eigenvalues of L (or its traces) are the familiar conserved integrals. But we we need to compute matrix-function Q(t) explicitly. Let us observe that the type-I Lax pair L = P iX, and M = 2 Y of the previous section (v = q-'), does satisfy (4.13). But the initial data a, b can not be chosen arbitrary in 36 to be able to project to a typeI trajectory in the q-space. Such data must be constraint by the angular momentum oper at or1O ,
+
+
J = i[z;51 = i[a;b]= iu[Q;L]u-'. Operator J can be computed for L and Q of the previous section, and is found to be a matrix with all off-diagonal entries equal to constant g and the diagonal 0,
J = g ( ( @ C ' - I ) ; t = (1;l;...1). Now we can write down the solution-curves of the equations of motion. Initial position variable b = z(0) = Q0 can be assumed (without loss of generality) diagonal (ql;...qn). Initial momentum a is taken to be
a = Lo = Po
+ ix,,
where Po denotes the initial momenta, and matrix X has entries 2. .(O)
t3
=1 . Qj - Qj'
+
(so that J = [a;b] is of the right type!). Then z(t) = Q, tLo describes the free evolution in the extended space 36, whose "eigenvalue-projection" on the "Q-space" W" gives the hamiltonian dynamics of type I. Thus we get a solution '"The angular momentum operator arises from the Noether SU(n)symmetry of the free motion in 36. Indeed, the Lagrangian L = I i I = Ir(i i*), is invariant under conjugations, z+uzu-', u E SU(n), whose infinitesimal generators ( = ((2) = ;[.;(I are identified with elements ( E.).(ts Since the dual space of 36 is identical to itself via pairing: ( p ; z ) - ( p 12) = tr(pz*), we get the conserved Noether integral, J = (p I((.)) = ( p I [ ( ; 2 ] )= (( I [ p ; z ] )= const, for all (. Hence, the angular momentum J = [ z ; p ] , thought of as a 4 .)-valued map on the phase-space 36x36 is constant. Let us remark, that SU(n) is a subgroup of yet larger symmetry-group S q N ) , N = n2-1, so J consists of 4")- components of the total angular momentum of L = I it I '!
38.4. Integrable systems related to classical Lie algebras
+-
Q ( t ) = (...qj(t)...) = "eigenvalues" of (Qo tLo) =
401
;
(4.14)
it
where { q i ; p j } are initial values of { q ; p } . In other words, the initial data { q ; p } is lifted from phase-space W" x W" to a point {(Q;L): L = P i X ( q ) } in the extended space 36 x M, then the free evolution in 36 is applied: (Q tL;L), and the result is projected back to W". For n = 2 the eigenvalues of Qo tLo can be computed explicitly (problem
+
+
+
1). Systems of type II and III. The role of the flat space
36 will be played now by the
hyperbolic (symmetric) space A = SL(n;C)/SU(n).The latter can be realized by all hermitian positive-definite matrices of det = 1, via the map, z+z*z = r. The free motion on space A is generated by all G-invariant vector fields, "7.)
= z*toz,
where r = z*z, and toE 36, identified with the tangent space of A, at ro = { I } . The projection consists once again of diagonalization of r E A, r = ueaQu-l. Differentiating the latter we can recast it into the Lax form: ii; = [ M ;L], with matrices
L =p
+ i(e-'aQMe'aQ 4a - e'aQMe-'aQ); M
= -iu -1(t)c(t).
(4.15)
Solution of (4.15) satisfying natural consistency condition are given by matrix L = P + ZX,with entries x i j = a coth a(qi - q j ) . We can lift up the initial data to a pair of matrices b = e"Q (diagonal), and matrix I , which solve the equation: 2aL0 = b2lb-l b-"Ub.
+
Thence we find the entries of CU, CUjk = apjbjk
+ ia2(1-bjk)sinh-'a(qj
-qk).
4.6. Poisson structure on cuadjoint orbits; the Toda lattice. We remind the definition of Poisson bracket on Lie algebra (5 (86.3). Let F ( z ) ; H ( z be ) two observables (functions) on (5. Then {F;H}(z= ) (z I [aCaH]),or (in local coordinates)CzkCfjaiFajH, (4.16) i jk (5, relative to some basis {el;%;...} c 0,and {zi}-coordinates of z E @* relative to the dual basis {e:;G; ...}. The corresponding hamiltonian dynamics on (5 is given by where { C f j } denote structure constants on algebra
$8.4. Integrable systems related to classical Lie algebras
402
j.= adgF[z].
(4.17)
Let us remark that Poisson bracket (4.12) on 6 is highly degenerate, as all Ginvariant functions, I ( a d i ( z ) )= I(z),for all g E G, equivalently, ~ d ; ~ ( , ) [ z=] 0, for all z E 6* yields trivial flows (4.17). So such { I } form trivial integrals of (4.17), Poisson commuting with all F on @! The degeneracy of (4.12) can be resolved by restricting the dynamics from 6* to co-adjoint orbits of group G ,
0 = { a d i ( z ) : zE G} cz G,\G, where G, denotes the stabilizer subgroup of z. Clearly, stabilizer 6, coincides with the null-space of the Poisson structure:
J,((;dF) = (z I [ ( ; d F ( z ) ]=) 0, for ( E gZ;and all F . So J becomes nondegenerate on the quotient @/GZ N T,(O)-tangent space of 0 at {z}, and furnishes 0 a symplectic structure. It turns out that many interesting examples of hamiltonian systems (classical and newly discovered) arise in this way. To begin we remark that the ad-invariant (Killing) product on 6 identifies Lie algebra 6 with its dual space O* and transforms (4.17) into the Lax form, k = [ M ;21, with M = d F .
There are several ways to construct Poisson-commuting integrals on @*, and on co-orbits 0 c @*. We shall outline 2 of them. 1-st method to produce commuting hamiltonians on the Lie algebra S is to look for a larger algebra @ 3 8. We assume that 6 is decomposed into the direct sum: 8 N B @ R. Our goal is to build commuting integrals on S, starting from @-invariants.
Theorem 2: Any pair of @-invariant functions F , H E I(@), restricted on S, Poisson commute on !2. We denote two projections from 8 to 8;R by IIl;II,, and call F' = F I Q, H' = H 18 the restrictions of F , H on 8. Their dual spaces 8* = R function F on
(9
and St* = 8'.
-
For any
its gradient BF is decomposed into the sum of Q and R-components,
OF = OF, +OF,. We take point z E Q* and observe that for any Binvariant F , (2
I [B,F;B,HI) = 4. I [ 4 F ; 4 ~ 1 ) ;
since (2 I [BF;...I) = 0 . But H i s also @-invariant,hence
98.4. Integrable systems related to classical Lie algebras
403
I [azF;6J1) = -(. I [6,F; 6 , m
(2
and the latter is 0, since the commutator belongs to
[R;R]c R, while z E S* is
orthogonal to R. So the Poisson bracket {F’;H‘} = (zI [ a , F ; O , H ] ) = 0, for all F ; H in
I ( @ ) , QED.
Once again the inner product on 0 allows to recast the hamiltonian dynamics 5 = “d;11(aF)[“17 on an orbit 0 c B* into the Lax form,
L = [L;MI; where L = s;M
= alF(z).
The Toda lattice. As an application of Theorem we shall discuss now the Toda lattice. Here algebra 0 = 4n;R); B = 9- consists of lower-triangular matrices and
$3= d n ) . A @-invariant product is given by
(A;B)+tr(AB);A , B E 4n). consists of symmetric matrices, Sym(n). We take So the dual space B* = R hamiltonian H = $tr(L2)on 0,and the evolution equation in the Lax form,
L = [ M ;L] = [8,H;L], where 8,H denotes the !&(antisymmetric) part of 8 H ( L ) = L. If
L = L+ + L-
+ D, (L- = TL+),
denotes a decomposition of L into the upper, lower and diagonal parts, then M = L + - L-.
+*;
The Toda lattice corresponds to a special choice of L. Namely, we pick a point
Lo=[
Y.
1 0
(4.18)
and take its co-orbit 0 = { L = ( u - ~ L ~ u ) ~ ~ E ,N, -, }: ~c B*, which consists of all tridiagonal matrices
4 L=
a1 *
“1
..
a,-,
The Lax “mate” of L is found to be
1
: trL =
Cbj = 0
(4.19)
$8.4. Integrable systems related to classical Lie algebras
404
The reader can verify the Poisson bracket relation for coordinate functions { a i ; b j }on 0,
(4.20) Those are, indeed, the Lie bracket relations for &-matrices, representing { u j ; b j } :
]
u p [ ’
b j * diag(0;... 1;O;...-1).
Variables { a i ; b j } are not canonical, but one can easily pass to a canonical set, via the coordinate change,
(4.21) The hamiltonian H then takes the form
(4.22) It is precisely form (4.22) that the Toda lattice was first discovered, namely as a chain/ lattice of n particles” on interactions. Functions
W with the nearest-neighbor exponential (nonlinear!)
{Ik = tr(Lk):k = 2;3;...n} make up
a complete set of commuting
integrals. The corresponding flows are given by the Lax operators,
Mk = (Lk)+- (Lk)-; “upper triangular” - “lower triangular” parts of Lk. This result was discovered by P. van Moerbeke. Another general method to cook up commuting integrals involves shifts in vector space 0 of certain G-invariants functions (polynomials). One starts with any F in the algebra of ad-invariant functions, I ( @ ) , and forms a shifted family
= F(z+Xa):all X E R}. The shifts turn out to form a complete, involutive family on 8, i.e. hamiltonians
FA,a; F p , a Poisson commute,
IF.\,
a; Fp, a}
= 0, for P ~ X ,
and any function, Poisson commuting with all
(4.23)
belongs to the algebra, generated
by them. The argument ([Per], chapter 1) exploits the structure theory of semisimple Lie
+
“Notice that coordinates { q j } are determined by (4.21) only modulo constant shift, qj-+q. c. ? Constant c can be identified with the center of mass coordinate = fCqj, that undergoes a uniform constant-speed motion, T = at b, due to momentum conservation: P = C pi = const, for hamiltonian H. The removal of the center of mass (conserved momentum) reduces the number of degrees of freedom from n to n-1.
+
$8.4. Integrable systems related to classical l i e algebras
405
algebras ($5.1-2).
The are many other modification of the shift construction, we shall skip further details, and just remark, that both methods, as well as other known constructions of commuting (integrable) hamiltonians turn out to be special cases of a general R-matriz (recursion operator) method (see [Per]; [Olv] for details). Let us mention that some classical problems (like “rigid body in an ideal fluid”) can also be brought into such framework.
$8.4. Integrable systems related t o classical Lie algebras
406
Problems and Exercises: 1. Show that the first 3 of Lax traceintegrals x1 = C p j - momentum;
{xm} of (4.19) are
+ 2 c v j k = 2H - Hamiltonian (Noether integrals), x 3 = c P! + C ( Pj - P k ) jk' x2 =
pg
i l k
2. Show that the only steady state (time independent) rational solutions of KdV are sums Of{*]
3. Solutions of Functiondaerential equation: [.(C+r]){Z(O
- z(r])l = z(Oz'(r])- z(r])z'(€)l
(4.24)
We are interested in odd solutions {z(()}, and even { z ( ( ) } . i) Show that z can not be regular at {0}, unless z(() = 0 (take hence z=O);
r]
= 0, and
z
= const,
ii) derive an asymptotic expansion of z(() at small (, +(I) a(-1 P€ ...). Hint: changing r]+-r], the equation is traniformed into,
-
+ +
4(-r]){z(O-z(r])} = z(€)z'(r]) + z(r])z'(O. Let (-q, and expand both sides in powers of small 6 = (-7.
