TOPICS IN THE THEORY OF ALGEBRAIC GROUPS
Notre Dame Mathematical Lectures. Number 10
Topics in the Theory of Algebraic Groups
James B. Carrell Charles W. Curtis James E. Humphreys Brian J. Parshall Boris Weisfeiler
University of Notre Dame Press Notre Dame London
Copyright© 1982 by University of Notre Dame Press Notre Dame, Indiana 46556
Library of Congress Cataloging in Publication Data
Main entry under title: Topics in the theory of algebraic groups. (Notre Dame mathematical lectures; 10) 1. Linear algebraic groups. 2. Algebraic varieties. I. Carrell, James B. II. Series. QA1.N87 no. 10 [QA564] 510s [512' .33] ISBN 0-268-01843-X Manufactured in the United States of America
82-17329
Preface In the fall semester of 1981/82 Professor A. Bialynicki-Birula from the University of Warsaw was guest professor at the mathematics department at the University of Notre Dame.
He gave a course entitled
"Algebraic group actions on varieties".
In conjunction
with his visit, Professors J. B. Carrell, C. W. Curtis, J. E. Humphreys, B. Parshall and B. Weisfeiler came for a week each and presented survey lectures on topics in the theory of algebraic groups.
In view of the superb
quality of both the course and the surveys, it was decided to make them available to a wider mathematical audience. This is the raison df§tre of this book.
Due to political
upheavals in Poland at the time, it was impossible for Professor Bialynicki-Birula to include a survey on the material of his lectures.
This is most unfortunate, of
course, and rather ironic for it was Professor Bialynicki-Birulafs course which served as catalyst for this volume. On behalf of our colleagues of the Mathematics Department of the University of Notre Dame, we wish to thank both the National Science Foundation and Professor 0. T. O'Meara, Provost of the university, for their
vi generous financial support.
Preface Professors Curtis, Humphreys,
Parshall and Weisfeiler were in addition supported by research grants of the National Science Foundation during the preparation of their lectures; Professor Carrell was similarly supported by the Natural Sciences and Engineering Research Council of Canada.
Finally, a personal word of
thanks to all the speakers for their visit and the cooperative way with which they prepared their manuscripts.
Alex Hahn Warren Wong
Notre Dame, Indiana July 1982
Contents
HOLOMORPHIC C* ACTIONS AND VECTOR FIELDS ON PROJECTIVE VARIETIES by J. B. CARRELL
1
1.
C* actions on protective varieties
3
2.
The B-B decomposition
7
3.
The Homology Basis Theorem
13
4.
A generalization of the Homology Basis Theorem
17
5.
Holomorphic vector fields and the cohomology ring
22
6.
Borelfs Theorem and holomorphic vector fields
25
7.
Holomorphic vector fields with one zero
29
8.
A remark on rationality
31
9.
Closing remarks
33
References
35
A DUALITY OPERATION IN THE CHARACTER RING OF A FINITE GROUP OF LIE TYPE by C. W. CURTIS 1.
Duality in the character ring of a finite Coxeter group
39 41
2. 3. 4.
Truncation and Duality in the character ring of a finite group of Lie type
48
Main theorems on duality
59 f
Applications to Springer s Theorem and a conjecture of MacDonald
64
Some additional results
68
References
71
ARITHMETIC GROUPS by J. E. HUMPHREYS
73
5-
1.
Lattices in Lie groups
73
2.
Finite generation and finite presentation
80
3-
Normal subgroups
86
4.
Normal subgroups (continued)
91
References
97
MODULAR REPRESENTATIONS OP ALGEBRAIC GROUPS by B. J. PARSHALL
101
1.
Elementary theory
102
2.
Induced modules
108
3.
Infinitesimal methods
113
4.
Rational cohomology
121
5.
Lusztig's conjecture
127
References
131
ABSTRACT HOMOMORPHISMS OF BIG SUBGROUPS OF ALGEBRAIC GROUPS by B. WEISFEILER
135
1.
Motivation
136
2.
Evident restrictions
140
3.
Isotropic semi-simple groups over fields
141
4.
Short remarks on OfMearaTs method
147
1
5.
Tits
results on homomorphisms of Lie groups
149
6.
Rigidity, strong rigidity, and superrigidity of lattices
152
7-
Quotient groups of lattices
163
8.
Lattices in PSO(n,l) and PSU(n,l)
167
9-
Concluding remarks
174
References
176
Holomorphic C* Actions and Vector Fields on Projective Varieties JAMES B. CARRELL
In this series of talks, I will discuss two ways of relating the topology of a smooth projective variety
X
(over ffi ) with the fixed point set of a one dimensional group of automorphisms (either X .
1 = 0., SL or I* = G m ) on These ideas are summarized in the following diagrams:
/-.x JPixed point set X I [of a £E* action on XJ
I Integral homology I (groups Hs(X,Z) J
Zeros of a holomorphic vector field (2) on X with isolated zeros
Complex cohomology ring H"(X,ffi)
If X
admits a (C* action with
X
finite and nontrivial,
then X
also has a holomorphic vector field with isolated
zeros.
The connection between the diagrams (1) and (2) is
not clear, however, and seems to be one of the basic open questions in this area (c.f. §2.5).
1
2
Holomorphic C* Actions and Vector Fields
This paper is divided into two parts, the first four chapters deal with
E
actions, and the next five with holomorphic
vector fields.
I have tried to keep the presentation on
a nontechnical level. have been included.
Several examples but very few proofs A few unsolved problems have also been
mentioned. I would like to thank the University of Notre Dame for support under the Kenna Lectureship Series.
J. B. Carrell
1. ffi* ACTIONS ON PROJECTIVE VARIETIES
A good place to begin a discussion of
£
actions is
with the fact that a holomorphic representation of ffi* on a finite dimensional complex vector space V , say
p : IE* •*•
GL(V) , induces a holomorphic action of ffi* on V , that is a holomorphic map
u : ffi x v -»• V
and p(x1X2,v) » u(X1,u(X23v)) . X-v
such that
y(l,v) = v
(We shall often write
for y(x,v) when speaking of affi*action.) The fact
that
p
is a linear representation means that each
X e ffi*
preserves lines through the origin in V , so u descends to give a holomorphic action of ffi* on 3P(V) ,
V :ffi*x jp(v) -*• I>(V) . A basic result about finite-dimensional representations of ffi* says that
V
decomposes uniquely into a direct
sum of weight spaces V, 9 kg 2, i.e. V = © Vk(k e S) , # where v € V. if and only if u(X5v) = Xkv for all X € ffi The k e S of the
1C
such that V, £ {0} are called the weights action on V .
By a holomorphic variety
1C
action on a complex protective
X , we mean a holomorphic map
n : OJ* x x •*• X
satisfying the properties mentioned above. It is well known that any holomorphic action of a one parameter subgroup
1C
on fl33Pn arises through
X : ID* •* 3PGL(n,E) , hence up to
4
Holomorphic C* Actions and Vector Fields
projective transformation a
I*
action on ffinPn is of the
form
x.cz 0 ,z 1 ,...,z n ] = cx a °z 0 ,x ai z 1 ,..,x an z n ]
(i.D where
aQ,a.,,...,a
c Z2 .
One frequently encounters the situation in which
X
is an invariant subvariety of a ffi3Pn with respect to ffi* action of the form (1.1) on map
01 x x •* X
defines a
1C
Example 1. The variety ffiUP3 with action Example 2.
Q23Pn . In this case the natural action on
X.
V(Zg5 + Z^Z*0 + Z-^zb
in
X-[Z^Z-^Z^Z-] - [X3Z0,X10Z13X5Z2,Z3] .
Grassmannians. Any ffi* action on IDn
permutes k-planes through the origin, hence defines a 03* action on the Grassmannian see that the image of Plucker imbedding is
Gk(ffin) .
Gk(ffin) in 03
IP(Affi )given by the representation X •* Akx . Notice that any ffi action on i.e. points
x
so that
IP(AkEn)
under the
invariant with respect to the ID*
n
action on
It is not hard to
X-x = x .
k
exterior power
CE3Pn
has fixed points,
Indeed the connected
components of the fixed point set are the linear subspaces of
Q3IPn
which correspond to eigenspaces of the induced
linear action on ffin .
Clearly any closed invariant
J. B. Carrell
subset
K of ffi]Pn has fixed points: namely if
x eK
then
lim X-x and lim X-x are both fixed points in K . X-K) X^« For convenience we set xn0 = lim X-x X-M)
and x
= lim X-x °° X->»
What is suggested by this construct is to consider the m*
connected components of the fixed point set X these are labelled X.
(suppose
X,,...,X ) and for each such component
its "plus and minus cells"
X. and X" , namely
X* - {x e X : XQ c X±} , and X~ - {x€ X : x^ e X±}
These "cells" turn out to be the fundamental objects that lead to connections between the topology of X and the ffl» topology of X . We will frequently refer to them simply as B-B
cells after A. Bialynicki-Birula who first proved
the main structure theorem fo'r them
CB-B] (which will be
discussed in §2). Example 3- Let ffi* act on ffilP2 by CZ0,XZ1,X2Z2] .
X-CZ0,Z-L,Z2] =
Clearly the fixed points are [1,0,0] ,
[0,1,0] , and [0,0,1] . Then, [1,0,0]+ = E3P2-V(Z0) , [0,1,0]"*" = V(ZQ) - {[0,0,1]} and [0,0,1]+ - [0,0,1] . In each case the plus cell is an affine space-
Holomorphic C* Actions and Vector Fields
Example 4. Consider the action on X = Gp(ffi ) induced by the action ii on 03 where vectors
X- (z0,z.,,z2,z.-) = (X zQ,X z-^X aQ > a.^^ > a2 > a^ .
z
p» x
z
o)
For a pair of independent
jh
u,v e ffi , let
denote the 2-plane they
span. We will compute <e.,,e«>
9
where
denotes the standard basis of ffi4 . lim X-V for 2-planes of the form
{e. : 0 £ i £ 3}
It suffices to consider V = < auneun ^-cue., , 3neun + 1 1 - -
B-j_e.. + 32e2+ ^QeQ> where X-V
(an-a,)
,
(a, -a..)
Since
aun > axn > ad0 > JaQ , it follows that lim X-V = X-^0 = <e- L ,eo>. To give an invariant characterization of <e..,e~> , we recall the definition of Schubert cycles in G2(E ) . that
(See also CKL]).
If k-i* b 2
are
1 <; b-j^ < b2 <> 4 , set 1
Inte
se3?s so
J. B. Carrell
7 i
Note that if then
o
o
LL
li
33 c ffi c ffij c ffi is the standard flag in
where
c^ 5^ 0 , then
V / 0(2,3) •
V £ 0(1,4)
and if
3g ? 0 ,
Thus the above calculation shows that
<e 1 ,e 3 >
+
» 0 ( 2 , 4 ) - 0(1,4) - 0 ( 2 , 3 ) .
<e-,,eo>
T
= 0(2,4) .
Gp(E )
ID .
It follows that
The Schubert cycles
o(b-.,bp)
on
are ordered by inclusion via the lexicographic
order on the
(b.,,bp) .
By a similar argument,
This is shown in the diagram below. <e.,e.>
= 0(1+1, j+1)
if
0 <. i < J * 3 .
dimension
0
1 2 3 )-ffi-< 0(1,3) V
4
0(2,4) > 0(2,3)
A useful feature of this diagram (sometimes called the Hasse diagram) is that one can see a free homology basis of any 0(1,3) rnamely itself and the Schubert cycles that precede it in the diagram. 2. THE
B-B
DECOMPOSITION
The structure theorem of Bialynicki-Birula [B-B] describes the structure of the plus and minus cells on X when
X
is a complete, smooth variety with
over an arbitrary algebraically closed field. Sommese CCS-,] and Fujiki
G
action Carrell and
[Pu] showed that this theorem goes
through without change to compact Kaehler manifolds .
g
Holomorphic C* Actions and Vector Fields
Recently, however, Sommese has found an example of a compact Moishezon manifold
X with a
E
action for which the B-B
decomposition X = UX. exists but not all of the canonical j maps X. •> X. , x •* XQ , are continuous CS2] • For a smooth projective variety X
X with fixed point components
X^,...,
the theorem says the following: THEOREM 1. (i) For each i = l,...,r , the natural
map
p± : X* ->• X± , x * XQ , is the projection of a holomorphic
fibre bundle whose fibres are all ffi* equivariantly isom. morphic -to a fixed £ d . (ii) 1C
In fact, if x € X, , then p^Cx) is
equivariantly isomorphic to
TX(X)/TX(XI) with ffi*
action induced by the representation
X H- dXx of C* in denotes the differential of the map
GL(T x (X)) .
y •* X • y
(dXxY at x.) (iii)
Zariski closure.