Derive small-r] asymptotics of z(r]),
+ P);
4 7 ) a(r]-2
iii) Expand both sides of (4.24) a t a fixed ( in powers of small and show that the coefficients, r] -2: -az = -az; r] - 1: -az' = -ad go r]l
:(z-.P)z :(z-P).'
r]
(starting from v - ~ ) ,
Lz" == -ayz. ayz --pz"' 12
Here z = z(<);z = z((). The 3-rd equation (q0) yields z
= "z+ .(P+r), 2 '
constant term can always be made 0, i.e. P = -7, since z is determined up to constant, So we get z=a(4.25)
I;sl.
The last equation along with (4.25) gives a 3-rd order ODE for
+
2,
+
(all y)z' - %"' cry2 = 0.
22 6 iv) Multiply (4.26) by z-3 and integrate it to get, Z-~Z''
+ 6yz-' + c = 0;
From the boundary condition at {O}, z(()
-
a(-', it follows that c = - 2 c ~ - ~ .
v) Multiply (4.27) by z3z' and integrate once again to bring it into the form (z')2 = a - 2 2 4 - 2 / 4 2
Now the inverse function
((2)
+ A.
becomes an elliptic integral
(4.26) (4.27)
$8.4. Integrable systems related to classical Lie algebras
2) p = fb2; X = ( r 2 b 4 1 3
407
z(() = abcothbt; abctgb(;
In all other cases integral (4.28) is expressed In terms ot elliptic functions. bxplicit formulae depend on roots of the quadratic equation: w2 - 2pu2w Xu2 = 0.
+
We skip the rest of the analysis (see [Per]), and just remark that all cases the 2-body potential has the form, u ( ( ) = a2p(b() const.
+
4. Lax inkgrala of (I-IV), can be chosen either as traces of {L'"},
x,
characteristic coefficients { P j } of L. Show that
Find
eigenvalue
{g1;g2}
of
system
(4.18)
for
n = 2,
= tr(Lm), or the
and
show
that
P = ! j ( p l + p 2 ) denotes the center of mass coordinates, that undergo a free motion: Q ( t )= Qo + P t , while the relative motion in q1,2
= Q ( t ) f q(t),where Q = !j(q1+q2);
the %enter of mass frame", 9 = I/(PO+tPO),
+t2/g2.
Check that co-orbit of any tridiagonal Lo (e.g. (4.18)) under conjugations with lowertriangular subgroup N - consists of all tridiagonal { L } (4.19). Show the Poisson bracket relation (4.28). i) Split variable z = -(zP
P-A
+~
-
a )J-(z
P-A
ii) Denote by F,; F, functions FA,a; F p , a; by
+ pa) = z1+ z2. a,; a,
- gradients 8, ;Oz., and derive 1
= q ( z l I V,F,; aF,1) - q z 2 I [OF,; a,F,I). iii) Use G-invariance of F , , , to show ad* a q z l ) ( ~ l=) "d5F2(z2)("2)= 0. tF1;F , } ( Z )
408
$8.5. The Kepkr problem and the Hydrogen atom
58.5. The Kepler problem and the Hydrogen atom. In the last section we reexamine the classical Kepler 2-body problem and its quantum counterpart: the hydrogen atom, from the standpoint of hamiltonian dynamies, symmetry and quantization. Following Moser, we show that Kepler problem is equivalent to the geodesic flow on 3-sphere. This explains the source of the hidden (Runge-Lenz) S0(4)-symmetry, mentioned in $8.2, and also the high degeneracy of eigenvalues (energy levels) of the hydrogen atom. Then we suggest a direct approach to the hydrogen-spectrum problem by realizing it as a Laplacian on the 3-sphere. Going further along these lines one can discover even larger S q 2 ; 4)-symmetry of the Kepler problem. The resulting “Kepler manifold” then appears as a Weinstein-Marsden reduction of the minimal ceorbit of 00(2;4). The latter can be quantized according to the Dirac prescription: namely, one first canonically quantizes the extended system (ceorbit), then imposes symmetry-constraints on the quantum/ operator level. The net result is once again the exact hydrogen spectrum!
The Kepler problem in W3 describes the motion of a body about the fixed gravity center at the origin. Its hamiltonian,
H=’2-(*. ZP rt
(5.1)
has an obvious SO(3)-symmetry, whose Noether generators make up 3 components of the angular momentum J = p x q . Furthermore, we have shown in $8.3 (two-center gravity problem) that H possesses an additional (hidden) symmetry called the RungeLent vector,
L = p x J-%. In fact, the J and Gvectors combine together to form Lie algebra 4 4 ) (or 4 1 ; 3 ) ) , where J sits in the upper-diagonal 4 3 ) block, while L fills in the complementary row and column. Precisely, for any 3-vectors t ,E ~ R3 we take t and 71 components of both vectors, J ( J ) = < * J L; ( q ) = v * L . These scalar hamiltonians on the phase-space w6 = { ( q ; p ) ) satisfy the following Poisson bracket relations,
I
14thJ(71)) = J(tx 71); (40;L(71))= Yt x II); {YO;Y71)) = -2HL(t x v),
(5.2)
where H is the hamiltonian (5.1). Relations (5.2) imply that on any “energy shell” (level
58.5. The Kepler problem and the Hydrogen atom
409
surface) H = X, functions { J ( ( ) ; q y ) }form a 6-dimensional Poisson-Lie algebra (3 of one of the following types: orthogonal, pseudo-orthogonal, or Euclidian-motion, depending on the sign of A. Namely,
4 4 ) ; if X > 0; so(3;l); if X < 0; E, = @Ds0(3), if X = 0 The proofs of all Lese statements axe outlined in prc,.-m 1. Of course, both J and L commute with H. Furthermore, Newton potential V = 8 is the only one among central potentials {V = f(r)}, that makes Runge-Len2 L = p x J f ( r ) $ , a symmetry of H (problem 2).
+
In 58.3 we applied the J and Gsymmetries to establish complete integrability of both the Kepler problem and a 2-center gravity problem. Next we would like to explore further their implications the corresponding quantum problem: the hydrogen atom. This will bring us back to the subject of Quantization (§6.3), and its connections to the group representations.
5.2. Quantization of classical hamiltonian systems. A classical mechanical system is determined by its phase-space, a symplectic manifold T (e.g. cotangent bundle), with a canonical/Poisson structure j, or equivalently, canonical 2-form 0 = j - ' d x A dx. Points { x E T} describe classical states of the system, while functions {f} on Ep represent classical okervables. One particular observable h ( z ) represents the energy (hamiltonian) of the system. Any observable f defines a one-parameter flow on 9, {exptZ!}, generated by its hamiltonian vector-field Zj = j(af). The flow takes any initial state zo into the state z ( t ) at time t. Quantum system is usually described by the quantum (Hilbert) phase-space 36, whose (unit) vectors {4 E J6:ll$112 = 1) give quantum states, while operators { A } on 36 represent quantum observables. The act of observation consists in evaluating observable A at a state 4,
A+(A4I$). Often Hilbert space 36 consists of L2-functions { $ ( q ) } on the classical configuration space A. One requires J hI $ I 'dq = 1, and interprets J D 111, I 'dq, for a region D c A, as a "probability to find the quantum system in region D".There are two standard views of the quantum evolution. The Schriidinger picture considers evolution of states, generated by a one-parameter group of the quantum hamiltonian H,
410
$8.5. The Kepler problem and the Hydrogen atom
= eitR[$Ol, while the Heisenberg picture takes the corresponding evolution of observables $ O + W
Bo+B(t) = e-itHBoeifH.
(5.3)
Quantization, from a geometric standpoint, amounts to constructing a Hilbert space 36 = X ( 9 ) , and assigning quantum observables (operators) {F}, to classical observables {f} on 9, f+F. Typically, one would like to preserve all basic symmetries (conservation laws) of the classical system in the new quantum system. So the correspondence, f +F, should maintain the basic commutator (bracket) relation between observables. Precisely, given a classical hamiltonian h we denote by G its symmetry-group (of canonical transformations on T), and by (5 its Poisson-Lie symmetry-algebra, made of observables {f} on 9,that commute with h. We want to construct a representation of algebra (5 (or group G) by operators on 36, that would take all Poisson-Lie brackets on 9, into the commutator brackets of operators {A,: f E 8 } ,
{f;h}+ih[Af; A,]; f,h E (5. Plank constant h in front of the commutator (a small parameter depending on the choice of physical units) will be assumed 1. The standard example of quantization discussed in $6.1 was the Heisenberg canonical commutation relations, i.e. the Poisson-Lie algebra spanned by all position and momentum variables {qi;pj} in W’”, with brackets, {Pi;qj} = bij. Quantization of CCR led, via Stone-von Neumann Theorem, to essentially unique” quantum phase-space 36 = L2(W”), where the basic observables {qi; pj} were represented by the multiplications and differentiations, qi+Qi[$l=
qi$;
~j+Pj[$l=iaq,$; $ E L2*
(5.4)
Furthermore, we have shown that representation (5.4) can be extended from the Heisenberg algebra, 1-st degree (linear) operators in { q;p},
Wl= { C a j q j + b j ~ j + c } , to the Weyl algebra W = W(Rn), generated by all differentiations and multiplications. In particular, the 2-nd degree component
+
+
+
W2= {f = Caiq; bijqipj cjp; ...}, gave the metaplectic (oscillator) representation of the semidirect product
W,
D
Mp,.
lzPrecisely, Stone-von Neumann Theorem proves uniqueness of irreducible representations of
CCR, so the only source of non-uniqueness are passible multiplicities of {TA }.
$8.5. The Kepler problem and the Hydrogen atom
411
There are many different ways to extend the Heisenberg-CCR through a representation of W. One of them is the standard Weyl convention, which assigns each (ordered) monomial f = q i p j a symmetrized operator,
f+Af = 8QiPj
+ PjQi).
This convention maintains the Poisson brackets, so operators: A f = f ( Q ; P ) , Ah = h(Q;P) obey the relation,
[A!;Ah] = iA{f; h]’
(5.5)
The Weyl quantization rule also extends to higher degree polynomials, for instance, q2p2+ Q(Q’P’
+ QPQP + QP’Q + PQ’P + PQPQ + P’Q’).
But now it fails to maintain the Poisson bracket relation (5.5) (the Poisson bracket for operators gets replaced by a more complicated Moel bracket, which involves higher derivatives of ‘symbols” f , h to all orders (see 52.3). In fact, a general result of Griinwald-van Howe claims that there is no consistent quantization rule that would work for the entire Weyl algebra, and would extend the Heisenberg CCR for { q ; p } (see
[Chi, [GS21). Of course, any classical hamiltonian h = ip’ + V, is still consistently quantized to a Schrodinger operator, H = - 3v’ t V(q). With this in mind we turn now to the hydrogen atom. 5.3.
The hydrogen atom. The real (physical) hydrogen atom consists of single
proton in the nucleus and a single electron. The proton is about 2000 times heavier that electron, and densely packed at the center. So the “classical prototype” of the hydrogen atom” is the standard 2-body (Kepler) system, the role of the Newton’s gravitational potential being played by the electrostatic Coulomb force (both happen to be the same v=
h!).
So the quantum model of an electron orbiting nucleus13 is the Schrodinger operator l30ur model is clearly a hybrid of the quantum and classical principles, the nucleus being treated as a fixed classical point-charge, while the electron represented by a quantum +function. Yet such model makes a good first approximation to the real physical system. It accurately predicts the energy levels of hydrogen, as measure through the radiation (emiasion/absorption) spectra, explains chemical bonds, etc.
58.5. The Kepler problem and the Hydrogen atom
412
H
= -LA - 1 ; in 36 = ~ 2 ( d ) .