X^
is a Zariski open subset of its
Hence X.
a closed subvariety of X
(the topological closure) is containing
X. as a Zariski
open.
IP* X-j^ , of X"'
(iv) There exists a unique component, say + so that x£ is Zariski open in X . X][ is
called the source of X . A completely analogous result holds for the minus
J. B. Carrell
9
decomposition of X . that
X"
The distinguished component
is Zariski open in X
X. so
is called the sink of X .
We will always label the sink as Xr . COROLLARY. is rational.
Suppose either the source or sink of X
Then
X
is rational i.e. X
is birationally
n
equivalent toffi3P . For a proof see isolated source vector space
CCS.,3 •
x , then
X
*n "the case, say, of an is a compactification of the
N ({x}) .
An important, but easy to establish, fact is that if X
is a smooth invariant subvariety of JD3Pn , then there
exists a Morse function of increasing on the
f on X that has the property
3R
orbits in X .
In fact, let V
denote infinitesimal isometry associated to let f
n denote the Pubini-Study metric on by solving the equation
restricting
i(V)Q = dP
on
S
cffi*,and
D3IPn . One finds D2IPn
and then
P to X . Let O
"ET 7 ,...,£ 7 ~I — J?L4 J Un II
as long as coordinates
[Zg,...,Z ] have been chosen so that
X-[Z0,...,Zn] = Cxa°Z0,...,xanZn] .
The following are not
hard to verify using the contraction identity (i) f = P | X
Q
To I /Y/Ii7l ^Ij la..I 7\ £. J_ _L J.
i(V)fl = dP :
is a Morse function on X whose critical
10
Holomorphic C* Actions and Vector Fields
submanif olds are (ii)
f
X-, , . . . 3X ;
is strictly increasing on the
H+
orbits of
nonfixed points; (iii) then
if
X is not contained in a hyperplane of ffiIPn ,
X]_ = XnCsource of
DJ3Pn )
and
Xp = Xn(sink of ffi3Pn ) ;
and
(iv)
the Morse index of f on X±
is dim^N"^) ,
x e X., i where N~(X. x i ) denotes the subspace of T_(X) x generated by vectors of negative weight (it is actually a subspace of the normal space to X. at x ) . In the compact Kaehler casea assuming
XD3* 7* 0 , there
is a Morse function satisfying (i),(ii), and (iv) due to Prankel
CPr]
and Matsushima.
Its importance here is in
guaranteeing f that f there i is no * sequence of points lle in in X - X so that (x.) and (x) component for i=l,...,k-l and lie in the same component.
(XI)Q
anc
*
xI la...,x, K tne same (xir)« also
J. B. Carrell
11
Examples of such "quasi-cycles" are known in the non Kaehler case (see CJu] and
CS23) •
The Prankel-Matsushima a different manner in Example 5.
Morse function is applied in
[At] .
(G/B) .
Let
G
be a semi-simple
algebraic group,
B
torus in
W = NG(H)/CQ(H) be the Weyl group of
H
in
B
G .
and
a Borel subgroup,
G/B
a fixed maximal
It is well known (see e.g. [H]) that
a smooth projective variety and that on
H
by left translation:
(G/B)H = {gB : g e NQ(H)} Thus the correspondence
and
H
G/B
acts holomorphically
v(h,gB) = (hg)B . gB
g •> gB
is
Moreover
depends only on
g e W.
sets up a one to one IT
correspondence between
W
and the fixed point set
and we may unambiguously refer to subgroup
X : ID* •* H
under the action
G/B
y(t,gB) = (X(t)g)B
associated to
source of
X
is
A one-parameter
is called regular if
By a theorem of Konarski of
wB .
X
is
(G/B) ,
(G/B)ffi
= (G/B)H
of ffi* .
[Kon] , the plus decomposition B
invariant provided the
eB . This can be used to identify the
associated plus decomposition and the Bruhat decomposition. In fact, for each (1.2)
w e W,
(wB)+ = B(wB) (the B To see this note that
orbit of
BwB c (wB)
wB e G/B)
by Konarski!s
12
Holomorphic C* Actions and Vector Fields
result.
Since the plus cells
Bruhat cells
(wB)
are disjoint and the
B(wB) cover G/B , the proof of (1.2) is
complete. Another treatment of the Bruhat decomposition using 1C
actions appears in
[A-,] .
We now turn our attention to possibly singular projective varieties
X
invariantly imbedded in a ffilPn . For
example we can now consider actions on Schubert cycles and, more generally, on the generalized Schubert varieties which are closures of the plus cells.
Although the
X. B-B
decomposition is no longer always locally trivial, one can single out a natural class of actions (which always exist in Schubert varieties) on which the still nice enough.
B-B
To do so, suppose X
decomposition is is endowed with
an analytic Whitney stratification whose strata are ID* invariant.
(For example, the canonical Whitney stratification
of X
is always invariant
on
is called singularity preserving as
X
singularity preserving as x eA
implies
means that
XQ
XQ e A
CW]). The Whitney stratification (resp.
X •*• « ) if, for any stratum
(resp.
A,
x^ c A ) . Intuitively, this
is just as singular as x
Example 6. Let
X -*• 0
is.
Y
denote the cone in ffilP^ over a 2 smooth algebraic curve X c ffilP with vertex x - [0,0,0,1] € ffiE^ -ffilP2. The natural action of ffi*
J. B. Carrell
on Y
13
induced by the action
[ZQ,Z13Zp,XZ-] on ffiH?^ has source Y
can be stratified with strata
X
and sink
{x} and Y - {x} and
this renders the action singularity preserving as Since
{x} is an isolated singular point on
is not singularity preserving as stratification of Y . Y - {x} and
{x> .
X •* «
X •> 0 .
Y , the action
for any Whitney
Note that although the cells
{x} of the plus decomposition are locally
trivial affine space bundles, the minus cell
x~ = Y - X
is not .
The next theorem partially answers the question of what structure a singular invariant subvariety must have. The proof will appear in CCG] THEOREM 2.
If
X
is a
D3
invariant subvariety
ffiIPn whose ffi* action is singularity preserving as
X -> 0
with respect to some invariant Whitney stratification of X then for each connected component X. of X ffi* , 9
the natural projection
p. : X. •> X. renders xl" a j j J j topologically locally trivial affine space bundle. The fibres are biregularly (and equivariantly) isomorphic to m +) . some ffiJ1 (depending only on X. 3.
THE HOMOLOGY BASIS THEOREM
Recall that the classical Basissatz of Schubert
14
Holomorphic C* Actions and Vector Fields
calculus basis for
CKL] says that the Schubert cycles form a homology G, (ffin) .
To be precise, fix a flag ffi1 c ffi2 c
XV
... c JCn in
JDn .
Then for any increasing
(a,,...,a.) of integers so that
k-tuple
1 < a., < a2 < ... < a, £ n ,
set
(1.3)
0(a1,...,ak) = {V € Gk(En) : dimffi(V n ID*1) ^ 1}
The ft(a.,,...,ak) are projective varieties called Schubert cycles (or Schubert varieties) whose associated homology classes in H.(Gk(ffin),ZZ ) we denote by The Basissatz says: For each m with the of
[Q(a.j,... ,ak)H with
[&(a,,...,a. )] .
0 £ m £ k(n-k) ,
J. 1(a. - j) = m
form a basis
H2m(Gk(ffin),ZZ) . Even showing that
n(a.,,... ,ak) is a projective
variety is somewhat complicated (see e.g. [KL]).
However,
by a calculation similar to that in Example 4, there exists a ffi action on
Gk(ffin) so that
\ X. = fl(a.,,...,ak)
for some component
X. . Consequently, by the theorem of J Bialynicki-Birula, n(a-L,...,ak) is automatically a subvariety of Gk(fl3n) . A more interesting fact, however, is that there exists an analog of the Basissatz for any smooth (and many singular) — play # projective variety with ffi action in which the X.T j a role similar to the role played by the Schubert cycles
J. B. Carrell
15
(with respect to a fixed flag) in Xffi
is isolated, the
xt
Gk(ffin) .
In fact, if
form a homology basis of
For this reason, we sometimes refer to the X. as generalized Schubert varieties.
HB(X,ZZ)
(and X" )
Before stating this
generalization of the Basissatz, let us mention that using the Prankel-Matsushima Morse function in
CPr]
(see also CKob]) that
(i) bk(X) = IjtVx.^j5 «J (ii)
X
X,,...,X over
.
where
X
j=
has torsion if and only if
THEOREM 3 CCS2] . Let X variety with
f , Prankel showed
1C
dlm
]RNx(XJ)
XE
=
does.
be a smooth projective
action having fixed point components
Let m.
(resp. n.)
denote the fibre dimension
p. : X. •* X. (resp. q. : X" •* X. ) . Then there j j J J J J exist canonical plus and minus isomorphisms (1.4)
(C of
,k : •jHk.2llljUJ,B) -Hk(X,2Z)
and
(1.5)
vk : ®jHk_2n.(Xj'2;) *Hk(X'ZZ) J
By dualizing these isomorphisms to cohomology over ID and using the Hodge decomposition Hk(X,D3) = ® Hp(X,flq) p+q=k one obtains the following result. COROLLARY [CS23 .
The plus and minus Isomorphisms
16
Holomorphic C* Actions and Vector Fields
induce isomorphisms
p-m. q-ni. TT* : H p (X,fl q ) + § H J ( X j a f i °)
(1.6) and (1.7)
(Xj ,Qq-nJ )
y* : HP<X f a*> - 9^'^ By taking dimensions (over
IE ) we get
-.-
= I
J
J
which is a result obtained by several authors: independently by Luzstig and Wright
[Wr]
for isolated fixed points via
Morse theory and independently by Pujiki
[Pu]
and Iversen
using mixed Hodge structure. There are several consequences that relate the source and the sink to each other and to
X.
(a) H°(x,nq) s H°(x1,oq) = H°(X (b)
(c)
ir
there exist exact sequences 0 -> K* -> Pic(X) -^ Pic(X1) ^ 0 0 -> K" -> Pic(X) -> Pic(Xr) •> 0
J. B. Carrell
K+
where X
17
(resp.
K" )
generated by the
in
X?
is the
ZZ-module of divisors in
(resp.
X~ )
which are divisors
X. Another relationship between (d)
X and
X
Index(X) - Llndex(X.) J J
The proofs of (a) - (d) are contained in is also proved in 4.
is
CCSp] .
(d)
CPu] .
A GENERALIZATION OP THE HOMOLOGY BASIS THEOREM One can ask whether the homology basis theorem is
also true for singular invariant subvarieties inffi3Pn. The answer is, not surprisingly, no in general.
However,
for actions which we call "good", the answer is yes. Among the spaces with a good action are the generalized Schubert varieties
X.
in a smooth
of plus cells in
X
X
which are themselves unions
(i.e. there exist
!,,...,!.
so that
xt = xt u ... u X* ) due to the fact that the plus cells J _1 ^ in xt are xl" ,...,xt and the fact that, since X is J x H k smooth, the X. are locally trivial affine space bundles. 1 k The strategy for extending the (plus) homology basis theorem is to single out a class of actions with plus cells being locally trivial affine space bundles for which a plus homomorphism with natural properties can be defined.
The
18
Holomorphic C* Actions and Vector Fields
proof then uses the Thorn isomorphism.
It seems to us that
the class of good actions does not give the optimal generalization . X. of Xffi* , let r . denote the J J closure of the graph of p. : X. •> X. in X x x. 9 and let J J j j g. : r. •*• X. be the projection. 0 J J For any component
DEFINITION.
An action
ID* x x + X
if, for each connected component
is good as X -* 0 ffi* X. of X , the following j
conditions hold: (i) the projection
p. : X. •*• X. is a topologically J J J locally trivial affine space bundle, and (ii)
X. has an analytic Whitney stratification such
that for each stratum
A,
closure{(p.(x),x) e X. x x|x
where
A"1" = {x e X : X
e A+}
e A) .