(5.6)
IZI
The evolution of quantum states { $ ( z ) E L 2 } is governed by the Schrodinger equation,
ili, = 11141; +(o) = +o-initial state, whose formal solution
$(t)= e ' y l l , o ] . As always we want to analyze $ ( t ) via spectral decomposition of Operator
H (6) is one of
H
(chapter 2).
the best studied "model examples" in quantum mechanics. Its
spectrum is well known to consists of an absolutely continuous part: [O;m) (of infinite multiplicity), and a discrete sequence of negative eigenvdues {A, =
-$: k = 1;2;3;...},
accumulating to (0). In other words, space L2(W3)is decomposed into the direct sum,
L2 = gc
W
gk;
of eigensubspaces { g k } , and an absolutely continuous (spectral) subspace gc.
Eigenfunctions { $ k } of
H are called bound states,
since the quantum evolution takes on
a particularly simple form for such 4,
$(t)= a time periodic motion of frequency A,.
itx
k$k;
In contrast, states
11, E gC scatter in the sense
that probability to find a state in any finite region D c Wn diminishes in time,
t ) I ~ d 3 ~ + 0as, t+oo.
I,4(z; I
The eigenvalue problem for operator (5.6) can be solved explicitly (see any Quantum mechanics text, e.g. [Bo]; [LL]). For the sake of completness we shall briefly outline the solution. Spherical symmetry of the Coulomb potential allows one to separate variables in polar coordinates (r;d;8) in
@,
i.e. space L2(R3) is decomposed into the tensor product
L2(R+;r2dr)8 36, 36 = L2(S2),and operator
H=
-L(a2+% 2
r
r r
+LA 1-1. r 2 S ri
where
A, =
1 a2 a; + cotea8+ -
sin% 4' denotes the spherical Laplacian on S2. In chapter 4 (34.4) we examined spectral theory of
spherical Laplacian A,, and established a decomposition 36 =
W
ex, 0
into the sum of
spherical harmonics
36, = Span{Yy: -m 5 j 5 m}; dim%, = 2m+l. Each 36,
constitutes an irreducible subspace of the angular momentum algebra
58.5. The Kepler problem and the Hydrogen atom
413
4 3 ) = { J = z x iV}14, and the Laplacian (central Casimir element) becomes scalar on ’rn,
As136, = m(m+l). Accordingly, the entire quantum space L 2 ( d ) breaks into the direct sum of “7?“‘isotropic components of J“, respectively, eigenspacea of A,,
L, = {t/J(r;d,O):As[t/J] = m(m+l)$} N L2(Rt;r2dr)@36,, and the reduced operators H, = H I L, turn into an ordinary differential operators on R+,
H, = - (8: + @r
+7
A S ) - jz. 1
The eigenvalue problem H [ $ ] = E+ is reduced now to an ODE for the radial part R(r),
R”+ :R‘
m(mtl) + [2(E +f) --]R 2
= 0.
= -and r-p = $, brings it to the form
Change of parameters: E-n
&z
R”
+ $R‘ + [(; - i) - Tm ( m]t l ) r
R = 0.
(5.7)
Further simplification of (5.7) results from the substitution,
R = prne-P/’w(p). Exponential p/2 comes here from the “leading part” of ODE (5.7), a t p = OD,
R “ - i R = 0, while power
pm are due
to the homogeneous Euler-type ODE
a2 + $8 - m(m+l) (see problem 3). Then function w solves an ODE, P2
pw”+ (2m+2 - p)w’+ (n - m - 1)w = 0,
(5.8)
known as confluent hypergeometric, or generalized Caguerre equation. Solution of (5.8) is
a generalized Laguerre polynomial, L$mA1)(p), of degree n + m . In order to get a Uregular” solution of the “singular” ODE (5.8) (points p = O;OD are singular!), n must be integer and m 5 (n - 1). So the radial components of an eigenfunction ~t is ~,,(p)
= pme-P/2L$2’)(p).
For a fixed eigenvalue parameter n, the angular momentum number m takes on values m = 0,1, ...( n-1), tJ
and we have 1
= Rn,(p)Yr(q5;6’), { Y r E 36,)
+ 3 + ... + (2n-1)
= n2 “hydrogen eigenfunctions”
of eigenvalue (energy-level)
We immediately notice that the multiplicity of the n-th eigenspace, dime, = n2, is much higher than could be expected on the basis of the apparent so(3)-symmetry! Indeed, a generic rotationally symmetric hamiltonian, I4Let us remark that the angular momentum observables: J k , = P k z , - P , z k , belong to the 2-nd degree (symplectic) component W z of the Weyl algebra. Hence, the angular momentum can be consistently quantized: J k , = xkBm- Zrnak; so J = z x V = V x z.
414
$8.5. The Kepler problem and the Hydrogen atom
H = -~A+V((ZI), should have only (2k+l)-degeneracy of the "spherical-harmonic decomposition": J6= %X,. The abnormal degeneracy of specH indicates some other (hidden) 0 symmetries of the hydrogen problem. The previous discussion should prepare the reader to make a correct guess: the hidden symmetry of H is related to the "Runge-Lenr vector". However, the quantization of L is not straightforward, as L involves cubic terms:
P x J = I P I 2Q - (P.Q)P, which can not be canonically quantized in general. It can be shown, however, (the details are left to the reader) that the correct choice, the one that maintains the classical Poisson-Lie brackets (5.2), is given by the symmetrized Weyl rule
L = ;(Px J - J x P)++s where J = is x V = iV x x (problem 2).
= $(V x J - J x V) ++s,
(5.9)
If we now fix an 'energy leveln of H, eigensubspace = { $ : H $ = A$}, the commutator brackets (5.2) would yield a representation of Lie algebra15 4 4 ) in 8. But irreducible representations of 4 4 ) N 4 2 ) x su(2), are products {am8 a,: k, m = 1;2; ...} (chapters 4; l), of degree mk. In particular, 4 4 ) contains a series of representations { a k @ a a "of } degree k2, which should correspond to eigenspaces of H. So Runge-Lenz suggest a possible explanation of spectral degeneracies of H. However, to establish the precise correspondence between the eigenspaces { G k } of
H, and irreducible representations { a k @ a k }we , need to revisit the classical Kepler problem. The analysis will reveal the true nature of the S0(4)-symmetry of H. We shall see that H (more precisely, its inverse H - ' ) can be realized, as the Laplacian on the 3sphere S3! But S3 is the quotient of the orthogonal group SO(4) N SU(2) x SU(2)/{ fI}, whose representations were analyzed in ch.4-5. In particular, we know that the regular representation R on L2(S3)has multiplicity-free "spectrum" (true for any S"), made of the tensor products a k @ a k ,where { a k } are standard spin-k representations of SU(2) (see problems 7,8 of 55.1). So the Laplacian A,, has spectrum {A, = k'} of multiplicities d, = k2, and we get spec(H) at once. Thus the 'Runge-Lenz" explains the mystery of hydrogen spectral degeneracies.
To convert the hydrogen problem H into the S3-Laplacian, we shall follow a 15assuming negative energy value A = H 5 0, as otherwise one should take Lorentz algebra 4 3 ; 1). The assumption is fully justified, since all eigenvalues of H are indeed, negative.
58.5. The Kepler problem and the Hydrogen atom
415
geometric approach of J. Moser [Mos], based on the stereographic projection. 5.4. Kepler problem and the geodesic flow on
sphere
S" = {zi -t 12 I
S3.A
stereographic map takes
= 1) in R", with the north pole removed, onto the horizontal
hyperplane, @:(zo;z)+w = L
(5.10)
1-x0-
In polar coordinates {(p;4)} on R" and {(6;4)} on S" (fig.5), p = cot;; and
4+4. Fs5. Stereographic projection
@ takes a sphere punched at ihe north pole onto the plane (hyperplane) in R"+'.
Nt
Since @ is clearly a clearly a diffeomorphism (smooth 1-1 map), whose inverse @-l:W-t(zo;2); zo
=*- lw12-1
I w I 2+1'
2=
A. 1 UI I 2+1'
N
the associated canonical map @ = ( @ ; T @ f - l ) takes the natural symplectic structure (1and 2-forms) from T*(S")to T*(W"). Fig.6 Polar angle 8 on sphere S" i s taken into the polar radius p = on R".
COG
Furthermore, map @ is conformal with respect to natural Riemannian metrics on
S"
and W", induced from the ambient space
W"+'. So
its Jacobian
@'
is a conformal
matrix (scalar multiple of the orthogonal one), T@I. @'
= $1,
(5.11)
where p denotes the conformal factor. The matrix transposition in (5.11) refers, of course, to a choice of (Riemannian) metrics in tangent spaces of S" and
W".
The
58.5. The Kepler problem and the Hydrogen atom
416
conformal factor can be easily computed (problem 3), and is found to be
P = W - Iw12)ItI, at each point {(w;t)} on the tangent bundle T(R") of the hyperplane. To check conformality and compute factor p , one just write the Riemannian metric on W" and S" in polar coordinates: ds2 = dB2 t sin%d@ (on
S") + +{dp2
due to p = cot8/2.
(P + I )
t p2d&} (on R");
Then substitution of @ in the kinetic energy form (Riemannian metric) on the bundle T*(S")yields a new kinetic energy function on T*(R"), K = i ( 1 t I w I 2 ) 2 I ( I 2. Here we used a general fact: any diffeomorphisms (coordinate changes) 9 from manifold
Jb to X,
induces a canonical transformation on the phase-spaces,
@:z-y,
N
9 :T*(AJ)-+T*(N); N
9 :(2; 0
4~ 1' 1 = ; (@(z);TA - *(Oh
(5.12)
where A in the second term of the RHS of (5.12) means the Jacobian matrix 9=' of 9 at 2". In
particular, any hamiltonian f(z;() on T*(AJ)is transformed into N
f
(Y;'I)= f 0 :
= f(@ - '(y);*@'('I)).
In the study of hamiltonian dynamics ' h can be replaced with any function F = u(K), the corresponding hamiltonian vector fields being related by a constant (on any energy level) factor, z, = U'(K)ZK.
In particular, we can take
u ( K )=
a-1 = i ( l tI
w 12)
I ( 1 - 1.
The crucial observation of Moser was to notice, that the hamiltonian flow of Ii' (the geodesic flow on the sphere!), restricted on 0-th level surface of u(K), turns into our old friend, the Kepler flow. Precisely,
u(K)/ItI = H t & where
H(ru;t) = ;I w I 2 -1. l€l'
(yo.
is the Kepler hamiltonian with the reversed position and momentum variables The interchange between {w} and {(} could be implemented by a canonical transformation (involution), fx(w;[)-+(-(;w). (5.13)
58.5. The Kepler problem and the Hydrogen atom
417
The composition of 2 maps, ao3:T*(Sn)+T*(Rn),takes the level-1 set of the kinetic energy (metric) form ((z;q):2K= I q I = 1) - a unit cosphere bundle over S", into the level set {Hfw;<) = - f} of the Kepler hamiltonian,
'
H = H,= f I ( 1
2 - h .
Here subscript 1 refers to constant in the numerator of the Newton/Coulomb potential of H. More generally, we call H, = 4 I f I and observe, using simple ")! homogeneity properties of H, with respect to symplectic dilations: (w; +Ow;&), that level sets, {Ha= E } = {HAu = X'E}.
' p,
In particular, cosphere bundles of different radii in T*(Sn), are taken by 1 = D o
St(S") = {(Z;q):2K(...) = I q I = a'}, into the level set,
{(w;<): H,= -T}.1 2u
Clearly, hamiltonians K and H are inverse one to the other, via map 1, '=H 2 K 0 1. Following [Sou], we call the union of all nonzero covectors over S", T+(S") = { ( z ; q ) : 9# 0}, the Kepler manifold. Let us summarize the foregoing discussion in the following
Thwrem 1: The stereographic map (5.10) composed with symplectic involution D (5.13), 4 = Q o @, takes the Kepler manifold, Ts(S"), onto the negative energy shell 9-= {(w;(): H I< 0) of the Kepler hamiltonian, and t r a w f o m the kinetic (metric) f o r m K = 3 I q I on T+(S") into the inverse 1 / H on T-.