The condition (ii) means one can unambiguously write r. H
for gT (I) c r. . It is easy to construct a space X J J with a point XQ in the source X., of X having the property that Y = E3P1 x 1E3P2
g" (XQ) 3 closure{ (XQ,X) :x e XQ} . Let with the action
X- (lzQ9z^1'9 Cwo,w]L,w2] ) =
,Xz..];[w,w5W]) , and let X be Y with the point
J. B. Carrell
19
([0, !];[!, 0,0]) blown up. Now take
XQ = ([1,0]; [1,0,0]) .
The reason for condition (ii) is to allow us to construct a wrong way map to define XQ
H k (X..,ZZ) -> HHk+2m (r..,ZZ) g :: H(X..,ZZ) (r..,ZZ). . If we try
g( cycle) = closure
"~ cycle) , then the point g"~(
in the above example will certainly cause a problem.
We must therefore be able to stratify
X.
so that the set
of bad points in each stratum is a subvariety of the stratum and then consider only cycles on transverse to the strata.
X.
that are
Thus a nice complex of transverse
cycles is obtained on X. that admits a wrong way chain J map into the chains of r. . In the example above we may " JC * stratify the components of X with one stratum each. Nice
0-cycles and 1-cycles in X, will avoid XQ . When a wrong way homomorphism g# exists, the plus homomorphism is defined as the composition (1.8)
Hk(Xj,ZZ)-^H
(r^ZZ)* H
(X,S)
J
u
where the latter map is induced by the projection
r. •*• X . j
We then have THEOREM 4 [CG]. If the action ffi* x X •* X as
is good
X -> 0 , then the plus isomorphisms (1.8) are valid for
all k .
Moreover, for almost every
the class of 1
Z
k
cycle
PT,( ) is represented by the
p" (z) on X .
z on X. , J k cycle
20
Holomorphic C* Actions and Vector Fields
Examples of actions that are good as (i) if X
is smooth, then any
union of plus cells in X (ii)
any
X
X •»• 0 :
X* in X j with the induced E
in which each
X. J
that is a action;
is smooth.
It is hoped that a more general setting in which the plus isomorphisms are valid will be found.
At the present,
all the examples we know of singular varieties with a plus isomorphism have a good action.
Hopefully, it will
eventually be shown that the plus isomorphisms are valid whenever the plus cells are locally trivial affine space bundles. Example 7. Let vertex
Y be, as in Example 6, the cone with
o p x e DOT - ID3P
over a smooth curve
2 X in ffi]P .
Then the action defined in Example 6 is good as not good as
X -*• « .
X -> 0
but
The plus isomorphism takes the form
H0({x» * HQ(Y) , H±(X) * H±+2(Y) , 0 s i <; 2 . These are the well known Thorn isomorphisms
CMS] .
is no minus isomorphism however if the genus of
X
There is
greater than zero. Example 8. The Schubert cycle with action induced by
X = Q(2,4) in Gp(ffi )
X- (z-^z^z^z^) = (z1,z2,z3,x""1zlt)
J. B. Carrell
21
has two fixed point components: the source and the sink
X2
D2IP
is a ffiHP containing the singular point.
The plus decomposition of X
is locally trivial, and since
and X* s EIP2 , both plus cells are smooth.
X* is X - X2
Therefore this action is good as
X -* 0 .
decomposition fails to be locally trivial. can easily verify that if V
W € Xp •
The minus In fact one
denotes the singular point
ffi2 of X , then V" « ffi2 while plane
X-^ is a
W" a ffi for any other 2
The plus and minus homomorphisms take the
form H
k-4(Xl) ®Hk(X2} *Hk(X) * W
® Hk-2(X2)
The minus homomorphism is neither injective nor surjective. It would be interesting to know if there exist examples of actions that are singularity preserving as are not good as
X •*• 0 .
X •*• 0
that
We mention a partial result from
CCG]. THEOREM 5. preserving as X
X have an action that is singularity
Suppose that for any stratum A of and for any component X. of Xffi* , either A n X. = 0 _______
or
Let
(A n X.)
X -»• 0 . —
—_
= A n X. .
w
Then the action is good as
t)
X ->• 0
We close this chapter with two questions. 1. In the case of a good action, how does the mixed
22
Holomorphic C* Actions and Vector Fields
Hodge structure on on
X
relate to the mixed Hodge structure
X1*?
2.
If
X
has a not necessarily good action with
isolated fixed points, do the odd homology groups of
X
vanish? 5-
HOLOMORPHIC VECTOR FIELDS AND THE COHOMOLOGY RING
It is a basic fact that the cohomology ring of a smooth protective variety field
V
X
with isolated zeroes
To be precise let sheaf
admitting a holomorphic vector
Z
is determined on
Z.
denote the variety with structure
0Z = flx/iCV)^1 where
contraction of holomorphic i(V)
Z ^ 0
i(V) : flp -> flP"1 denotes the p-forms to
(p-l)-forms.
Then
defines a complex of sheaves
o. which is locally free resolution of
0^
since
V
has
isolated zeros. It follows from general facts that there exists a spectral sequence with H°(X,0Z) .
E"p*q = Hq(X,Qp)
The key fact proved in
[CL-j,]
abutting to is that if
X
is compact Kaehler, then this spectral sequence degenerates at
E., as long as
finiteness of
Z
Z / 0 .
and
As a consequence of the
i(V) being a derivation, we have
J. B. Carrell
23
THEOREM 6 CCL23. If X
is a smooth projective variety
admitting a holomorphic vector field
V
with
Z = zero(V)
finite but nontrivial, then (i) Hp(X,flq) =0 2p
P
if p / q
P
H (X,E) - H (X,Q ) and H (ii)
(consequently
2p+1
(X,ffi) = 0 ) , and
there exists a filtration H°(X,0Z) - Pn = P^
where
n = dix X , such that F-jF.1
= ... =P1 c P
j_+-|
and
=F0.
having the
property that as graded rings
s
(2.D For example, if V
V2p(x-ffi)•
has only simple zeros, in other
words if Z is nonsingular, then
H (X,0Z) is precisely
the ring of complex valued functions on algebraically,
Z.
Thus,
H (X,0Z) can be quite simple. The difficulty
in analyzing the cohomology ring is in describing the filtration
P.
Example 9- For each holomorphic action of ffi* , one also has the infinitesimal generator, i.e. the holomorphic vector field
V
with respect to
obtained by differentiating the action X:
24
Holomorphic C* Actions and Vector Fields
Clearly, the fixed point set of ffi coincides with zero set of V .
One can easily show that the infinitesimal E3P11 in local affine
generator of the ffi* action (1.1) on coordinates
^ • Z-^ZQ,...,^ =zn/zo
at the flxed
point
[1,0,..., 0] is the holomorphic vector field (2.2)
V = IJ. 1 (a ± -a 0 )c 1
on Let us continue this example by exhibiting the filtration. The holomorphic vector field (2.2) on zeros if of
aQ < a.. < ... < a .
n
JC3Pn
has isolated
Also, the cohomology ring
n
DJIP , © H "l3I3P ,03), has the structure of a polynomial
ring, on one generator of degree two, truncated at degree 2n .
That generator is in fact the cohomology class of
the closed two form ft on singular and finite, H (ZjO™) o Z .
(^z\ *ffifor
each
Z
? £ Z
is nonso
is the ring of all complex valued functions on
We will let
value at
Q3IPn . Now since
(XQ,...,An) denote the function whose
[1,0,..., 0]
is
XQ
etc.
Then it can be shown
that P0 - < (!,...,!)> * H°(ffi]Pn, 03) Px and
=<(!,...,!),
(aQ,...,an)>
(a0,...,an) is sent to ft under the isomorphism.
J. B. Carrell
25
(2.1) . In general,
For example, the linear independence of the for
0 < i £ n
(aQ,...,a J
follows from the van der Monde determinant
det
n (a. -a.)
a
n
1^ 4
Example 10. Vector fields on the Lie algebra of
H .
-1-
J
G/B .
Let
We call a vector
k
v e h
denote regular
if the set of fixed points of the one parameter group exp(tv)
of
H
acting on
H
exactly
(G/B) .
G/B
Set V =
H
zero(V) = (G/B) .
by left translation is
exp(tv) | txB() so that
Clearly, the zeros of
V
Z-
are all
simple. 6.
BOREL'S THEOREM AND HOLOMORPHIC VECTOR FIELDS
For of
W
w € W
on
k.
and W
v e k , w.v
will denote the action
thus acts effectively
h* in the Usual way: w-f(v) = f(w every character line bundle (gb,a(b
)z)
-v)
on for
k
and on
f e fe* .
To
a e X(H) , one associates the holomorphic
La = (G x QJ)/B on and where
a
G/B
where
(g,z)b =
has been extended to
B
by the
26
Holomorphic C* Actions and Vector Fields
usual convention. the
Now
da e h*
and since
da , for all a e X(H) , span
well defined linear map the condition
h .
for any a e X(H) .
also denote the algebra homomorphism
algebra of ft* . Iw)
W
3 , where
3 : R = Sym(ft*) -*
acts on R , so denote by
f(0) = 0 ) .
Let 3
Sym(fe*) is the symmetric
the ring of invariants of ¥
such that
Thus there is a
0 : /i* + H (G/B,E) determined by
3 (da) = ^(1^)
H*(G/B,JC) extending
G is semi-simple,
in R
I,, (resp.
(resp. f e I,,
Borel proved that 3 is a surjective
homomorphism whose kernel is R I,, . Consequently, since R I™
is a homogeneous ideal,
3 induces
an isomorphism of graded rings
3:
(2.3)
The purpose of this section is to show how Borel!s theorem relates to vector fields. Note that H (G/B,0«) ej = w C for any vector field on G/B generated by a regular vector in ft .
We will begin with a more detailed description
of
H°(G/B,0Z) .
Define a linear map ^v : h* + H°(G/B,(?Z)
by
¥y(w)(w) = -wa>(v) .
algebra homomorphism let
I¥
such that
Then
can be extended to an
¥y : R -»• H (G/B,0Z) .
denote the ideal in R f(v) » 0 .
¥y
The ring
For any v e h ,
generated by all R/Iy
$ e ITFW
is only graded when
Iv is homogeneous, i.e. only when v = 0 and R/IvW = R/RI,w+T . However R/IV is always filtered by degree. Namely, if
J. B. Carrell
27
p - 0,1,...,
set
(R/Iv)p = Rp/Rp n Iy
{f e R : deg f <, p> .
Notice that
where
Rp -
Iy c ker¥y ; for if
(j> e I,r and 4>(v) = 0 , then for all w e W , #__(*) (w) = w » (w#)(v) - <|>(v) = 0 . In fact it is shown in EC] that for
v
in a dense open set in h9
(2.4)
¥
induces an isomorphism
?v :R/Iy + H°(G/B,0Z)
preserving the filtration, i.e. ?y((R/Iv) ) = F . Consequently, for each
p , the natural morphism
F-.p -* P
is onto. The first step in the proof is to identify elements in
H (G/B,0L7) that determine the Chern classes c., JL (L a ) for a e X(H) . To accomplish this we recall the theory of V-equivariant Chern classes. on
X
is called
V-equivariant if the derivation
lifts to a derivation map satisfying s e 0X(L) .
A holomorphic line bundle
V(f) = i(V)df , V
(i)
CCLp]
A
f e 0X ,
V e H°(X,0™) Z .
that
V e P.. and has image
c., (L)
under the isomorphism
(2.1) , and (ii)
Z * 9-
A
defines a global
section of End(0Xv(L) ®0 n 0^) s 0 ; i.e. Z X Z It is shown in
V : 0V •*• 0Y
V : 0X(L) + 0X(L) ; i.e. a ffi-linear
V(fs) = V(f)s + fV(s) if
Since
L
every line bundle on
X
is
V-equivariant if
28
Holomorphic C* Actions and Vector Fields
The calculation of
c
i(La)
is
provided by the following
lemma Lemma. of
V
Given
to
V
0(La ) so that in H°(0/8,0,, ii) , V a (wB) = where v € k is the regular vector corresponding
-da(w~ -v) to
a e X(H) there exists a lifting
V. In other words, ty (da)w = -(w-da)(v) = -da(w~ -v)
so, since the da
span
of the proof is outlined in appear in
^v(^ ) c P^ •
k ,
[C] .
The remainder
Complete details will
[A23 .
To prove Borel!s theorem (2.3), note that we have, for each regular
v c k , a commutative diagram
H2(G/B,E)
where
1
is an isomorphism, and
Consequently
3
is surjective.
a commutative diagram for each
¥
Moreover, this results in p > 1
-" F~/F.
(2.5)
is surjective.
J. B. Carrell
where
29
V
is surjeetive and
i
is an isomorphism.
3 : R •*• H*(G/B,C) is surjective. one must show that
ker 3 = R I™ .
To complete the proof, But because
dim R/R Iw =
dirnH0 (G/B,0Z) = |W| , it suffices to show that and this is surprisingly easy. then
^v(f) <= F
Thus
In fact, if
-L due to the fact that
Hence, by commutativity of (2.5),
RI* c ker 3,
f e R *v(Iy)
c
n RI^ > PO -
3(f) = 0, and Borel!s
theorem is proved. 7-
HOLOMORPHIC VECTOR FIELDS WITH ONE ZERO
So far we have considered only vector fields with simple isolated zeros, i.e. vector fields with the maximal number of zeros.
At the other extreme are vector fields
with exactly one zero. at
p € X
Suppose
and let V = £a.3/3z.
coordinates near
p.
so the cohomology ring
V
has exactly one zero in holomorphic local
Then
H (XjO,,) o L = l[zn±,... ,zn I/, \a^,...,a n H*(X,ffi) is the graded ring
associated to a certain filtration of lECz.,,... ,z ]/(a.,,... ,a ) Let's consider a basic example. Example 11. Let V on ffi]Pn generated by matrix
be the holomorphic vector field exp(tM)
where
M
is the
(n+l)x(n+l)
30
Holomorphic C* Actions and Vector Fields
The unique zero of coordinates
hence
V
is
C-, *•••,£„
[1,0,...,0] , and in the affine
at
[1,0,...,0],
H°(03]Pn, 0^) = ffiU-^/U^4"1) .
cohomology ring of ffi3Pn .
This is already the
In, fact using the theory of
equivariant Chern classes, it is shown in corresponds to
c.,(0(l))
under the isomorphism of Theorem 6 H (JDIPn, 0Z)
The existence of the grading on the fact that the
D3
CCLo] that c-j
action
follows from
A- [ZQ,Z, , . . . ,Z ] =
[Z0,AZ1,. . .,AnZn] on ffi3Pn has the property
dA-V - A-1V
which implies that the functions that define the ideal i(V)fl
are homogeneous (with respect to the action) and
hence that
H (Q3]Pn, 0Z)
is graded.
In general we know
the following THEOREM 7 CACLS] .
Let
X
be a protective manifold
having a holomorphic vector field with only isolated zeros but having zeros.
Suppose there exists a
01
(A,x) •*• A-x
X
for some integer
k 7* 0 .
Then
on
so that
H°(X,0Z)
d A - V = \f
action
is a graded ring in which the
filtration by degree coincides with the filtration of Theorem 6.
Consequently,
cohomology ring of
X
H (X,(?z)
and
P.
H*(X,Q3) , the
with complex coefficients, are
]. B. Carrell
31
isomorphic graded rings. Applications of this theorem to the algebraic homogeneous spaces
G/P
will appear in a later paper.
In the
case a regular unipotent one-parameter subgroup of
G/P G
will
generate a holomorphic vector field with exactly one zero and this subgroup will imbed in an Jacobson-Morosov Lemma SL(2,ffi) Thus
provides the
H" (G/P,02)
CJa] .
SL(2,JD) c G
by the
The maximal torus in this
02* action of Theorem 7 where
can be viewed as an analytic ring.
k = 2. Its
relations will be reflected in the structure of an infinitesimal neighborhood of the zero.
It would be interesting
to know if the generalized Schubert cycles on
G/B , i.e.
the closures of the Bruhat cells, admit an intrinsic characterization in the ring duals of these classes in explicitly in
CBGG] .
H (G/B,0Z) .
The Poincare
H* (G/B,ID) are calculated
We will return to this question in
§9. 8.
A REMARK ON RATIONALITY
The condition of Theorem 7 that vector field
V
some integer
k / 0
X-^CtJ-X"
1
k
= 4>(X t)
$ : E •* Aut(X) Aut(X)
and a
X
admit a holomorphic
02* action so that
dX-V = XV
for
is equivalent to requiring "that for all
X e 02* , t e 0] , where
is the one parameter subgroup (of the group
of automorphisms of
X ) generated by
V .
When
32
Holomorphic C* Actions and Vector Fields
the identity component
AutQ(X) is semi-simple and
a unipotent one parameter subgroup, i.e. a GdQ [H] , the Jacobson-Morosov Lemma existence of an
<J> is
action
CJa] guarantees the
SL(2,ffi) c AutQ(X) in which
$(03) =
(fo l]:t £ ^l ' If (X*x) "* X"x denotes the E* action on X induced by the maximal torus in SL(2,ffi) , then X"1 = $U2t) .
Using this fact, it is possible to
prove a result of Deligne THEOREM 8 Suppose X such that
Aut(X)
CD] . is a smooth projective variety
is semi-simple.
Suppose that there
exists a holomorphic vector field on X
generated by a
Ga. Q
action whose fixed point set is rational (as a projective subvariety of X ) . Then Outline of proof.
X
is rational.
By a theorem of Sommese
if AutQ(X) is semi-simple, then any
E
CS.,] ,
c AutQ(X) has
fixed points on X .
It follows, by Blanchard's theorem
CM, p. 25] , that
can be imbedded in some ffilP so
X
that each g e SL(2,Q3) c Aut(X) transformation.
is induced by a projective
By Theorem 7.1 of
CCS«] , V
is tangent
to the fibres of the plus cells in X , hence the sink of X that X
is contained in zero(V) . X
N
Therefore, assuming
is not contained in any hyperplane of
= X n L = zero(V) n L
X
HIP ,
for some linear subspace L of
ffi]P . It follows that the sink
X
of X
is rational,
J.B.Carrell
so
X
33
is rational, by the corollary to Theorem 1.
The question of whether the existence of a holomorphic vector field on X having isolated zeros implies X rational has been considered by Lieberman in and by Deligne
CD] .
is
[L-.ljCLp] ,
By the induction argument in [Lp]
one can reduce this problem to showing the Conjecture; A smooth protective variety that admits a holomorphic vector field with exactly one zero is rational. 9-
Borel's Theorem
CLOSING REMARKS
R/RIy
H
H*(G/B,ffi) has another
interpretation due to Kostant can be seen to be the ring (nonreduced) variety cone in
Q.
[Kos] .
Namely,
R/RI,,
EC firm] of functions on the
h n n , where
n
is the nilpotent
A problem of Kostant is to understand in an
intrinsic manner how Schubert calculus works in
EC firm] •
The isomorphism of Theorem 7 may shed some light on this problem since we now have available the fact that
H"(G/B,ffi)
is isomorphic to H (G/B,0Z) for the vector field associated to any regular element in
n•
In the same spirit as Kostant,
one may ask
Question.
Suppose V
is a holomorphic vector field
with one zero having an -associated
E*
action so that
34
dA-V = XCk
Holomorphic C* Actions and Vector Fields
£ 0). Find intrinsically the elements in
H^CX.O™) associated to the *
xt by Poincare* duality. J
J. B. Carrell
[A..]
35
REFERENCES ^ E. Akyildiz, Bruhat decomposition via G action, Bull. Ac ad. Pol. Sci. . Ser. Sci., Math., 28(1980), 551^547. _ , Vector fields and cohomology of G/P , to appear.
CACLS] E. Akyildiz, J.E. Carrell, D.I. Lieberman, and A.J. Sommese, On the graded rings associated to holomorphic vector fields with exactly one zero, to appear in Proc. Symp. in Pure Math. [At]
M. Atiyah, Convexity and commuting Hamiltonians , Bull London Math. Soc.3 14 (1982), 1-15-
CBGG3
I.N. Bernstein, I.M. Gelfand, and S.I. Gelfand, Schubert cells and the cohomology of the spaces G/P , Russ. Math Survs . , 2JJ (1975), 1-26.
CB-B]
A. Bialynicki-Birula, Some theorems on actions of algebraic groups, Ann, of Math. 9J3 (1973), 480-497-
CC]
J.B. Carrell, Vector fields and the cohomology of G/B , Manifolds and Lie groups , Papers in honor of Y. Matsushima, Progress in Mathematicsa Vol. 14 Birkhauser, Boston7 1981.
CCG]
J.B. Carrell and R.M. Goresky, Homology of invariant subvarieties of compact Kaehler manifolds with ID* action, to appear.
CCL.,]
J.B. Carrell, and D.I. Lieberman, Holomorphic vector fields and compact Kaehler manifolds, Invent . Math. 21 (1973), 303-309-
_ and _ _ , Vector fields and Chern numbers, Math. Annalen, 225 (1977), 263-273-
_ , Vector fields, Chern classes and cohomology, Proc. Symp. Pure Math . 3 Vol. 30 (1977), 251-2PT [CS-]
J.B. Carrell and A.J. Sommese. Scand. Jj-3 (1978), 49-59-
E* actions. Math.
36
Holomorphic C* Actions and Vector Fields
_ Some topological aspects of W actions on compact Kaehler manifolds. Comment . Math. Helvetici 51 (1979), 567-582. CCS.,]
_ SL(2,E) actions on compact Kaehler manifolds, to appear in Trans . Amer. Math. Soc.
CD]
P. Deligne, Letter to D.I. Lieberman
CPr]
T.T. Frankel, Fixed points and torsion on Kaehler manifolds, Ann, of Math. 70. (1959), 1-8.
CFu]
A. Fujiki, Fixed points of the actions on compact Kaehler manifolds, Publ. R.I..M.S.., Kyoto University 15 (1979), 797-826.
CH]
J.E. Humphreys, Linear algebraic groups , SpringerVerlag, Berlin - New York (1975) -
CJa]
N. Jacobson, Lie algebrasa John Wiley and Sons, New York - London (1962).
CJu]
J. Jurkiewicz, An example of algebraic torus action which determines the nonfilterable decomposition, Bull. Ac ad. Polon. Sci. , Ser. Sci. Math. Astronom. Phys"- 2TT19 76) 667-674.
CKL]
S. Kleiman and D. Laksov, Schubert Calculus, Am. Math Monthly 79 (1972), 1061-1082. ~
CKob]
S. Kobayashi, Trans formation groups in differential geometry, Springer- Ver lag, New York Tl972) .
CKon]
J. Konarski, Properties of projective orbits of actions of affine algebraic groups, to appear.
CKos]
B. Kostant, Lie group representations on polynomial rings, Amer. J. of Math. 8£, 327-404.
CL^]
D.I. Lieberman, Rationality and holomorphic vector fields, to. appear. _ Holomorphic vector fields on projective varieties, Proc. Symp. in Pure Math 30 (1977), 273-276.
J
J. B. Carrell
37
CM]
Y. Matsushima, Holomorphic vector fields on compact Kaehler manifolds . Regional Conference Series In Math. No. 7, A. M.S. (1971) -
CMS]
J. Milnor and J. Stasheff, Characteristic Classes, Annals of Math study no. 76. Princeton University Press, Princeton N.J. (1974).
CS.,]
A.J. Sommese, Extension theorems for reductive group actions on compact Kaehler manifolds. Math. Ann. 218 (1975), 107-116. , Some examples of
E*
actions,
to appear. CW]
H. Whitney, Tangents to an analytic variety, Ann. of Math. 81 (1965), 496-549.
CWr]
E. Wright, Killing vector fields and harmonic forms, Trans . Amer. Math. Soc. 199 (1974), 199-202.
A Duality Operation in the Character Ring of a Finite Group of Lie Type CHARLES W. CURTIS
INTRODUCTION
The subject of these lectures, while perhaps not a major theme in the representation theory of finite groups of Lie type, nevertheless cuts across the representation theory of these groups in interesting and sometimes unexpected ways.
The main results reported on here are
due to Alvis (El], C2]), following an earlier paper C73 by the author.
Some of Alvis's main results were obtained
independently by Kawanaka, and a homological interpretation of the operation has been given by Deligne and Lusztig CIO] In order to describe the contents of this paper, we first require some terminology. group, and let
ch(IDH)
denote the ring of complex valued
virtual characters of H. the
Let H be a finite
The elements of
ch(EH) are
2Z-linear combinations of the elements of Irr H,
the set of irreducible characters afforded by the simple 39
4-0
Characters of Groups of Lie Type
left
EH-modules.