'
Finally, it remains to quantize both problems and to compare their spectra. The geodesic flow of hamiltonian, K = I q I on S" (in fact, on any manifold A) yields the Laplacianl' A.
'
A straightforward approach to relate the "negative part of Schrodinger operator u H" to AS' would be to quantize the canonical map 1 = g o @ , i.e. assign a unitary O -n symmetric spaces A, like the n-sphere, the "quantized Laplacian" can be derived from symmetry considerations. Lie algebra elements must be represented by invariant vector fields on A, generators of point-transformations g:z-+z, g E G. Classically, invariant vector fields {X} on A, are in 1-1 correspondence with invariant hamiltonians { F x ( z ; ( )= X -(; E Tr)on the phase-space T * ( A ) . Quantizing algebra { F x } we get a family of 1-order differential operators { i a X } ,acting on Lz(A). But the hamiltonian K IS the sum of squares of invariant vector fields, which upon quantization yields the invariant Laplacian.
418
58.5. The Kepler problem and the Hydrogen atom
operators U:L2(S")+L2(W"), to the stereographic (symplectic) map 5,and to quantize involution u (the latter is clearly given by the Fourier transform 9:L2(W")-+L2(W")). Then one has to verify that the product W = 9 U takes L2(S") into the (negative) discrete component of H - ', and intertwines both operators,
Since, spectrum of the Laplacian A on
S3is well known (chapter 5):
Id,];
with multiplicity this would yield the result, as s p e d (its discrete part) is the inverse of specA! Another formal "derivation" would be i) to take the "stereographed" spherical Laplacian
ii) to show that the eigenvalue problem
K[$I = E2$; is reduced to a "square root problem", a ( 1t I w
1 ')[$I
= E$;
iii) to observe that the latter turns into a hydrogen eigenvalue problem: (w' -&I$ = $; with interchanged position and momentum {w;GI}.
(5.14)
Then (5.14) becomes an eigenvalue (A = 1) problem for the Schriidinger operator H E . Using the obvious scaling properties of the hydrogen hamiltonian: spec(HE) = 1 s ec(H,) E2
would yield the requisite relation between the k-th eigenvalue of H,and eigenvalue E of A,:
Once one could get the discrete hydrogen spectrum from spec(Ag)! Neither of 2 direct arguments, has yet been establlished rigorously. The known methods involve an extension of the Kepler problem to a higher dimensional manifold, W4 x W4* (Kepler represents a U(1)-symmetry reduction of the latter), and the Dirac formalism for imposing constraints in quantum systems with symmetries. The Dirac's procedure [Dir] involves quantizing first the extended (large) system, then imposing "quantum constraints" as operators on the extended quantum Hilbert space 36, so that the
419
58.5. The Kepler problem and the Hydrogen atom
“physical states” become null-vectors of quantum constraints. This procedure could be implemented in many different ways (see [Simm]; [Kum]; [Soul-21; [Mla]; [GS2]; [Hurl). We shall outline an approach, due to [Simm]; [Kum] (see also [GSp]), where the Kepler problem is “conformally regularized”, embedded into a system of 4 constraint harmonic oscillators. 5.5. Couformal regdarktion and the Dtac quautktkm. We consider a system of 4
oscillators described by the hamiltonian
& = fcI ajI2 in the
phase-space C4 equipped
with the natural symplectic 2-form: w = C d a j A d 6 j;
(5.15)
and impose the constraint, b1I2
+ b 2 I 2 = 1631’ + b4I2.
(5.16)
Notice that (5.16) defines a twistor-space II, made up of all complex null-lines in C4, quotient modulo the one-parameter subgroup { e i f : a + e i f a } - a flow of h,. So the twistorspace 17 (a 6-D manifold with the canonical form (5.15)) arises from the WeinsteinMarsden reduction of the constraint oscillator system. Let us also remark that the conformal group G = Sy2;2) acts naturally on II (since G preserves the (2;2)-indefinite form (5.16)on C4, hence it takes null-lines of (5.16) into null-lines). In fact, II represents a minimal co-adjoint orbit of SY2;2), and the constraint system can be obtained by
geometrically quantizing orbit II in the sense of 06.3. 2”. Embedding. We shall embed the Kepler problem, {T*(R3\{O});w = d p A dq; h = :p2
- i}
into the constraint oscillator problem. We fix energy-levels h = -E; (E > 0) (for the
*
Kepler hamiltonian), and h, = 1 (for the oscillator). The embedding procedure (of “Kepler” on the energy shell, h, = -E to the “oscillator”), will exploit the familiar conserved integrals of h,
J = q x p - angular momentum; and L = J x p +A. I91
We remind the basic Poisson-bracket relations for { h ; J ; L } ,
{ h ; J } = {h;L} = 0; {Jj*J 9 3’} = - 6 i3k ’ ’ J kr { J , ; L3’} = -6;jkLk; {Lj;Lj} = 6 83k ’ ’ 2hJk
(5.17)
in other words h commutes with J and L, and the latter form a Lie algebra 4 4 ) (for negative energies h < O!).
Alongside (5.17) we shall use 2 other geometric relations
(problem 4):
J . L = 0; I L 1 2 - 2h I J I 2 = 1. We set p = &%
(5.18)
(constant for a fixed energy level), and define a new set of vector-
58.5. The Kepler problem and the Hydrogen atom
420 variables
z=pJ+& y=pJ-L.
Relations
(5.17-18)
show
that
+ ILI
I z I = Iy I = p2 I J I
= 1,
so
map
!F:(q;p)+(z;y), takes a negative (fxed) energy shell qE= { h ( p ; q )= -E} of the Kepler
phase-space T * ( R 3 ) into the product of %-spheres, 3:4pE+s2 x s2 = ( ( 2 ;y)}
N
CP' x CP'.
Furthermore, the symplectic structure (5.15) on the shell 9, goes into the sum of the natural (volume) forms on the spheres (problem 4), (5.19)
One could check that variables {z;y} do, indeed, Poisson-commute (problem 4). The latter could also be written in terms of the complex (stereographic) parametrization of S2:
z+x = ---1-(z+ 1t1112
'
I z I - 1); similarly
w+y
= ...; z,w E C.
Then (5.20)
Thus the reduced Kepler problem yields the product of two projective lines CP', with a 1parameter family of symplectic forms { w E } , depending on the energy-level E. Next we remark that the projective space CP' gives the energy-reduced phase-space for a pair of oscillators. Indeed, restricting the hamiltonian h,, = I z1 I
+ I z, I
= 1/-
(fixed
energy), and factoring the energy-shell modulo its flow, z = (zl;zz)+eitz, we do recover space CP' with the 2-form17 (5.20).
3'. Now we turn once again to the constraint 4-oscillator (5.16) and fix its energy-shell, 4
C I z j I 2 = 1Along with 1
&E
(5.16) this relation gives the same pair of projective spaces
and 2-forms, as the above energy-reduced Kepler system, 1z1I2+
IZ,I~=
1z3l2+ I z 4 1 2 = l
&G
So 2 hamiltonian systems: Kepler (on the negative energy region), and the constraint 4oscillator produce the identical (symplectomorphic) family of reduced energy-shells. Hence, both systems are equivalent (better to say the Kepler system is embedded in the constraint-oscillator, as the latter includes the "collision Kepler trajectories", so it gives a conformal regularization of the Kepler manifold). 4'. Finally, the constraint oscillator problem can be quantized according to the Dirac 17By the same pattern projective space CP" gives (n+l) oscillators, energy-constraint oscillators,
~z,,
I t ...+ I zn12 = Const.
s8.5. The Kepler problem and the Hydrogen atom
421
prescription (see [Hurl). Namely, we first quantize free oscillators (see 56.2) and get 4 + commuting operators {aj= - 8: :z
j = 1;2;3;4} in the Fock-space of symmetric
tensors (polynomials) in 4 variables, 36 21
m
ex,, 0
36,
-
spanned by the products of the
i+ j+m+l = k} (p.2). Each of oscillators
Hermite functions {+i(z1)+j(z2)+,(z3)+,(z4):
spectrum of eigenvalue (f6.2), spec[aj] = {k +i:k = 0; 1; ...}.
{ a j } has an $integral
Following the Dirac procedure we impose constraint on the “quantum level”, i.e. consider solution of the joint eigenvalue problem,
+ ...+a, = A; + 01 = a, + a,.
a1
Ql
The latter boils down to a solution of a simple Diophantian system,
+ ...+
k, k4 = 2n-2; k1 + k, = k, k,
+
3
, -
(5.21)
Here 2n represents the n-th eigenvalue of the quantized constraint oscillator. Thus we get
k, + k, = k,
+ k4 = n - 1.
In other words, the 2n-th
eigenspace of the “constraint
oscillator problem (quantized Kepler)” is obtain from the n-th eigenspace 36,,(12) of the 2-D oscillator a l + a 2 , tensored with the n-th eigenspace 36,(34) X,,, = 36,(12)
@ 36,(34).
of a 3 + a 4 :
Spin variable n for a 2-D oscillator (a,+az) takes on all
integral and $integral values. Indeed, in 56.2 we learned that eigenspaces of the n-D oscillator coincide with irreducible representations
{T”}
of Sqn),
correspond to S q 2 ) . Thus we get the dimension of eigenspace 36,,
80
2-D oscillators
t be. n.n = n2, the
“constraint-oscillator-eigenvalue” A, = 2n, and the corresponding “Kepler eigenvalue”,
Additional results and comments: The material of 58.1 is fairly standard and could be found me-
.L1
,>n classical
mechanics [AM]; [Am]. The same applies to $3.2 (the Marsden-Weinstein reduction appeared first in [MW). Classical problem from the standpoint of Lie symmetries are treated in many sources ([Am], [Oh]). The recent book [Per] by A. Perelomov contains a comprehensive survey of the classical results, as well the recent developments in “integrable hamiltonians” (see also [FM]). This book along with the review article [OP] was our main source in 58.4. The “hydrogen quantization problem” goes to the very onset of quantum mechanics. W. Pauli (1926) and V. Fock (1935) first discovered the So(4)symmetry of the hydrogen hamiltonian on the Lie algebra level (see [LL]). J. Souriau
[Sou] and J. Moser [Ma] reviewed the classical Kepler problem, and applied the “stereographic projection” method to it. The higher S q 2 ; 4)-symmetries were analyzed in
88.5. The Kepler problem and the Hydrogen atom
422
[Simm];[Sou];[Kum];[Hur];[GSp], and more recently in [GS2]. The general review of geometric quantization and the references could be found in [Sn], [Hurl.
Problems and Exercises: 1. Verify that commutation relations (5.2) with constant E define one of 6-dimensional Lie algebras: 4 4 ) ; 4 3 ; l ) or e3= R3 D 4 3 ) , depending on sign of H. 2. Check that the "combined angular momentum"-"Runge-LenZ" quantum vectors (5.9) obey the commutation relations (5.2), and both Operators commute with H.
3. Laguerre polpomiab and the hydrogen bamilbnian. Generalized Laguerre polynomial L g ( z ) , of degree n and order a, can be be defined 88 a regular solution of the ODE, zy"
+(a+l-
z)y'
+ ny = 0.
They form an orthogonal family on Rt with weight w(z) = z"e-=, and can be generated by the Rodrigues formula, ~ g ( z= ) Const z-aeZ[za+ne-7(n).