The operations of addition and multi-
plication of characters correspond to the operations of forming direct sums and tensor products of the corresponding a
EH-modules.
A duality operation in
7L -automorphism of
Ch(ffiH)
of order
2,
ch(EH)
is
which pre-
serves the inner product (f,g)HH - iHl"1 I f(x)IO£T xcH of class functions on
H.
Such an operation clearly per-
mutes, up to sign, the elements of
Irr H.
example of a duality operation is the map £ € ch(EH), where
£
A familiar £ -*• "£,
is the complex conjugate of
£.
This map corresponds to the operation of forming the contragredient module of a given
EH-module.
Another
example is given in § 1, for finite Coxeter groups, and consists of multiplying a character by the sign character.
The duality operation described in § 2, for virtual
characters of a finite group permutation of
Irr G
G
of Lie type, defines a
(up to sign), with corresponding
characters not necessarily having the same degree. degree of
£ € Irr G
j£* 6 Irr G,
and its dual
always have the same
differ only by a power of
p,
c* e ch(EG),
The
with
f
p -part, and hence
where
p
is the character-
istic of the finite field associated with
G.
In § § 1-4, we have given a self-contained exposition,
C. W. Curtis
41
often with complete proofs, of the main results.
In § 5,
we survey without proof some other results, with references to the literature. It is a pleasure to acknowledge the hospitality and interest of the Notre Dame Mathematics Department during the time these lectures were presented. 1.
DUALITY IN THE CHARACTER RING OP A FINITE COXETER GROUP.
Let
(W,R) be a finite Coxeter group with dis-
tinguished generators
R = {r-,...,r },
with the pre-
sentation W =
= 1, 1 < i, j <> n>.
e : R •* ID* defined by
preserves the defining relations of
eCr.^) = -1, 1 £ i < n, W,
and therefore
can be extended to a homomorphism
e : W -* E* which we shall call the sign representation of example, the symmetric group with generators
defining relations ^riri+l'
= (n,n+l)
(r±r.) - 1 if
=» 1> 1 ^ i ^ n.
For
S +., is a Coxeter group
r1= (12),...,r 2
W.
and
|i - j| > 1, and
In this case
e(cr),
for
a € S , is the usual signature of a permutation, and is 1
if
a
is even, -1
if
a
is odd.
42
Characters of Groups of Lie Type
It is easily checked that the map li € ch(IDW),
is a duality operation in
y -»• ey, ch(EW)3
for and it has
traditionally been used to organize the character tables of the Coxeter groups. Our first aim is to give a geometric interpretation of
e.
It is a standard result (see C33) that
W
can
be identified with a finite group generated by reflections on a real Euclidean space n = |R|.
Each reflection
pointwise fixed, and if
r eW a
E
of dimension leaves a hyperplane
H
is a vector orthogonal to
H,
then
where on
E.
(
, ) is the positive definite scalar product
The vectors
{+a}
orthogonal to the hyperplanes
fixed by the reflections in
W,
with their lengths
suitably normalized, are permuted by the elements of and are called the root system There exists a set properties that
II of roots
{(^,...,0 }
$ associated with {ou, . . . ,an>,
W,
W.
with the
form a basis of E over
K , and every root a can be expressed in the form n a = Z c. a. 3 where the coefficients {c.} are either 1 1-1x x all non-negative or all non-positive. Such a set of roots is called a fundamental system, and there exists a fundamental system II such that the distinguished
43
C. W. Curtis generators
tr.K
reflections
{r
< i
Qf
^
coincide with the
} e II, where r is the reflection a i ai i defined as above, fixing the hyperplane orthogonal to a.. a
The subsets , J
J 5 R
all define subgroups
Wj =» <J> =
which are themselves Coxeter groups, and
u
are called parabolic subgroups of
W.
In this set-up, the sign representation is defined by
e(w) = det w,
vector space for
E.
the determinant of
w 6W
We are interested in one more formula
e, which arises from the action of
unit sphere
S
on the
in the Euclidean space
define a natural triangulation of of the root system.
S
The action of
W
E.
on the We shall
defined in terms
W
on the simplicial
complex defining the triangulation yields a representation of W (1.1) group with
on the rational homology
H#(S),
and we have:
PROPOSITION.
Let
(W,R)
be a finite Coxeter
n = |R| > 2.
Let
r
be the abstract
simplicial complex whose simplices are the cosets {wWj : w e W,J c R} W,
of all proper parabolic subgroups of
with order relation (defining the faces of a simplex)
given by the opposite of inclusion. (i)
the geometric realization
homeomorphic to the unit sphere (ii)
Then:
S
the rational homology
in H#(D
|r|
of
r
is
E. =S = : H ^
is
44
Characters of Groups of Lie Type
given by: H^CD • 0 except in dimensions and
0
and n-1
HQ(r) = Hnn(r) s $ as rational vector spaces. (iii)
the rational homology group
the trivial representation of W, group
H r n_i( )
HQ(D
affords
and the homology
affords the sign representation
We shall give a sketch of the proof.
e.
For, more
details see Carter C4] or Bourbaki C33- Let II = {a.,...3a } be the fundamental system of roots such that the reflections and let C
}-
.
C = {£ € E : (£,0^) > 0
coincide with for all
R,
o^ e n}.
Then
is an open simplicial cone called the fundamental
chamber of subsets J
{r
W
acting on
Cj c (J, where
E.
€E:
C
(J ig" the closure of
ranges through subsets of
U
The walls* of
n,
are the C,
and
defined by
(5,a ± ) - 0, a± e J , ( C 5 a ± ) > 0, a± e E - J
It can be proved that for each subset stabilizer of of
Cj
in
J <= R,
the
{r
W-translates
bijective correspondence with the cosets Moreover, each vector interior of a unique chamber
the
W is the parabolic subgroup W
W, generated by the reflections
For a fixed
"J c n,
5 ^ 0
in
W-translate
C or one of its
faces.
: a. € J} . aj J {wCj> are in {wW_}.
V belongs to the wC JS J 5 R,
of the
It follows that the
45
C. W. Curtis
intersections of all the translates with the unit sphere
S- in
subdivision of the sphere simplex
wCj n S
on
S
E,
S.
{wCj : w e W,
J c R}
defines a simplical It is also clear that a
is a face of
w'Cj1 n S
and only if wW- ^ w'W', proving that
if
.S is the
geometric realization of the simplicial complex r defined in the statement of the proposition, and proving the first statement.
The second statement is
immediate, from the properties of the rational homology1 of a sphere. For the third statement, it is easily checked that HQ(r)
affords the trivial representation
order to prove that
H n_i(r)
the Lefschetz character A tion of W of
W
on
H^(r);
affords
e,
1^ . In we consider
of the homology representa-
then A
is the virtual character
given by A *
S (-D^rC^H.Cr)). 1 i=o
For the action of a finite group (in this case W) a finite simplicial compex
r,
on
it is well known that
for each w € W,
A(W) = x(lr|w), where
X ( | r [)
fixed point set
is the Euler characteristic of the |r| w
of
w acting on the geometric
46
Characters of Groups of Lie Type
realization
|r| of
r. For a reflection
fixed point set of
r
dimension than
and we have
S,
r,
the
is a sphere of one lower
A(r) - 1 + (-l)n~2. On the other hand, by definition of A that
H.(T) = 0
and that
r
except in dimensions
and the facts 0
and n-1,
acts trivially on HQ(r) we also have
Mr) - 1 + (-l)n~1Tr(r,Hn_1(r)). Combining these results, we have for each reflection
r,
Tr(r,H
and hence
Tr(r,H
-,(r)) = -1 .(I1)) = e,
completing the proof of the proposition. As a consequence, we obtain: (1.2)
COROLLARY (L. Solomon Cl6]).
representation
e
of
W
e=
The sign
is given by w
r (-i)lJh JcR
W
,
J
W LWr is the induced permutation representation of J on the left cosets of WTj, for J c—R .
where W
For the proof of the corollary, we first note that for
J c R,
GJ n S
JII - j| - 1 = i;
is an
i-simplex
if and only if
for example the vertices of the
simplicial subdivision correspond to the faces J
a maximal subset of
R,
and their
CjT,
with
W-translates. We
C. W. Curtis
47
then have, for Cj
w e W, since
for each
W T is the stabilizer of d
J,
T r ( w , C ± ( r ) ) - #1 simplices fixed by
where
C±(r)
is the
i
basis consisting of the |n-j|
-1-1}.
h
chain group of i-simplices
r
T, with a
{xCj : x e W,
We then apply the Hopf trace formula
to the Lefschetz character of the homo logy representation of
W on
H$(T),
A(w) =
and obtain
n-1 I (-1)1 i=o
T r ( w , C1. ( r ) ) -
JcR
"Xj
On the other hand, from what has already been shown in part (ill) of the Proposition, A(w) - 1 + (-D^^-eCw), w € W. Combining these results, we obtain Solomon's formula. A final corollary gives an expression of the duality operation y c ch(IDW),
y •*• ey,
for virtual characters
as an alternating sum of induced characters,
which will point the way toward the duality operation we shall define for characters of finite groups of Lie type.
48
Characters of Groups of Lie Type
(1.3)
-COROLLARY. For each virtual character
U s ch(EW), we have ep - Z (-D(u|Ww ), JcR J where
u|Wj is the restriction of y e ch(IDW) to w
WJT.
By Solomon's formula we have W e = Z(-l)lJlo« . W J
Multiplying by
ji, we have W
).
J
A well-known identity for induced characters then gives 'J
"J
completing the proof. 2.
TRUNCATION AND DUALITY IN THE CHARACTER RING OF A FINITE GROUP OF LIE TYPE.
In this section,
G denotes a finite group of Lie
type. Such a group may• be described in various ways. For example, we may assume over a finite field
IP
G is a Chevalley group (or a twisted type of Chevalley
group) as in Carter [4] or Steinberg [183. Alternatively, let G be a connected reductive affine algebraic group over an algebraically closed field K of characteristic
49
C. W. Curtis
p > 0, with a rational structure defined by a surjective endomorphism
a : G -> g such that the subgroup
elements of G fixed by a, is finite.
Then
Sa,
of
G - 5a
is a finite group of Lie type, and the two descriptions are essentially equivalent (see Steinberg [193). As examples to keep in mind, let let q » pa, K.
Then
a > 0, where
p
g = GLn(K), and
is the characteristic of
G has two rational structures, defined by
the maps ax : (x^) * (x±jq) and a2 : (x±j) -> t(xijq)"1. In the first case, the resulting finite group of Lie type is the untwisted group GL (IP ), while in the second case, §a is the unitary group U (IP2)> "~ 2 q twisted type of Chevalley group.
a
In this section and the next, the properties of G we shall require can be summarized in the statement that G has a split a fixed prime
(B,N)-pair of characteristic p, with Weyl group W,
p > 0, for
and satisfies
the Chevalley commutator relations (see Richen [143 or Curtis [63). Thus subgroup B, axioms.
G has a
(B,N) pair with Borel
and a subgroup N, satisfying the usual
In particular, we have:
(i) G - ; (ii)
If H = B n N,
then H < N,
and the group
50
Characters of Groups of Lie Type
W = N/H
(the Weyl group of
G)
is a finite group
generated by a set of involutions
R.= {r.^,...,^},
satisfying the additional conditions: (iii) notation where
r.Bw £ BwB
u Br.wB
(where we use the
Bw, for w e W = N/B n N,
n € N
corresponds to
w
to denote
Bn,
under the natural map
N •* N/H) , and ^B^. 2 B
(iv)
for all
r^ e R.
From the axioms, it follows that
(W,R) is a
Coxeter system, with distinguished generators we have the Bruhat decomposition of G =
with
w -»• BWB
R,
and
G:
u B w B, weW
a bisection from
W
to the double cosets
B\G/B. The split splitting of
B
(B,N)-pair
in
G
is defined by a
as a semidirect product:
B ='UH, U ± B, U n H - { 1} ,
with of
U = 0 (B) , B,
and
the unique maximal normal p-subgroup
H an abelian
pf -group.
For example, example, in in the the case case of of (B,N)-pair B
GL (JP ) , a split
is given as follows: (upper-triangular matrices),
51
C. W. Curtis N « {monomial matrices, with exactly one nonzero entry in each row and column}, H - B n N * {diagonal matrices},
{unit upper-triangular matrices}.
Then the Weyl group
W = N/H,
to the symmetric group pair of rank
in this case, is isomorphic
S , and defines a split
(B,N)-
n-1.
The standard parabolic subgroups of
G
general case) are the subgroups containing
(in the B,
and are
defined in terms of the Bruhat decomposition by P, - BWTB -
where
Wj
u
BwB,
is a parabolic subgroup of W,
over the subsets of
R.
and J
It follows from the
ranges
(B,M)-
properties and the commutator formulas (see Curtis £63) that each standard parabolic subgroup has a decomposition as a semidirect product (called a Levi decomposition) Pj = LJVJ»
with
Vj - Op(Pj) ^ Pj,
where
Lj,
split
(B,N)-pair with Weyl group
VT
called the Levi factor,, is a group with a (Wj,J).
The group
is often called the unipotent radical of
PT,
and
52
Characters of Groups of Lie Type
is the group of
3P -rational points on the unipotent
radical
of a parabolic subgroup
over S
RU(£J)
3P , in case
G
gj
defined
arises from an algebraic group
as described earlier. In case
PT = B, d
the Borel subgroup, with
J = 4>,
the Levi decomposition is given by B - UH, with
V, - U, L. - H,
and is part of the definition of split The parabolic subgroups of bolic subgroups ments of
G.
{PTd}., 05 d CJTl
G
(B,N)-pair.
are the standard para-
and their conjugates by ele-
In the case of
GL (IP ) , they are the
stabilizers of all flags in the underlying vector space on which
GLn(3P )
acts, where a flag is a chain of
subspaces W--L c W0tL c ... c w S , s * 1.
Thus the Borel subgroups are conjugates of the stabilizers of complete flags (with
dim W. = i, 1 <s i £ s
and
s = n - 1).
The definition of the duality operation involves restriction and induction from parabolic subgroups
PT, d
J £ R. The usual definitions, however, have to be modified to take into account the unipotent radicals
VTd,
that we really are setting up relationships between characters of
G
and characters of the Levi subgroups
so
C. W. Curtis
Lj
of
53
Pj,
for
J £ R.
The process can be described in
a general way as follows. subgroup of
G,
V < E z G. £.
and
Let
Let
V a normal subgroup of
X be a left
Then the restriction
characters of
where
G be a finite, H,
£[„ = resH£
is a sum of two-
H,
H
having
V
£|H
from irreducible
in their kernels, and C" n
is the contribution from those that do not. £' £1
can be viewed as a character of
hand, let
trivial representation of
since
V
V 5 H,
V;
then
X
affording the
inv^(X)
and, in fact, is a
afforded by
for each character
of
H
natural map
with
X V
is a flJH-
E( H/V) -module
acts trivially on inv,,(X) . Let
the character of H/V
acter of
Note that
H/V. On the other
inv,_(X) be the sub space of
module since
so that
EG-module with character
£,1 is the contribution to
characters of
H a
£(H/V)
be
invv(X) . Finally,
H/V, let
X
denote the char-
in its kernel defined by the
H •+ H/V.
These ideas are connected by the
following result, whose proof is left as an exercise. (2.1)
PROPOSITION.
Keeping the above notations,
the following characters of H/V
coincide :
?' = CrH/vx - (5L)/w/vx E U,XG)X. H (H/V) »H (H/V) X£lrr(H/V)
54
Characters of Groups of Lie Type
An easy lemma from the Felt-Thompson paper (Cll3, p. 783) implies that for an element C Q (x) n V = {!},
x c E such that
we have
which is a kind of reduction theorem for character values on certain elements in terms of characters of the smaller group
H/7. From this point on, we shall often be following the
approach of Alvis [13 (see also Curtis C71). (2.2)
DEFINITION.
Let
type, with Coxeter system operation of truncation homomorphism of character
G be a finite group of Lie
(W,R),
and let J £ R. The
Tj : ch(EG) •*• ch(IDLj) is a
E- modules9 which assigns to each virtual
\i e ch(EG)
the virtual character
TTuu € ch(ELjT), where TJTuU) = |V J"1 J The fact that of
Lj,
TjU
Z y(v*), ^ €JLT. is in fact a virtual character
and connections with Proposition 2.1, are given
by:
(2.3)
PROPOSITION.
ter afforded by some
"
Let
\i € ch(IDG)
EG-module M.
Then
be a charac-
C.W. Curtis
55
For the proof, let e TjM = invTrv .M, normalizes
and for
VjT .
6j = |V J |" 1 Z V€V v. Then J 2, e L T3 J &eJT = e TJJl since L Tj
Then
and hence
T^-y = u,p ^ > . J J Corresponding to the operation of truncation, we
have a second operation, which is a homomorphism of 2Z- modules Ij : ch(ELj) -»• ch(EG),
given by X
IjX = X ,
is the lift of
previously.
for each character X from
LdT = PJT /VJT
The operations
I_
and
X of to
T_
PJT
Lj,
where
defined
are adjoint
with respect to the scalar product of characters (see (2.5)). REMARK.
These ideas can be used to describe Harish-
f
Chandra s organization of the character theory of An irreducible character
£ c Irr G
G.
is cuspidal (or
discrete series) if
TT£ = 0 for all J <= R. It is an j ^ easy exercise to show that C is cuspidal if and only if
U,IjX) = o
other words
c
for all
J c R
and all
X € Irr Lj.;
is missed by the process of lifting and
induction from all characters of Levi factors of proper parabolic subgroups of
G.
A basic result (which we do
in
56
Characters of Groups of Lie Type
not prove here) asserts that every character is either cuspidal, or is a constituent of irreducible and cuspidal character where
X
and the Levi factor
mined (by
£) up to conjugacy.
in the character theory, of
G
Lj,
X
5 € Irr(G) I T X, for some u
of
LT, for J c R, J ^ are uniquely deter-
Thus the main problems are to find all cuspi-
dal characters, and to decompose the induced characters IjX
from cuspidal irreducible characters of
L,.
Returning to our main theme, we have: (2.4)
DEFINITION.
endomorphism
5 •* c*
The duality operation is a
of
ch(EG)
2-
defined by
( - l ) l J ' l TT T TT£ , C € c h ( E G ) . J J
JcR
REMARKS AND EXAMPLES,
i)
Note the parallel between
this definition and the duality operation in Corollary 1.3). The fact that operation in ii)
ch(lCG)
C •*• 5*
ch(IDW) (see
is a duality
will be proved in § 3-
The dual of the trivial character is the Stein-
berg character of
G, 1 * = StQ, where
a character of degree
|G| , the
StQ € Irr(G) is
p-part
of the order of
G. iii)
If
c e Irr G
since all the truncations
is cuspidal, then TT£, for J
J c R,
5* = +£ , are zero.
7*
We next derive some properties of truncation and
C. W. Curtis
57
induction, which will be needed to prove the main results about the duality operation. For subsets
K £ J £ R, P
J,K
= L
J
let n
V
these are the standard parabolic subgroups of taining the Borel subgroup
Lj T, conTheir Levi
BTj = B n LjT.
decompositions are: P
J,K = V V J,K> V J,K
= L
J ° VK>
and we have VK - Vj KVj (semidirect, with Then we have additive maps, for
Vj < V R ).
K £ J £ R,
T£ : chClCLj) -> ch(!CL K ), 1^ : ch(EL K ) -»• ch(ELj), defined as above. The following result is easily proved, and is left as an exercise: (2.5)
PROPOSITION.
Let K £ J £ R. Then we have:
(i) (Transitivity) IK - *J*K>
(ii)
and
(.Frobenius reciprocity) For
T
K = TKTJ;
5 c ch(fl2Lj),
n € ch(ELK), (e,i£n)L = (.T^5,n)LK. J
For the next result, let J,Jf £ R,
and let DJJt
58
Characters of Groups of Lie Type
be the distinguished double coset representatives for WAW/WJ, . For a subgroup. H £ G
and
X e ch(EH), and
x € G, we define the conjugate virtual character X
X € ch(£(xH)) by
x
\(*h) = X(h), h € H,
where
xhx"1. We then have: (2.6) f
Let
J,J
PROPOSITION (Intertwining Number Theorem). c R,
X € ch(ELT) , X1 e ch(ELTI).
and let
d
E (IJX,IJIX')0 - E d 6 JJ Df K « J n djf » d K f ,
where
J
Then
,
x . X ' , K
so
LK -
d
^K,.
For the proof, we first apply the usual intertwining number theorem, and obtain (I X,I A') - S (A| ,dX'| ) d d deDjj, Pj n dPJt 'PJ n \, Pj n PJt
We have a factorization of P
J
n
= L (L
%'
K J
Pj n
n dv
j'^VJ
n
n
p 1» p J
J
n
p
v 6 v
j K»
y € V
j» Kr>
and the expression becomes
n
\«>.
Then
J'
|PT n ""P,,!"1
where
(see Curtis C6],§ 2),
S.J^J
with uniqueness of expression.
J
PJt
2
so
A(Avdyz)dX'(Avdyz),
yz € V
J* vz
€
59
C. W. Curtis
completing the proof. 3.
MAIN THEOREMS ON DUALITY.
In this section,
G
denotes a finite group of Lie
type as in § 2. (3-D
where
THEOREM.
Let
£ e ch(IDG)
and
J 5 R.
Then
(TjO* is the dual of Tj£ in chdDLj) . (In
other words the operation of truncation intertwines the duality operation.) We give a sketch of the proof. For more details see Curtis C73- We have to prove that
z ( - i ) l J l l T TJi TJI T JT f c = £ (-I) |K| I£T£T T C. K KJ
Jt£R
K£J
A typical term on the left hand side is
by Mackey's sugbroup Theorem.
for all
(T,5)
We then have p
PROPOSITION. T T ( d (T T , c) H
(3.2)
d
J,J
f
c R, d € D,,
pJ )
d
T (E
J
where
QTJ
*J'
nn
J
)
^J
d f
K= J
P
n J.
60
Characters of Groups of Lie Type
This result is Prop. 2.1 of Curtis C 7 H , and will not be proved here. Applying ( 3 - 2 ) to the left hand side, we obtain TJT U*) -
Z (-l)IJ'l(TT(ITITTI£)|p )
J'cR ~"
J
J
TJ | d T,,e|. J
J'cR
=
'Pj
P
E ( - 1 ) I J ' I 3E
=
J
J
)
z (-i
J'cR
,
Q
J f n J=K
Z ( E (-l)I J l la T ! J K )l£T K ^ KcJ J'cR J JK K K
(by ( 2 . 5 ) ) ,
where 1
1 aJ'JK nJ = T , -rtr ~ card{d c D T TfI : J JJ
K} .
The proof is completed using the following result. (3-3) for
PROPOSITION.
J',J,K £ R.
Let
aJIJK
be defined as above,
Then
•Pci/"1^
a
J'JK = ^"^
For a proof, see Curtis ([71* Lemma 2.5). As an immediate consequence of Theorem 3-1, we have: (3.*0 where
St«
COROLLARY. For all J £ R, TjStQ = StL and
St,
are the Steinberg characters of d
G
and
Lj, respectively.
C. W. Curtis
61
(3-5)
THEOREM.
(Alvis C 2 ] ) ,The duality map
C •*• C* is a self-adjoint isometry of order ch(3DG) .
2, in
Thus
) and e* = c, for all
s,n € ch(EG).
To begin the proof, we recall (prop. 2.3) that for* all
J £ R,
and
C e ch(lDG) ,
Lj T
9elrr LuT
and by Theorem 3-1* we also have
Let
Can ^ ch(iDG) . Then
(c*,n) - s (-D^'djTjC.n) J J JcR
- z C-i)lJl(iJ T JcR —
JcR —
9 € Irr L j T
(pelrr L. J
by symmetry. We now have, for
^* -
C e ch(EG)
S (-l)IJll (T ( * JT J T C JcR
62
Characters of Groups of Lie Type
( - l ) l J l l j. ( . TjC ) *
(by (3-D)
JciR
2 (-D JcR
S (-l)lT KcJ J K
£ C-1)IJ' Z C-Dl^Wc JcR KcJ *•A
(by C2.5))
2 (.-D|K|( E ( - D ^ b l c
KC.R
since
I (-1)IJI= 0
if
K ^ R.
Finally, to prove that have, for
C •»• S*
is an isometry, we
5,n e ch(EG)
completing the proof. (3-6)
COROLLARY.