Show that the reduced hydrogen hamiltonian, R"+:R'+[($-;)--]R
= 0.
m(m+l)
could be brought to the Laguerre form. Steps:
(5.22)
r2
1' Verify the following commutation relations for differential operators: L = 0' and M = z2a2 bza c (Euler-type);
+
+
+ + + +
+
= eAz[(a A)2 6(a A) c] = eAZ[L 2A8+ ( 6 1 ( i ) L[e (ii) M[Z....I = Z.[M + 2sza + s2+(6-1)s]
+ 60 + c,
+ A2)]
2' Take the reduced "hydrogen operator" (5.22), and apply (i) to the product R = e-r/2u(r), to get an ODE, m(mt1) )u = 0. M[u] = a%+ ($- 1)au - (++r2
Break M into the sum of 2 Euler-type operators, M , (M,
+ ~ , ) [ r ' v ] = r8{a2+
(v -
1)8 -
+ M,, and show that + s ( s t l ) -r2m ( m t l ) "1-
For the singular term (...)r-2 in the potential to cancel out index s must be equal rn, and the resulting ODE becomes a generalized Laguerre equation! 4. i) Verify relations (5.18) for { h ; J ; L } ; ii) show that the canonical 2-form on 9, c T * ( p )is taken into the form W , (5.19) by map I; iii) check that variables {zi}, and {yj} Poisson-commute, {zi;yj} = 0. Find the commutation relations among {zi}.
Appendix A: Spectral decomposition of self adjoint operators.
423
Appendix A Spectral decompasition of setfadjoint operators.
Any selfadjoint operator A in Hilbert space 36 (bounded or unbounded) admits a spectral decomposition, which generalizes the notion of an eigen-expansion of a symmetric matrix,
A
N
r 1
...
J
Here {A,; ... A,} = spec(A) denotes the eigenvalues of A , and the k-th diagonal entry/block corresponds to the k-th eigensubspace of A. In other words space 36 is decomposed into the direct sum of E,; and operator A I E,= AJ, i.e. (A.1) where Pk means the orthogonal projection from 36 to E,. Diagonal matrices can be thought of as multiplication operators A: (fk)+ ( X k f k ) , on spaces of (scalar/vector) ntuples {(fk):l _< k _< n}. In other words there exists a unitary map %: 3 6 4 @ Ek, that conjugates A to a multiplication operator,
I %A%-’
= A:
f(A)+Af(A)
I
(A.2)
Both results: spectral decomposition (A.l) and diagonalization (canonical form) (A.2), can be generalized for all selfadjoint Hilbert space operators. Precisely, spectrum of a selfadjoint operator, spec(A), is a closed subset of R. The role of eigen-projections {P,} is played in general by the family of spectral projections { P ( A ) } , associated to closed subsets of A c R, or equivalently by the spectral measure (resolution) dP(A). Projections { P(A)} form a commuting, monotone (P(A,)
+
AdP(A).
Spectral subspaces {E(A)} of A are images of spectral projections {P(A)}. If operator A has discrete spectrum { A , } (e.g. finite rank or compact), then
E ( A )= @ E , - direct sum of eigensubspaces sitting “inside A”. A, E A I
The proper analog of the diagonal matrices (A.2) are multiplication operators
m+x
A: f@), (A.3) on spaces LZ(R;dp)of scalar/vector valued functions on R, with a Bore1 measure d p .
Appendix A: Spectral decomposition of self adjoint operators.
424
It turns out that multiplications provide canonical models of all selfadjoint operators. The basic result of spectral theory of selfadjoint operators states that any such A is unitarily equivalent to the direct sum of multiplications (A.3). Namely, Theorem Al: There ezists unitary operator 'U: 36+ @ L2(dpm),such that m
In the canonical realization of A spectral subspace E ( A ) consists of all functions vanishing outside A, E(A) = { f : s u p p ( f )c A}, whereas spectral projection P(A) becomes a multiplication with the indicator function xA: f + xAf. Canonical realization also yields some natural properties of spectral subspaces, e.g. spectral subspace E(A) is invariant under the commutator of A , Com(A) = { B : A B = BA}. Indeed, any B, that commutes with a multiplication by X on L2-space of scalar/operator functions on W, must itself be a multiplication with a scalar/operator valued function, B: f(A) +B(X)[f (A)]. The proof of spectral decomposition (spectral measure) exploits some basic harmonic analysis on commutative group, R and 2. We shall demonstrate it for a single unitary operator to give the reader a gist of the arguments involved. Function called positive- definit e for all n-tuples {<';
... (,}
cd(€j-
€&j%
d(<) on
R or
Z is
2 0,
jk
c R and all coefficients {al;
... a,,} in C.
Bachner's Theorem : A n y positive-definite sequence
{dk} on Z
coefficients of a positive measure dp on the dual group
1, i.e.
coincides with F o u r i e r
The proof is outlined in problem 12 of 52.1 (chapter 2). Now given a unitary operator CJ in Hilbert space 36, we pick any vector (I, E 36, restrict U onto a cyclic subspace 36, spanned by do,
M, = span{o'(d): L = 0; f 1; f 2; ...I, {dk = (Uk$I d ) } is positive-definite on 2.
and observe that the sequence of matrix-entries Hence, by Bochner's Theorem
4k - E 1d2 =e
-ik6dp o( 6 )Y
(A.5)
for some measure p = p,,. Let us remark that any spectral measure {dP(A)} in Hilbert space
M defines a family of scalar measures { d p } associated to vectors {(I 36}, (I dP$,(4 = ( d P ( 8 d I d ) .
Appendix A: Spectral decomposition of self adjoint operators. We claim that Bochner measure
425
(A.5) is the spectral measure of U 136,.
Indeed, space 36, is identified with LZ(T;dp,),via map
b p i k z ; bk = (Uk$Jo1$J); and operator U acts by multiplication on Lz-functions, $J-$(z)
=
U: $(f?)-+eiO$(0). Indeed, space 36, contains a dense subspace of vectors $J =
cajUj[$,J = J(U)[$J,I,where f = c akeikz$
and operator U acts by the shift, { b j } + { b j - , }
on the space of Fourier coefficients.
It follows now from (A.6) that any “function of U”,
M = f ( U ) = CajUj, also acts by
multiplication on L’,
M : 3(z)-f(z,$(z). The latter holds for all polynomials {f(U)}, hence all continuous and even bounded (Lm) functions {f} on T. Hence spectral projections { P ( A ) } are nothing, but multiplications in L2(dp) with indicator-funcfions,
XAh)= i.e.
2 E [O; A]. 0; 2 4 [O; A]’
{ 1;
P(A):$ ( + X A ( Z ) 3 ( 4 ,
$ E LZ.
This yields spectral measure of U on a cyclic subspace 36,. To get it for the entire space
36, we take an orthogonal complement 36’ = 36 8 M,, find a cyclic subspace 36, c 36’, resolve it into L z [ d p l ] , and continue the process. Eventually entire space 36 is resolved into the direct sum of Lz-spaces, m
36 2: @ 36,; 0
36,
J,
N L2[dp
and we establish spectral decomposition for any unitary operator U .
Remark 1: After spectral decomposition is proven for unitary { U } one can easily pass to self-adjoint operators { A } (bounded or unbounded), via the Caley transformation: A+U = ( A - i ) - ‘(A+i), that relates both classes. Thence one proceeds to more general cases, e.g. joint spectral resolution for a commuting family (representation) of selfadjoint operators, or unitary groups, etc.
Remark 2: Spectral resolution allows to construct “functions of self-adjoint operators” { F ( A ) } ,used often in differential equations (see $2.3). We define
+
W)= F ( w J ( Y , a clear generalization of F ( A ) = @ F ( X k ) P k , for a matrix A . k
426
Appendix A: Spectral decomposition of self adjoint operators. Our last result concerns spectral decomposition of compact selfadjoint operators.
Theorem A 2 Any compact selfadjoint operator A in Halbert space 36 has purely discrete spectrum of real eigenvalues Xk+O; all eigenspaces Ek = { ( : A [ = X k ( } (Ak # 0 ) are finite-dimensional, and the system of eigenvectors is complete, i e . @ E(X) = 36. X E specA Proof: We recall that compact operators are characterized by the property: ( i ) A-image of fhe unit ball B = {li(ll< l}, K =closure{A(B)}, is compact in 36. Let us also remark that (ii) compact projections P are always finife-rank (since a unit ball B in any Banach space 9 is compact iff d i m 3 < m). Discreteness of specA is proven by taking its upper bound A, = supp{(A(
I ():I[<11 5 l},
and
&., Such tois clearly an eigenvector > 0, the corresponding eigenspace E , is finite-dimensional by (ii).
using (i) to show that the bound is attained at some vector of the eigenvalue A,.,
If A,
Next we peel off subspace E,, i.e. restrict operator A (respectively. quadratic form ( A t I ()) onto the orthogonal complement of E,, 36, = 36 8 E,, and repeat the process. This yields the next eigenvalue A1
= sup{(A€ I €I:€ E 36,;11€
II I 11 < “0;
and its eigenspace El.The process continues until the entire Hilbert space 36 is exhausted, i.e. resolved into the direct sum of eigenspaces: 8 E X ,QED. X
427
Appendix B: Integral operators.
Appendix B: Intqral operators. Throughout the book we often encounter integral operators,
K f b ) = jXK(GY)f(Y)dY, acting in various function-spaces: L2;LP;C(X),etc., and need to estimate their operator-norms, and analyze other properties, like positivity, compactness, HilbertSchmidt, trace-class. In this section we shall briefly outline some basic facts of the integral-operator theory. Integral operators could be thought of as %ontinuous matrices", with "entries" {Ii'(z;y)} labeled by points of X . Many natural features of matrices extend to integral operators fairly straightforward, for example, self-adjointness of K means K(z;y) = K(y;z) - kernel of the adjoint operator A'*; positive definiteness (KfI f) >_ 0, for all f , is equivalent to positive definiteness of any matrix {(aij):ai= j K(z.*z.); q;...zn E X}. I' 3 The operator-norm of K: LP+Lq could be estimated via Minkowski inequality,
whence follows,
119
( I( J I
I ~ . fI +; y) I ~ r ) p l / q d y ) L ' P ; where p' = 2- the dual Hiilder index.
fllP;
P-1
I
The constant in the RHS of (B.l) called m u e d Lq;P-norm of K(z;y), is not optimal, but sufficient for many purposes. For L2-spaces ( q = p = p' = 2) we get
I Kf Ib II1K IUIf Ib; the mixed norm of K becomes its L2(dzdy)-norm. Compact operators in Hilbert/Banach spaces take bounded sets {[I C} into compact (the latter could be verified in LP-spaces by Ascolli-Arzella Theorem').
fl/<
Self-adjoint compact operators are easily described in terms of their eigenvalue spectrum: specA = { A k } - purely discrete, Xk-+O, as k+w, and all nonzero X's have finite multiplicities, dimE(X)< 00, for any X # 0 in specA. In genera! (non self-adjoint) case a convenient way to describe compact operators is terms of their 'We remind that according to Ascolli-Arzella: compactness of a subset Jb C LP; or equivalent to boundedness equicontinuity.
+
e
is
Appendix B: Integral operators.
428
modulw, I A 1 = (AA*)li2-a positive self-adjoint operator. Then
I A I - compact (j A(, I A I ) = sk(A)+O, as k + w . * The eigenvalues of modulus I A I are called s-numbers of A, and A - compact
($
I
- - \
the class of all compact operators 6,(36) is characterized by asymptotic vanishing of s-numbers, sk(A)-+O. Among all compact operator we distinguish two important subclasses (ideals of the algebra S(36)): 0
Hilbert-Schmidt: 6 , = {All A
lhs= C sk(A),= C A(,
AA*) < m};
Trace-class: = {A:IIA&.= E s k ( A )= EXk( I A I ) < 00); eigenvalues counted with their multiplicities. 0
To describe both we choose an orthonormal basis {el;...ek;...} in 36 and represent any bounded operator A by an infinite matrix ( u ~ ~ ) $aij ~ := (Aei I ej). We define the trace of A, t r A = z a j j - sum of diagonal entries (provided the sum converges).
Proposition 1: Operator A is Hilbert-Schmidt iff C . . I a i j 1 < 00, furthermore the sum of squares of matriz entries gaues the Hilbert-S&midt-norm of A, IIAllis = C I a i j I = tr(AA*). 81
The statement is easily verified by diagonalizing B = AA* and using invariance of trace under all conjugations: B+U-lBU,
with invertible U.This also shows that positive self-
adjoint operators with finite sum
11 A Ih, = trA =
Caij < 00,
belong to the traceclass, and
&(A).
Let us remark that Hilbert-Schmidt-norm defines the inner product on 6, ( A I B) = tr(AB*); and the (operator) product of two Hilbert-Schmidt-operators belong to the trace-class. Space 6, equipped with such product turns into the Hilbert space, isomorphic to the tensor product 36 @ 36. Next we want to study compactness, trace-class and Hilbert-Schmidt properties of integral operators. It is difficult to completely characterize compact and trace-class operators in terms of the kernel K(z;y), although there are many sufficient conditions. The Hilbert-Schmidt class proves to be more manageable.
Appendix B: Integral operators P r o p i t i o n 2: (i) Operator K E 6, iff
JJ
429
I K(r;y ) I 2dxdy < 00,
and the
Lz-
norm of the kernel is exactly the Hilbert-Schmidt-norm of K . (ii) Trace of an integral operator: trK = K ( x ; x ) d x - integral over the diagonal of
J
xxx.
The proof easily follows by choosing an orthonormal basis of functions {$k(z)} writing
6=
Dirac 6(2;y) = C $ ~ ~ ( z ) & ( y ) , or in the physicist
I $k(z))($k(Y) I.
bra-ket
and
convention:
In the mathematical convention we represent the identity m
operator in 36 = L 2 ( X ) , as the sum of orthogonal projections: I = @ (... I $k)$k. Then trK =
c(K$k l$k)
0
= jK(z;y)6(z;y)dzdy = JK(z;z)dz,
and (i) follows from (ii) by writing 11 K ]pHs
= tr(KK*).
It follows from Proposition 2, that continuous integral kernels K ( z ; y )on compact spaces X yield compact (even Hilbert-Schmidt) operators. In fact, one has, Proposition 3: Integral operators with continuow kernels K ( x ;y ) on compact
manifolds X are trace-class with t r K =
J
K(x;x)dx.
It suffices to check that self-adjoint operators with positive kernels K(z;y) = K(y;z) 2 0 are trace-class. But any such K is the square of an Hilbert-Schmidt operator M with continuous kernel M ( z ;y) = 4 -j.
As a corollary, we immediately get compactness of the convolution-operators,
Rf:u+f*u, on torus T",or any compact group G.
430
Appendix C: A primer on Riemannian geometry. Appendix C: A primer of Riemannian geometry: g u n i k , connection, curvature.
Riemannian manifolds typically come equipped with 2 structures. One of them is metric-temor {gij(z)} on tangent spaces { T , : z ~ 9 6 } ,along with the dual metric .. {(g'j) = ( g j j ) - * } on cotangent spaces. Metric brings in a host of other interesting geometric structures.
C1. Geodesics is defined as a shortest path 7 = {x(t):O 5 t 5 I}, that connects 2 given points z,y E 96. So 7 solves the variational problem, minimizes the length functional
I 1
1
L [ ~=I z g i j ( z ) k G d t = e(z;k)dt. 0
0
Hence 7 satisfies the Euler-Lagrange equations (see $8.1),
u=a e-.Q.e=o, 67 z dt 2
or
The latter can be written in terms of the so called Christoffl symbol of metric g,
by differentiating the LHS (C.2), using symmetry of metric tensor gij in ij, and multiplying the result by the inverse matrix {g'j} (see Proposition below). This yields fk +
crFj&
= 0.
(C.3)
ij
As a simple illustration we shall obtain geodesics on the Euclidian space, Poincare half-plane W,and the 2-sphere. Euunples. 1) Geodesics in the Euclidian space R" are straight lines: x = O+x(t) = a + bt.
2) The Poincare plane H = {z+iy:y (gij)
=
[
Y-2]:
its dual (g'j)
=
> 0) carries metric
[* } Y2
d s2
-d z 2 + d ~ 2 ,so
tensor
Y2
The Euler-Lagrange equations then become:
To solve system (C.4) we note that i/y2 = a - const2. Introducing an auxiliary variable 21n chapter 8 the reader will see the meaning and reason for conservation of i / y 2 . This quantity represents the p,-momentum (conjugate to the 2-variable) of a hamiltonian system asciociated to Lagrangian L. Its conservation reflects a translational symmetry of L in the z-direction.
Appendix C: A primer on Riemannian geometry.
431
u = y/y2, we rewrite (I)-(II) as a system
Variables y and u in the first 2 equations (C.5) separate and we get
whence follows
On the other hand the last 2 equations yield
!&". a, and substitution function u = u(y) from (C.6) results in
But the latter has a general solution
that represents a family of all semicircles in the upper half-plane perpendicular to the real axis!
3) Sphere S2 carries metric ds2 = dd2 + sin24d02, in polar angles 4,O. The reader is asked to show that geodesics are great circles: C = C(40;80),that make angles Qo with z-axis, and 8, with z-axis. Circle C(q5,;O0) can be parametrized in terms of the arc-length
(
COSQ
= cos~ocos8
Geodesics on a Riemannian manifold 96 allow to introduce two other important structures: ezponential map, and parallel transport.
C2. Exponential map exp.'Tz+96, takes tangent vectors at a point z onto the manifold (in a neighborhood of {z}). Point zo = z(0) gives the initial position for a 2-nd order system (C.3), while tangent vector ( = k(0) plays the role of initial velocity. The combined initial data (zo;[) produces a unique solution-curve, 7 = z ( t ; z o ; [ )- a geodesics issued from zo in the direction (. Map (-m(t;[) is called exponential and denoted, exp,,(t[), sends ray { t [ }c Tzo into a geodesics 7. One can show that exp takes a neighborhood of (0) in T onto a neighborhood of {zo} in 96 in a 1-1 way, its inverse is often called log.
432
Appendix C: A primer on Riemannian geometry.
C3. Covariant derivative and parallel transport. Parallel transport on surfaces consists of moving tangent vectors { ( ( t ) }along a geodesics {y = z(t)}, in a way that preserves their length I[(t)l, and the angle between ( and 5. In other words we get a family of linear maps { U ( t ) }on tangent spaces {Ti= T z ( t ) }along y, Ut:To+Tt, which preserve metric g IT,. Clearly, U iis nothing, but a Jacobian matrix 4' = (&) a4 of the map ,
4,: "o+"(t;
"0;
0 = =PZo(Y).
Covariant derivative then represents an infinitesimal generator of U,.The formal definition of covariant derivative (affine connection) involves a family of linear transforms {V,:X E T , } on tangent spaces, , labeled by vectors X E T,, that satisfy
V , ( f Y )= @,f )Y + fV,Y, (C.7) for any pair of tangent vectors X , Y E T,, and any function f (here axf denotes the derivative of f along X ) . Covariant derivative (C.7) is uniquely determined by its values on basic vector fields {aj},for a chosen set of local coordinates {xj}. Namely,
vaj(aj) = Crfja,, where coefficient {I'fj} form the so called Christoffel symbol of connection V . Alternatively we can describe V as a linear map that takes vector-fields { X }on 96 to 1st order differential operators on sections of the tangent-bundle,
X+V, = a, t r(x;...):Y - + [ xY; It r(x;Y ) , where a x ( Y )= [ X ; Y ]means Lie derivative of field Y along X (or commutator of 2 vector fields), and F(X;...) -the Christoffel operator (matrix) on T,. Covariant derivative gives rise to the parallel transport of tangent vectors/fields along curves y c 96. Namely, field X is called parallel along 7, if its derivative
V i . ( X )= 0, (C.8) where -j = d7 denotes a unit tangent (velocity) vector. The geometric content of (C.8) becomes apparent when one first looks at a linear OD system,
dds X t A(s)X = 0; X ( 0 ) = Xo; (C.9) whose trajectories { X = X(s;X,)} define a family of "parallel lines". In other words, the fundamental matrix-solution {U(s;t):($ A)U = 0 (s 2 t), U(s,s)= I},implements the parallel transport of initial states (vectors) { X o } , at time t to terminal states { X ( s , X o )= U(s,t)[X,]}, at time s. In our geometric setup we move tangent vectors ...) along y, via ODS d i- A(s), where ds - arc-length along y, and A = r(+(s); Christoffel operator in the direction of -j(s). An interesting situation arises when 2
+
Appendix C: A primer on Riemannian geometry.
433
structures (the metric and the connection) are consistent, i.e. parallel transport preserves metric (norms, angles between tangent vectors). So
( U ( s ) € l W h ) = (W,for any Pair ( 3 9 E Tm equivalently,
"(x ds I Y )= (v+(x) I Y )+ (XI v+(Y)),for any pair X , Y ,
along any path 7 c 96. Here
((7.10)
(I) denotes the Riemannian product on tangent spaces.
We introduce torsion of connection V as a vector-valued bilinear form on T,,
T ( X ; Y )= V X ( Y )- V y ( X )- [ X ; Y ] . In local coordinates T represents an anti-symmetric part of Christoffel tensor I' in lower indices,
~ p" .3.)3=
c(rfj- r$)ak.
Propmition: There ezists a unique torsion-free couariant derivative (parallel transport) V on 96, consistent with metric { g i j } , whose Christoffel symbol can be written ezplicitly in terms of the metric tensor. We take a coordinate system {zj} and compute the covariant derivative Vi = V product
(ajlak)= gjk, via (C.10).This yields a,(sjk)= Cm( r z g r n , + Wmj)
(1)
Then we write 2 similar relations for all cyclic permutations of the triple { i j k ) ,
Using symmetry of tensor
r in lower indices (torsion-free), we observe that first term of
(I) is equal to the 2-nd term of (11), 1-st of (11) is equal to 2-nd of (III), and 2 - nd of (I) is the first of (111). So 3 equations (1-111) are ofthe type
B+C=r Here A (resp. B,C) represents C r Y ) J m k (resp. its permutations in {ijk}), and we get m
the Christoffel tensor, contracted with the metric tensor {gjk} expressed in terms of {gjh} and its derivatives. Multiplying (contracting) it with the inverse metric tensor {gj'}, we get the final formula
434
Appendix C: A primer on Riemannian geometry.
So we recover the Christoffel symbol of the Riemannian metric (C.2) determining the equation of geodesics, P
+ r(i;i)= O.
Parallel transport along a closed path U(t;O), where z ( t )= zo, gives a linear map in TZo called holonomy. In case of Riemannian connection operators { U ( t ) } clearly preserve the norm 1 [ in T,, so they define isometries of the tangent space.
18
C4. Curvature is closely related to the parallel transport and covariant derivative. We start with 2-D manifolds. Given such 96 we pick a tangent vector ( = to
at point zo E 96, and transport it along a small loop 7 = boundary of “small region D”. On return to the initial point vector [ will rotate by an angle q5 = d ( D ) (angle q5 gives the holonomy matrix U g= -sing ‘OSg cos4 sing Functional 4 ( D ) is easily seen to satisfy the relations: q5( D , U D2)= q5( 0,) q5( D2);for any pair of regions D,; D,, while the change of orientation: D-r-D (7traversed in the opposite direction) yields
I).
[
+
d4-D) = -4(D). Such q5 clearly represents a differential 2-form (curvature-form) integrated over
D, The value of w on a pair of tangent vectors [;r/ can be found from the following procedure. We take a small parallelogram P, in space T , spanned by (;q compute holonomy angle q5 along P,, then set
The reader is asked to compute w on the 2-sphere of radius R and the Poincare plane (the former has: w = R-2dS, the latter: w = -dS, where dS is the natural Ftiemann surface area element). For a general Riemann surface w = KdS, with respect the (Riemannian) area element, where function K gives the Gauss (Riemann) curvature of the surface (on S2 and W,function K is constant f 1).