(Alvis)
The duality operation
permutes, up to sign, the irreducible characters of Let
C e Irr G.
Then
virtual character of Theorem 3-5- Thus
G
(5,5) = 1,
such that
+C* € Irr G,
so
C*
is a
(£*,£*) = l, by as required
The next result shows, how the duality operation interacts with Harish- Chandra's "philosophy of cusp forms", discussed in § 2.
G.
C. W. Curtis
63
(3-7)
COROLLARY.
character of K c J).
*
LdT,
Let 9
for some
J "c"""R
Then
Moreover, for
cp
(so T^9 = 0 j\,
for all
i_i
£ e Irr(G) t
5
U,Ij so
be a cuspidal irreducible
U*,Ij
* permutes the irreducible components of
Ij
as above.
We have, for
C e Irr G,
) * , C ) - dj^5*) a - (9,Tj(c*)) L J - (9 J (T J O*) L
(by ( 2 . 5 ) )
Cby ( 3 - D ) J
= (cp*, TjC) L
(by ( 3 . 5 ) )
J
- (-l)lj|(
= (-l) | j | (Ijq),5)
(by ( 2 . 5 ) ) ,
J
using also the fact that if
9
is cuspidal in
then 9*
This completes the proof of the corollary.
Irr(LjT ),
64
Characters of Groups of Lie Type
4.
APPLICATIONS TO SPRINGER'S THEOREM AND A CONJECTURE OP MACDONALD.
All the results in this section are due to Alvis [13.
Let
G be a finite group of Lie type, and
prime associated with
G
G = g g,
the
(i.e. the characteristic of
the field, for a finite Chevalley group). the set of all
p
p-elements
in
G.
Let
V
be
(Note that in case
for a connected reductive affine algebraic group
as in § 2, then
ments in
G.)
V
is the set of all unipotent ele-
We shall call
V
the unipotent set in
It is necessary to extend the operations £ -* £*,
etc. to the vector space
class functions on
G;
cf-G
G.
Tj, Ij,
of complex valued
then the usual results continue
to hold, by linearity, since
Irr G
is a basis for
The first main result is a variation by Alvis of a theorem of Springer [173. (4.1) G,
and
THEOREM.
Xv
V).
where
p be the regular character of
the characteristic function on
function which is of
Let
1
on V
and 0
«*
—
v
V
(i.e. the
on the complement
Then
I^Li
is
the
In I
p!-part
of the order of G.
C. W. Curtis
65
We first require: (4.2)
LEMMA.
For all
J 5 R, TjXv = Xy| p
We first note that for a class function 1
Is the class function on
P,
f(fcu) = £(&) 3 £ c Lj, u e Vj. £ € L,
U €
J
£
on
Lj,
defined by We have to show that for
V,
which is the same as
J"
for is a
g € Pj.
On the right hand side, we get
p-element,
and zero otherwise.
1
if
On the left side,
it suffices to show that for
u € Vj, g € Pj, gu
p-element
is a
if and only if
g
g
p-element.
is a
This is
easily checked, and the Lemma follows . For the proof of the Theorem, we also require the facts that on
StG(l) = I G I P *
and
hence that
StQ
vanishes
p-irregular elements (see Curtis C53) . Then we have * x v* st a
^ by direct verification)
- E(-l)l J 'x-l vp
G
J
)G
(since
StQ - Q 1*)
Characters of Groups of Lie Type
completing the proof. As a more or less direct consequence, we obtain the following result, the last part of which is related to conjecture of MacDonald which remained unproved after the others were settled by Deligne and Lusztig (see Alvis [2] for further discussion). (4.3)
THEOREM.
Let
5 e Irr G.
Then the following
statements hold
(i)
Z
(ii)
U*(D|pl - C(Dpi.
(iii)
e(l) Z z;(u) is, up to sign, a power of ueV
p.
For the proof of (i) , we have Z 5(u) - |a|(cv,0 ur ueV
=
l a l < 5 *v, x )
(by (3-5))
- |G|(^MG|^P) (by (4.D) = |G|nU»,p) • |GDU*(D
(since +£* € Irr G) .
sr
P
For part (ii) , we first obtain C(D where
l
KU(V S 5(u) - Z— FTTT^ (C
{C}
ments in
€
alg-lnt 1C n Q = 2Z ,
ranges over the conjugacy classes of G,
and
u^ e C,
for each
p-ele-
CC. We have also used
C. W. Curtis
67
the standard result from character theory that
.
— -7-=-x— € alg-int.lD
(the algebraic integers in I D ) ,
for any irreducible character
£
and conjugacy class CC.
Thus
by part (i). Since replace
5 by
£*
±Z* e Irr G, by (3.5), we can in the above argument, and obtain
also
Combining these results we obtain (11). Finally, (iii) follows at once from parts (i) and (ii) . As a further application, we obtain another proof of a result of Steinberg C193(4.4)
COROLLARY.
|v| = |G|2.
The proof is immediate, upon applying (i) to the trivial character, since
1* = StQ, and StQ(l) = |G| .
68
Characters of Groups of Lie Type
5.
SOME ADDITIONAL RESULTS.
We shall describe briefly, without proofs, some other results about the duality operation. 5.a.
THE PERMUTATION OF THE IRREDUCIBLE COMPONENTS OP Ij9,
FOR 9
CUSPIDAL IN Irr Lj.
By Corollary 3.7, the duality operation permutes (up to sign) the irreducible components of
Ij
for
J —c R and q> a cuspidal irreducible character of Lj. T The precise nature of this permutation has now been determined.
To describe the main idea, we begin with
the simplest case, with J = c|>, q> = 1, so the permutation character of the action of cosets
1,1 = 1Q , G on the
G/B. Using the theory of Hecke algebras, it
is known, in this case, that there is a bisection i|j •+• 5, (C,1
from
Irr W
to the characters {5 € Irr G:
) ^ °}> with the property that
for all J £ R. Let e be the sign character of W. We then have: (5.1) we have
THEOREM (Curtis C71). For all f e Irr W.
C. W. Curtis
69
Thus the duality operation, for the characters in IB ,
corresponds exactly to the duality operation of
ch(lDW)
described in § 1.
to components of
A_
This result has been extended
by McGovern [13],
general case, for components of Hewlett and Lehrer 5.b.
Ij9
and to the
as in (3-7)* by
[12].
HOMOLOGICAL INTERPRETATION OP DUALITY In § 1, we interpreted the duality operation in
ch(EW) on
in terms of the homology representation of
Hg(r).
A similar interpretation of the duality
operation is possible for a finite group
G
Let
G;
A
W
be the combinatorial building of
of Lie type. then
A
is
the finite simplicial complex whose simplices are the proper parabolic subgroups of
G,
with
G-action
given
by conjugation, and with the order relation given by the opposite of inclusion.
Thus the vertices of
the maximal parabolic subgroups of (assuming the rank (5.2) homology n-1,
n
of the
G.
BN-pair
in
G
1Q
is
0
and
respectively. Since
£2),
The rational
is zero except in dimensions
and in these dimensions affords
are
We first have
THEOREM (Solomon-Tits [16]). HS(A)
A
StQ = 1Q , this result suggests the
and StQ,
70
Characters of Groups of Lie Type
possibility of a homological interpretation of the duality operation, in the general case, using a suitable coefficient system over the building
A.
Such a result
has been obtained by Deligne and Lusztig [10].
(See
also Curtis-Lehrer [8] for a proof of (5.2) in terms of a comparison of End^H^A)) with
EndCQW(Hx(r)) ,
and [9] for extensions of this idea to the homology of the building over certain coefficient systems.)
C. W. Curtis
71
REFERENCES 1.
D. Alvis, Duality in the character ring of a finite Chevalley group, Proc. Symp. Pure Math.a No. 37 (1980), 353-357, Amer. Math. Soc., Providence, R.I.
2.
D. Alvis, Duality and character values of finite groups of Lie type, J. Algebra, 74 (1982), 211-222.
3.
N. Bourbaki, Groups et_ algebres de Lie, Chap. 4-6, Act. Sci. et Indust. No. 1337, Hermann, Paris, 1968.
4.
R. W. Carter, Simple groups p_f Lie type, Wiley, New York and London, 1972.
5.
C. W. Curtis, The Steinberg character of a finite group with a (B,N)-pair, £. Algebra, 4 (1966), 433-441.
6.
C. W. Curtis, Reduction theorems for characters of finite groups of Lie type, J. Math. Soc. Japan, 27 (1975), 666-668.
7.
C. W. Curtis, Truncation and duality in the character ring of a finite group of Lie type, J. Algebra, 62 (1980), 320-332. ~~
8.
C. W. Curtis and G. I. Lehrer, A new proof of the theorem of Solomon-Tits, Proc. Amer. Math. Soc.3 to appear.
9.
C. W. Curtis and G. I. Lehrer, Homology representations of finite groups of Lie type, Contemporary Math., 9 (1981), 1-28, Amer. Math. Soc., Providence, R.I.
10.
P. Deligne and G. Lusztig, Duality for representations of a reductive group over a finite field, to appear.
11.
W. Feit and J. G. Thompson, Solvability of groups of odd order, Pacific J. Math.., 13 (1963), 775-1029.
12.
R. B. Hewlett and G. I. Lehrer, A comparison theorem and other formulae in the character ring of a finite group of Lie type, Contemporary Math., 9 (1981), 285-288, Amer. Math. Soc., Providence, R.I.
72
Characters of Groups of Lie Type
13.
K. McGovern, Multiplicities principal series representations of finite groups of Lie type, £.• Algebra, to appear.
14.
F. A. Richen, Modular representations of split (B,N)-pairs, Trans. Amer. Math. Soc. , 140 (1969), 435-^60.
15.
L. Solomon, The orders of the finite Chevalley groups, J. Algebra 3 (1966), 376-393-
16.
L. Solomon, The Steinberg character of a finite group with a (B,N)-pair, Theory of Finite Groups (ed. by R. Brauer and H. Sah), W. A. Benjamin, New York, 1968, 213-221.
17-
T. A. Springer, A formula for the characteristic function on the unipotent set of a. finite Chevalley group, £. Algebra 62 (1980), 393-399.
18.
R. Steinberg, Lectures on Chevalley groups9 Yale University, 195TI
19.
R. Steinberg, Endomorphisms of linear algebraic groups, Mem.' Amer. Math. Soc.a 80 (1968).
Arithmetic Groups JAMES E. HUMPHREYS
1.
LATTICES IN LIE GROUPS
Arithmetic groups arise naturally as discrete subgroups of Lie groups, defined by arithmetic properties.
In this
lecture I want to describe some of the possibilities, especially when the Lie group is semisimple.
For a compre-
hensive treatment, RaghunathanTs book [22] would be a natural starting point (to be supplemented by more recent research papers). (1.1)
Let G be a connected Lie group.
meant a discrete subgroup
T
for which
measure (induced by Haar measure on G). different cases to consider: if
G/r
T
By a lattice in G is G/r
There are two very
is uniform (or cocompact)
is compact, nonuniform otherwise.
the standard lattice
1
f
in
is a nonuniform lattice in
n
R
has finite
For example,
is uniform, while
SLn(Z)
SL (R).
Both of the examples just mentioned have an obvious arithmetic flavor.
To be more precise, we have to consider 73
74
Arithmetic Groups
a Lie group G which arises as the topological identity component
G(FO° in the R-points
G(R)
of an algebraic
group G defined over Q (or other number field). Many familiar Lie groups do arise in this way. in some general linear group
GL , then
If G_ is embedded G n GL (Z) is
often a lattice in G(e.g., when G is semisimple, by results of Borel and Harish-Chandra). Whether it is a lattice or not3
G n GLn(l) or any commensurable subgroup of
called an arithmetic subgroup of G.
(KQ)
is
(Recall that two groups
are commensurable if their intersection has finite index in each.) We will stick to the case of groups defined over Q; the process of "restriction of scalars" often makes this the essential case. Several questions can be posed right away: (1) Does a given Lie group G contain both uniform and nonuniform lattices? (2) If G has the form
G(JR)°
for a Q-group G, are its
arithmetic subgroups actually lattices?
If so, is every
lattice in G of this type? (3) What group-theoretic properties does a lattice (or arithmetic group) r have?
Is
r
finitely generated
(f.g.)? finitely presented (f.p.)? torsion-free? its normal subgroups?
What are
(These questions, or others of a
cohomological nature, can often be studied effectively in the context of G and its homogeneous spaces.) Remark.