In higher dimensions the role of angle q5 is played by the holonomy matrix U = U ( 7 ) . As above any pair of tangent vectors [,q defines an infinitesimally-small parallelogram P, = Pc((;q).We take the holonomy U, = U(P,)in the orthogonal group SO(T,). Also higher-D ( n 2 3) orthogonal groups are non-commutative (so U(D,U D2)is no more a simple product of two U’s), we can still expand U,([;q) in small E and find,
Appendix C: A primer on Riemannian geometry.
435
+
Uz(t;7]) = I t E2R(h) O(e3>, where R is a 2-form with values in the Lie algebra so(n) of skew-symmetric matrices in the tangent space T,, called the Riemann curvature tensor of 96. The curvature tensor gives rise to sectional curvatures of 96 in all geodesic planes, passing through x. The latter means all 2-D surfaces C in 96 (near z,,), spanned by the exponential (geodesic) images of all 2-planes in T,C = ucp(g). The Gaussian curvature of C is expressed in terms of the Ftiemann tensor, as
q q= (m;7])eld* These are sectional curvatures of 96. The formal definition of the Riemann curvature tensor is given in terms of the connection
V, namely,
R ( X , Y ) = VxVy
- V y V x - VLX;yp
an operator-valued bilinear form on tangent space {Tz}.Here [ X ; Y ]denotes the Lie bracket of vector-fields X,Y. We can also write it as a quatric form, ( R ( X ;Y ). V I W ) , or rank-4 tensor of type (1,3),
RCk = ((ViVj - VjVi)aJa,). Riemann curvature tensor is shown to satisfy the relations:
.R(X,Y) = -R(Y,X); g ( R ( X ,Y ) Z ; V ) = - g ( R ( X , Y )* V ;Z ) . In other words operators R ( X , Y ) on T, are skew-symmetric relative to metric g, and the bilinear operator-valued map ( X , Y ) + R ( X , Y), is anti-symmetric. 0
If torsion T = 0, then
R ( X , Y ) .Z + R ( Y , Y ) .X
+ R ( Z , X ) .Y = 0 (Bianchi identity)
Remark: The Christoffel, torsion and curvature tensors are often convenient to recast in the language of differential forms. We choose a basis of vector fields {XI;...X " } , and consider the corresponding dual set of 1-forms: w = ( w J ) , e.g. ( d d ) for w = (a .). Then one can introduce a connection 1-form: R = w, i.e. 3 of= xrtjwk. k
One can think of w, R as vector-valued and operator-valued 1-forms. Clearly, R determines ,'I hence covariant derivative V. On the other hand both can be described in terms of the torsion and curvature, via the Cartan structural equations:
Appendix C: A primer on Riemannian geometry.
436
dR = R A 0 + i R w A w Cartan’s equations generalize the Frenet equations for the orthogonal moving frame {T;N;B} (tangent, normal, binormal), along curve 7 ,
in terms of curvature between derivatives
ti
and torsion r of 7 . For 2-sphere they give the standard relations
{a,@,}
of the orthogonal frame {u;u;w } ,
Fig.1: Polar angles (q5;e) on the 2-sphere Sz and the orthogonal frame { u;u;w}.
C5. Geodesics and m a t u r e on Lie groups and homogeneous spaces. A Lie group G always possesses a right (or left) invariant Riemannian metric3 g: it suffices to fix g at the identity {e}, and then translate it by right/left multiplications, &:y-+yz (or z-ly), that maps tangent spaces T,+T,.
Lie algebra 6 of G can be identified with the
tangent space T,, and also with the algebra of left-invariant fields on G (see 51.4).Such fields are considered parallel, as left translates on G form parallel transports. So all geometric structures: metric, covariant derivative, curvature tensor become leftinvariant. In particular, geodesics through the identity are all one-parameter subgroups (7 = z(t)}, holonomies are Jacobians of left translates {&(t):6 = Te--+Tz}, and curvature will measures a “degree of non-commutativity of G”. It suffices to compute 3Let us remark that G typically may not possess a bi-invariant Riemannian metric. In chapter 5 we show that all simple/semisimple Lie groups G do have a bi-invariant (Killing) form B ( X ; Y ) ,but this form is positive-definite only for compact G.
Appendix C: A primer on Riemannian geometry. them at {e}. The Riemannian product
(I)
437
on tangent spaces defines a bilinear map
B:(5 x 6-45,
(B(X;Y)IZ)=([X,Y]IZ); all X , Y , Z € O , which plays the role of the Riemann curvature tensor. The covariant derivative at {e} then becomes,
V X ( Y )= , K [ X Y ]- B(X;Y)- B(Y;X ) h while sectional curvatures takes on somewhat more complicated form,
The latter simplifies for bi-invariant Riemannian metric on compact Lie groups,
Next we turn to homogeneous space % = K \ G . Space 96 has a G-invariant metric, iff the stabilizer subgroup K of point zo, acts by isometries on the tangent space To = TzO,i.e. Jacobians, {4’u:u E IC} preserve metric go = g(zo) on To. The latter is always the case, when stabilizer K-compact (there any metric go on To could averaged over K (see $3.1) to make it K-invariant). In particular, symmetric spaces K\G (semisimple Lie group, modulo maximal compact subgroup) possess a G-invariant metric. Manifolds Rn; Sz; W provide examples of symmetric spaces, as we explained in $1.1,
R” = Mn/SO(n); sz= SO(3)/SO(2);
w = SLZ(R)/So(2).
Obviously all geometric structures (covariant derivative, Riemann tensor) on symmetric spaces are G-invariant, and could be computed in terms of the so called Cartan decomposition, discussed in $5.7. Also geodesics on % are transformed one to the other by symmetries (isometries) of 96.
There are plenty of references to the basic differential geometry, of which we mention only a few, that were used most often: [Am]; [Sp]; [DFN]; [Car]; [He1,2]; [Cha].
This Page Intentionally Left Blank
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List of frequently used notations: - reals; complex numbers; quaternions; integers
Manifolds/domaina - manifold -Poincare half plane {Sz> 0); -Poimare disk { I t I < 1) - Minkowski space - rank-one hyperbolic space of dim = n - projective spaces of dim 1,k; also - projective spaces over reals, complex spaces - the n-sphere {IIZII = 1) in Wn+' - root system (positive roots) of a semisimple Lie algebra - Weyl chamber
SP-
- vector/Hilbert
SMn
space - tensors of rank m over r;also T €4 N('T). - symmetric tensors of rank m - polynomials of degree m - algebra/space of d x d, n x m matrices - harmonic polynomials/sphericd harmonics of degree m - m-th anti-symmetric tensor power of space Y - symmetric n x n matrices
G;21
- group and its dual group (dual object)
Tn
- torus - permutation group of n elements;
Matd;Mat,
xrn
nrnm
Groups and algebras
Wn An
w
- alternating group (of even permutations) -Weyl group of a semisimple Lie algebra (root system) The standard notations in the upper-case bold are reserved for the
classical Lie groups: Sq3) S q p ;q); S q n ) ;SU(n);9 4 2 ) ; SL,(C); SL(n); SL(3);GL, etc., and the corresponding lower-case bold denote their Lie algebra En = Wn b S q n ) - Euclidian motion group; (3, - its Lie algebra Wn - Heisenberg group - Cartan (diagonal) subgroup/subalgebra of a semisimple Lie group H;8
448 B*;B* (li = st @
List of notations - Bore1 subgroup/subalgebra of upper/lower triangular matrices in SL, - Cartan decomposition
Function spaces; algebras; distributions - space of continuous functions e(A) P(31L); em(&)- rn-smooth (m-smooth) functions - compactly supported functions - Lebesgue spaces of integrable functions - Sobolev space of order s - distributions - algebra of all (bounded) operators of a Hilbert/Banach space 'Y - Weyl algebra on Wn, generated by all multiplications differentiations Transforms; Operators - Fourier transform and its inverse
- Radon transform and its dual - operation of symmetrization,
Tm('Y)+!fm('Y)
- wedge product of vectors, tensors, differential forms - Laplacian (Laplace-Beltrami operator)
- spherical mean of function f at {z}, of radius r Representations - intertwining space of representations S and T - the commutator algebra of T - tensor (Kronecker) product of representations T and S - direct sum of representations { T 3 } - direct integral of representations (2'': z E %} - adjoint representation of Lie algebra/group
and
Index.
A ............................... ............................................
Action-angle variables 373 Action functional (Lagrangian) .................369 46 Adjoint action Algebra: Lie ....................................................... 46 249; 256 enveloping (of Lie algebra) group (convolution) algebra 24 commutator algebra ............................. 37 Angular momentum (operator) 176. 276. 379
.......... .................. ..
Classical mechanics system .................369. 385 Clebsh-Gordan coefficients ........... 130.181. 240 151 Coboundary .............................................. Cocycle ...................30,58,138,150,275,285,306 150,271,292,302 Cohomology ........................... Commutator 38,40,45,138,258 algebra, Com(T) 48,52, 154 (derived) group G' ...................... (derived) Lie algebra ................ 53,192.207 47.50, 283 (Lie bracket) [A; B ]..................... Compact form (Lie algebra) .................. 193,208,248 19,25,41,63,126,157,170 group 39,63,425 operator Completely integrable hamiltonian system 374 Conformal map ................................... 13,109,365,377 group/algebra ...................... 15,21,305,366 432 Connection (affine) ................................... form ............................................. 296. 435 Conservation laws ..................................... 378 Conserved integrals ................373,378,385.393 Continuity (of representation): .................... 28 27 Convolution ................................................ Convolution algebra .................................... 28 389,411 Coulomb potential .............................. 353.432 Covariant derivative ........................... 241,353,358,434 Curvature ..............................
............
................... .....................................
B Bessel function .................... 73.78.82.89.96. 265 89. 103 differential equation ........................ potential .................................... 73.74. 111 Bore1 subgroup ............ 20.141.210.269.305. 361 subalgebra .............................. 207.268. 364 Bundle. vector. G-invariant .......... 138.149.258. 267 115.227.344. 359 tangent/cotangent holomorphic line .............. 267.271.293.296 Burnside-von Neumann Theorem ................ 39
...........
C Canonical commutation relations (CCR) ...... 275.375 variables ................................. 291.370.374 295.302. 371 2-form .................................... 373.398 transformation .............................. 372 volume ............................................... Cartan automorphism 241 167.176.189.204. 310 basis .......................... .192. 242 classification ................................ .318. 322 decomposition ............................. subgroup/subalgebra ....... 174.197.206. 242 Casimir (central element) ...... 250.330.344.413 52 Central ideal ............................................... extension (of group. algebra) ..150.274.285 131.223 projection ..................................... potential (force)............................ 379.385 Central Limit Theorem ............................... 85 Character (dual) group ........................... 37.61.69. 83 of representation ........................ 38.40. 125 infinitesimal 250.332 formula ..................... 261.273.280.298. 314 Characteristic ideal (of Lie algebra) Classical Lie groups: .................... 21.41.54. 191
....................................
.................................
D Decomposition: 40. 43 primary (of repn) ............................ spectral (operator) ........... 184.342.412.423 of representation ................ 39.130.142. 175 130. 181 tensor (Kronecker) product 39.225.235. 249 Degree (of representation) Derived subgroup. subalgebra. 53.191. 202 series ....................................... Derivative: covariant ...................................... 431. 437 296 exterior ............................................... fractional .............................................. 78 47. 302 Lie ................................................. Direct sum/integral ................................ 33. 40 Discrete groups .................................................. 22 308.314.339.362 series representation 17.147.149 Dihedral group Dual object (of group) 41.59.134.152.260. 317 (commuting) pair of algebras ........177.225 120 pair of manifolds ................................. pair of Lie algebras/symmetric spaces 241
...........
........
......... ............................... ...
.
Index.
450
E Envelope of Lie algebra ................. 249.256. 330 Equivalence (of representations) 42 Euler angles 161. 171 -Lagrange equations ...................... 369. 430 386 rigid body problem 49 Exponential map
.................. ........................................... ............................. ........................................
F
.................................. .....................................
Factor-algebra/group 53 62. 79 Fourier analysis algebra A ............................................. 64 64. 69 integral/series .................................. 62 transform ............................................. Frobenius reciprocity ............. 142.223.252. 34 1 Function 81.111.117. 184 harmonic positive-definite 86.88. 424 characteristic (of random variable) ....... 86 173.250.328 spherical ................................. Function spaces: L2;Lp; e; e, e, em;e" 27 Sobolev 36, 36. .......................... 27.57. 73 Fundamental solution (Green's function) 94 348 Fundamental region .................................. Fundamental (homotopy) group .... 51.349.358
........................... .........................
..
.....................
....
G Gauss d e c o m p ition ........................... .31. 211 Gaussian function (normal distribution) 66.71.87 kernel (semigroup) ............. 96.116.188. 283 Generator (infinitesimal) of group action ....48 Generating function (for polynomials) 189.237 49.120.188.350.377. 430 Geodesics Geodesic flow 344.358.393.408. 415 Green's function (see Fund solution) Group: 61. 65 abelian ............................................ 17.133.141.147 alternating ......................... affine ........................ 14.20.31.147.153. 264 classical (Lie. matrix) ............ 21.41.54. 191 crystallographic .................................... 22 17 cyclic .................................................... 17.147.149 dihedral Euclidian motion ........ 14.35.41.95.265. 294 finite 17.83. 199 Lorentz ............... 14.2 1.4 1.147.266.305. 361 20.53.191.274.292. 303 nilpotent
.....
..................... ...................
.
.
................................... ......................................... ................
one parameter (of operators) .................49 orthogonal (see Orthogonal) Poincare (see Poincare) 18.143 polyhedral ...................................... 54.191.363. 367 simple; semisimple 13.46.54.192.305 special linear. SL symmetric (permutation) .....13.34.199. 218 .........21.55.193.243. 284 symplectic. 20.53.191.274.292. 303 solvable transformation ...................................... 13 unitary (see Unitary) Group algebra (see Convolution)
............. ...........
wn).
..................
H Hamiltonian: classical ................................. 291.359.370 quantum 115.276.291.358 370.372 vector field/flow ........................... Hamilton's principle of minimal action 369 71 Hausdorff-Young inequality ......................... Harmonic 281.285.373 oscillator ................................ function (see Function) 175.184.253 polynomial ............................. Heat equation ............................................. 92 semigroup/kernel ..................... 96.185.353 invariants ..................................... 352.357 Heisenberg commutation relations (CCR) 276.278.411 20.41.53.154.274. 294 group ........................ uncertainty principle ...................... 83.272 Hermite polynomials/functions .................. 281 Huygens principle ................................ 101. 187 Hydrogen atom ................................... 411.420 Hyperbolic (Poincare) plane/space .. 15.25.123.244. 348 elements (in SL, ) ................... 313.321.335 (wave) differential equation ........... 93. 104
..........................
......
I Imprimitivity system (Mackey) .... 149.303.307 Indefinite (Minkowski) product ................... 14 Independent Random variables ................... 85 Induced action/representation .............. 30. 137 holomorphically induced ........ 267.296. 309 Inertia tensor/moments ............................. 387 Integral (conserved) 383.378.388.394 428 operator 71 Interpolation (Riesz) ................................... Intertwining: operator; space Int(T;S); .......42
......................
..............................................
Index . number ................................................. 42 Invariant (Haar) measure .24.162.227.308.319 subspace ............................................... 37 vector field (right/left) 48 polynomials ............................ 121.251.256 Inversion/Plancherel formula. measure commutative groups 63.65 compact ............................................. 131 general ......................................... 297.300
...........
.........................
........................
942);S q 3 ) ...........................................
........................................ ............................... .......................................... ......................................... ......... Sl42);S q 3 ) ...........................................
302
affine group 265 Euclidian motions 266 Heisenberg 280 SL, 318.326.362 Irreducible representation ............................ 37 of compact/finite groups 126.132.144
165 compact Lie groups ............................ 206 Heisenberg and semidirect prod ....259.278 SL, ......................................... 307.311.364 Iwasawa decomposition .................312.318.363
J Jacobi identity (in Lie algebras) ...................... polynomials ........................................
46 172
451 functions .......................... 173.178.180.360 transform (canonical variables) 370.377 Levi-Malcev decomposition ..................55.193 Lie group/algebra .................................. 20.46 classical 21.54.393 bracket 46.49.435 derivative (of v. field) ...................302.432 Theorem ............................................. 207 Log (inverse of expmap) 49 Lorentz group 14.21.41.266.305.361
.....
..................................... ......................................
............................. .................
M ........ ......................
Matrix entry (of repn) 29.40.126.171. 340 Mellin transform 88.306.327.337 Metric: G-invariant ........... 227.241.245.331. 388437 hyperbolic 13.147.266.394 Riemannian 13.15.430.434.353. 372 Pseudo-Riem . (Minkowski) ...... 13.147.266 26.273 Modular function ................................. Moment urn angular .......................... 176. 183.276. 379 380.389 map .............................................. operator ........................... 430.276.287.381 variable ................................. 370.379.381 Multiplicity (of irreducible i~ in T) ............... 40 Multi-polar coordinates ............................. 244
.......................... ............
N
K Kernel: reproducing ........................................ 288 integral kernel (of operator) 126.428 Poisson ........................... 81.97.98.106.346 Kronecker (tensor) product ................................... 33
..........
Nilpotent algebra/group (see Group) Noether Theorem ...................................... 378 Norm: Hilbert-Schmidt 33.128.428 trace-class 33.36.126.428 Sobolev (see Sobolev)
....................... ............................
0
L ............................................... .......... ........................................ ............................... .......... .................. .............................................
Lagrangian 369 Laguerre polynomials/functions 413.420 Laplacian: 73.105 on R";T" spheres S2;S" 176.184 symmetric spaces.................... 244.247.255 hyperbolic (Poincare) plane 332.344 Riemann surfaces r\H 352.358 Laplace-Beltrami operator .................. 243.332 Lax pair 394.400 Legendre: differential equation ...................... 173.178 polynomials .................................. 172.189
Operator compact 39.74.426.428.346 differential ........58.92.112.121.171.176. 432 elliptic 92.98.113.261.354 Fredholm ............................................ 114 Hilbert-Schmidt (see Norm) integral (see Integral) intertwining (see Intertwining) pseudo-differential ............................... 112 SchrGdinger..................93.225.276.281.411 trace-class (see Norm) unbounded (closed) 56.58.332 Orbit
........................ ..........................
..........
Index.
452 co-adjoint ............................... 290.294.297 method ............................................... 290 Orthogonal groupsla1gebras:qn); S q n ) ; S q p ; q )
Quaternions ................................... 13.161.163 Quaternionic-type representation .......... 38.157
.................. 14.21.54. 161.174.196.227.259. 434
R
Orthogonality relations (for matrix entries/ characters) 126.134.149.173.232.340
Radical (of Lie algebra) ........................ 55.193 Radon transform ................................. 120.182 Random variable 85 walk 84.85.87.145 Rank of Lie group/algebra ............................... 197 symmetric space 121.241.246 tensor ........................................... 217.368 Reducibility (complete. of representation) .193 Representation adjoint .................................. 34.46.52.192 co-adjoint ................................... ..256.293 contragredient (dual) ............................ 34 induced .................................... 30.137.257 holomorphically induced 269 irreducible ............................................. 37 primary (see Primary) projective (see Projective) regular 24.29.33 tensor (Kronecker) product of ...............33 unitary ................................................. 29 Root (of Lie algebra); ................................ 197 system ................................................ 197 vector ................................................. 197 Runge-Lenr symmetry ................. 389.390.408
.................
......................................... .....................................
P
.................................
Paley-Winer Theorem 79 Peter-Weyl Theory ................................... 126 Plancherel formula/measure (see Inversion) Phase-space 370.374.380 Poincare group P. ................. 14.41.55.147. 266 (Lobachevski) half plane ........ 15.25.51.430 disk ...................................................... 16 Poisson kernel (see Kernel) summation formula ..... 66.109.187.326.354 (Lie) bracket (see Symplectic) Polynomials (orthogonal) Legendre; Jacobi (see Legendre. Jacobi) Gegenbauer 185.188 Hermite 281 Laguerre (see Laguerre) Potential Bessel (see Beasel) energy .......................................... 369.379 Riesz ............................. 78.82.122.311.362 of Schriidinger operator (see Operator) 393 N-body Primary decomposition (see Decomposition) projection ............................................. 40 subspace ................. 40.43.130.223.329.344 representation ................................. 40.130 Product: direct of groups/algebra 21.43 semidirect (of groups/algebras)
..................................
.................................. .............................................
....................
.....................
.........................................
...............................................
...................
............ 15.21.35.46.52.55.133.147.200.260. 411
tensor (see Tensor) Kronecker (see Kronecker) Projective space representation
............................ .........................
........... ....................................... .............................. ......................................
121.141.296 150.284.289
115.276.385.409 Quantum hamiltonian state 276.291.409 observable 276.291.409 Quantization (procedure) .291.296.409.411.414 condition 292.293
S Selberg (trace formula) .............................. 338 Schur's Lemma ........................................... 37 Type-criterion ..................................... 157 Semisimple Lie group/algebra .............. 54.192 Semidirect product (of groups) ............. 15.147 (of algebras) .................................. 55.193 Signature (of representation) ..................... 212 Simple Lie group/algebra 54.192 Schriidinger operator (see Operator) Solvable group/algebra ........................ 53.191 Space: homogeneous 24 intertwining (see Intertwining) Sobolev 27.57.73 symmetric (see Symmetric) Spectral decomposition (unitary. s-a operator).424 subspace/projection .................38.279.424 Spherical functions 173.250.328 harmonics 143.174.184.253
.....................
........................................ ........................................
.
....................... ........................
Index. Subspace: cyclic ............................................... 40. 44 37 invariant of smooth vectors ................................. 56 primary (see Primary) spectral (see Spectral) 15 Stabilizer (isotropy) subgroup ..................... 290 subalgebra .......................................... Stereographic map ......................... 16.365. 415 92.112. 183 Symbol (of differential operator) Symmetric space 13.243. 437 Symmetric (permutation) group .13.44.200.218 Symmetry group of regular polyhedra 17.18.144 Euclidian space (see Group. Euclidian) Minkowski space (see Poincare) hyperbolic space (see Lorentz) Symplectic group/transform (see Group) (Poisson) structure .......... 290.296.371. 373 manifold .......................... 291.297.371. 409
..............................................
............................
....
......................
T Tensors: 159.164.175.214 symmetric/anti.sym 217 mixed ................................................. Tensor product of 428 matrices/operators vector spaces; algebraa .........33,45,175,428 34,43,152,181,218,223 representations 109,187,352 Theta-function
........
.............................. ....... ..............................
U Unitary operator/matrix/group .......... 16.20. 2 1. 161 representation (see Representation) Unitary trick (see Weyl)
V
.....................
Variation (of functional) Vector bundle (see Bundle)
378. 384
W Wave equation ............................................... 93 kernel/propagator 96.99 Weight in function-spaces ........................... 32.73 for orthogonal polynomials ..... 172.206. 420 (highest) of representation ............ 206. 231
............................
453 diagram ........................................ 206. 208 Weyl algebra W ........................ 174,183,284,410 chamber ................................. 228,230,246 character formulae 181,284 group 199,206,242 177,183,218,224 invariants 114 principle (volume counting) unitary trick ................................. 167,193
........................ ..................................... ........................ ................
Y Young inequality ......................................... Young tableau/symmetrizer ......................
72 219
2 Zeta function .............................................. 89 (of operator) .................................. 90.352
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