"Arithmetic groups" also arise in the setting
J. £. Humphreys
75
of algebraic groups over global function fields.
In another
direction, one can study "S-arithmetic" subgroups, where S is a finite set of valuations including all archimedean ones. (1.2)
Lattices in solvable Lie groups have been rather
thoroughly studied (cf. [22, Ch. II-IV]). To list a few of the key results, due to Mal'cev, L. Auslander, Mostow, and others, it is convenient to assume that G is simply connected (s.c.); the general case can usually be reduced to this one. (1) Let G be a s.c. nilpotent Lie group.
Then G has
a lattice subgroup iff the Lie algebra of G has a basis with rational structure constants.
(The idea of the proof
is to obtain a lattice by exponentiating the Z-span of such a basis.) (2) An abstract group
T
is isomorphic to a lattice
in some s.c. nilpotent Lie group iff
T
is f.g., torsion-
free, nilpotent. (3) All lattices in a s.c.. nilpotent Lie group are uniform and arithmetically defined. (4) All lattices in a s.c. solvable Lie group are uniform, but not necessarily arithmetically defined. (5) A lattice in a s.c. solvable Lie group is polycyclic (hence f.g.).
Any polycyclic group has a normal
subgroup of finite index which is isomorphic to such a lattice.
(Here the idea is to embed the given polycyclic
76
Arithmetic Groups
group in some in
GL (Z) and then study its Zariski closure
GLn(C).) (6) Given a lattice
T
in a s.c. solvable Lie group
G, there is a faithful representation which
f :G •*• GL (B)
for
f(D c GLn(Z). The results (2) and (5) suggest how Lie groups or
algebraic groups may be profitably used to study polycyclic groups.
(Cf. the recent work of F. Grunewald--P.P. Pickel-
D. Segal, S. Donkin, and others.) (1.3) The study of lattices in semi-simple Lie groups is in some respects far more complicated than in the solvable case.
Lattices still turn out to be f.g. (which allows
one eventually to conclude that all lattices in Lie groups are f.g. [22, 13-21]), but they may or may not be uniform. Borel showed that when G is noncompact, G has both uniform and nonuniform lattices (cf. [22, Ch. XIV]).
The proof
reduces quickly to the case of a simple group G isomorphic to its adjoint group.
Then the idea is to find an
auxiliary algebraic group G over Q and an epimorphism f:G(R)° = Gf -»• G
with compact kernel.
suitable arithmetic subgroups
f
r
f
of G
By locating with
compact (resp. noncompact), one gets lattices of the desired types in G. subtle.
G f /T f r = f(rf)
The construction here is rather
For example, to make
G'/F1 noncompact, it is
essential to have a nontrivial unipotent element in
Tf,
J. E. Humphreys
77
which depends on having a nonzero nilpotent element in a suitable Q-form of the Lie algebra. special case like
Of course, in a
G = SL (R), one might argue directly
that the arithmetic subgroup lattice (cf. [8] or [10]).
SL (2)
is a nonuniform
But even here it is difficult
to exhibit straightforwardly a uniform lattice, without use of a larger auxiliary group G1. As noted above, Borelfs proof of the existence of
(1.4)
both kinds of lattices in a semisimple Lie group is based on a construction of arithmetic groups.
The question
remains: Must all lattices be obtained in this way?
To
make the question precise (and to avoid uninteresting technicalities), we formulate a definition:
Let G be a
connected semisimple Lie group, G* its adjoint group, p:G •*• G* be
the canonical map. A lattice
T
in G is said to
arithmetic if there exists an algebraic group Gf over Q,
with an arithmetic subgroup f:Gf(|R)0 •* G*
T! c G/(Q) and an epimorphism
such that Ker f is compact and
f(Tf) has
finite index in p(F). For certain semisimple groups of JR-rank 1, such as S0(2,l) = PSLp(R), answer:
our question actually has a negative
There exist non-arithmetic lattices (both uniform
and nonuniform).
Examples involving
S0(n,l) (n<5) were
first discovered by Makarov and Vinberg, while Mostow [17sl8] has recently found others in the groups
SU(n,l)
78
Arithmetic Groups
.
It remains to be seen whether such examples also
occur in the groups
Sp(n,l) and whether they are limited
to low dimensions. (1.5)
Por semisimple Lie groups of R-rank>2, it was con-
jectured first by Selberg (in the uniform case) and later by Pyatetski-Shapiro (in the general case) that all "irreducible" lattices are arithmetic.
(A lattice is
irreducible if its projection to any nontrivial proper factor is non-discrete: this rules out obvious counterexamples involving products of rank 1 groups.)
The first
complete proofs of these conjectures were given by Margulis (cf. [12], niques.
[13],
[30]), using a dazzling array of tech-
Here is a very brief indication of how he proceeds
in [133.
ARITHMETIC ITY THEOREM.
Let G be a connected semi-
simple algebraic group over (R, of DR-rank £ 2, and assume G = G_(R)° lattice
T
has no compact factors.
Then any irreducible
in G is arithmetic.
SUPERRIGIDITY THEOREM. meticity Theorem,
T
Let G be as in the Arith-
an irreducible lattice in G.
Let k
be any local field of characteristic 0(R, C, or a finite extension of Q ) . Let P be a connected semisimple k-group without center,
$:T •* P(k)
is Zariski dense in P.
a homomorphism such that
Then: (i) If
k # R,C, *(D
is
J. E. Humphreys
79
relatively compact in the k-topology. then
F = FT*!^*
(ii)
If
k = IR or C,
a product of k-groups, where
is relatively compact in the k-topology and extends to a rational homomorphism
pr.j(<j>(r))
pr2 o$:T •»• Fp
G_ •*• Pp.
Although its formulation is technical looking, the second theorem implies the first and has other far-reaching implications (e.g., for the study of isomorphisms between simple algebraic groups over various arithmetic rings). For the proof, Margulis draws together a wide range of methods: ergodic theory, function spaces,....
Once estab-
lished, it can be invoked (in several different ways) in the proof of the Arithmeticity Theorem.
For example, at
one stage it is known that, for the given lattice
r,
there exists a centerless semisimple matrix group
H over
Q and a monomorphism
a:T -> H(Q)
with Zariski dense image.
After composing with the inclusion into
H(Q ),
rigidity Theorem forces the image of
to be relatively
T
compact in the Q -topology, for each prime p.
the Super-
This means
that the powers of p in denominators of matrix entries in a(T)
are bounded.
But
T
is f.g., so the denominators in
question can involve only finitely many primes. these statements,
a(T) n H(Z)
Combining
has finite index in
This is a major step toward proving that
T
a(T).
is arithmetic.
We should mention a further striking consequence of Margulis1 methods:
With G and
T
as above, each noncentral
80
Arithmetic Groups
normal subgroup of
T
is of finite index.
Earlier results
of this type mostly depended on having a positive solution to the congruence subgroup problem.
2.
FINITE GENERATION AND FINITE PRESENTATION
Given an arithmetic group whether
T
F,
it is natural to ask
is finitely generated (f.g.) and, if so, whether
it is in fact finitely presented (f.p.).
These questions
can sometimes be answered positively by exhibiting generators and relations; but in other cases only a qualitative or indirect proof is available.
And in a few situations,
negative answers turn up. (2.1)
Consider a very classical example: the group
T =
A
PSL2(2), VIII].
or its close relative
Let S=
£°
with respective images
r = SL2(Z), cf. [19, Ch.
J), T- (5-^ s,t,u
in
= (J j) m
T.
Note that
?,
T = SU.
A
From linear algebra one knows that U (or equivalently, by S and T). elements T
s, t
T So
is generated by S and T
is generated by the
of respective finite orders 2,3.
In fact,
is the free product of the cyclic groups they generate. A
To see this, it is easier to work in T. It has to be shown a e e b l n that A = + T ST S...T S can never reduce to + I (where a,b € {0,1} a
= b = 0,
and so
e. e {1,2} ). By rearranging, we may assume e e i n A = j ^ S T . . . S T . Now it is enough to show
J. E. Humphreys that no nontrivial word in the semigroup generated by \0
3J and
for each while
ST
* (l l) reduces to + I.
Z = fe °\ in this semigroup,
(ST2)Z
= /*
Z have like sign.
\ .
But note that (ST)Z = - fa*c b*~)
By induction, all entries of
It follows that if Z has a nonzero entry
off the diagonal (which ST and ST true for these longer words. (2.2)
ST =
both do) , the same is
So we can never reach + I.
Nielsen found a finite presentation for
which Magnus later reduced the case of
SL-(Z),
SL (20
for
In a modern guise, this fits into the computation of by Silvester-Milnor: SL (Z) (n £ 3) elementary matrices
to
n £ 3. KpZ
is generated by the
E..(i * j ) , where
E.. has 1 in the
(i,j) position and on the diagonal, but 0 elsewhere, subject only to the relations:
(EjM»Ek£) = 1 if j * k, i^£;(E1 .,E.k)
if i,j,k are distinct; (Ei2E2l"lE12^=
= E±k
two relations alone define a central extension SLn(Z),
with kernel
Iu
The
flrst
St (Z) •>
K2Z « Z/2Z. (The covering group is
called the Steinberg group.) Other Chevalley groups GKZ) such as
Sp2n(Z) were sub-
sequently studied by Klingen, Wardlaw, Behr, HurrelbrinkRehmann, culminating in the explicit presentations of Behr [6].
He views G(Z) as an amalgamated product of rank 2
parabolic subgroups, via the action of G(Z) on a simplicial complex introduced by Soule*.
Then the rank 2 cases
Spit(Z), and G2(Z) can be plugged in.
S
82
Arithmetic Groups
Relatively few groups over other arithmetic rings have been treated as explicitly as these groups over Z: mainly SLp
over rings of integers of imaginary quadratic
extensions of <8, cf. Swan [293.
In [20] O'Meara
proved that certain of the groups
GL -SL
over Hasse
domains (rings of S-integers in global fields of arbitrary characteristic)are at least f.g. (2.3)
If one is willing to settle for less explicit infor-
mation about generators and relations, far more general arithmetic (and S-arithmetic) groups can be shown to be f.p. by the reduction theory of Borel, Harish-Chandra (cf.[8], [10]).
The idea is to start with a semisimple group G over
Q, with
r = G(2)
or other arithmetic subgroup acting on a
homogeneous space X of the Lie group G(R).
There is an
open "Siegel set" in X approximating a fundamental domain for the action of
T.
Then a simple lemma produces (in
principle) a finite generating set in LEMMA.
Let X be a connected topological space, acted
on by a group U.T = X.
T(on the right).
Then the set
A,
so
y! € T ! 3
1
U.T
we get
forces U.T1 « X, That
A
Let U be open in X, with
A = { Y ^ r j U - y n U # <)>}
(The proof is easy: by
T:
is open.
Let If
ye Ay1 <= rf. whence
Tf
generates F.
be the subgroup generated
U.y n U.yf * $
for
yeT
,
Since X is connected, this
Ff = T.)
is finite in the case of our arithmetic group
J. E. Humphreys
83
is the hard thing to prove, and the proof gives little insight into the nature of
A
even for familiar groups
r.
(It is somewhat like proving that class numbers are finite without actually calculating them.) Behr [4] went on to show that the group
T
above is in
fact f.p., where the relations involving elements of the "obvious" ones (cf. [10,13-**])•
A are
By using the Bruhat-
Tits theory of reductive groups over local fields, Kneser and Behr w«re also able to treat S-arithmetic groups in a similar spirit. (2.4)
For arithmetic subgroups of algebraic groups over
number fields, the methods of Borel, Harish-Chandra, Behr lead to positive results about finite presentability.
But
for groups over function fields, there are some negative results, and in general the terrain has been less well explored. Here is a brief survey of the best studied situation: (* is a Chevalley group (scheme), usually assumed to be simply connected, e.g., SL
or
s
P2n-
field in one variable over F , such as scendental. Og
K is a function Fa(t)
with t tran-
S is a finite nonempty set of primes of K, and
is the ring of S-integers in K(e.g., IP [t] or F [tjt"1]
in case |S | = 1 or 2, K = Fq (t)). the S-arithmetic group in question.
Finally,
T = G(0g)
The known results
depend crucially on two numerical invariants:
is
84
Arithmetic Groups
r = rank G(e.g., n-1 for SLn and n for sP2n^»
s
" Is!•
example, Behr [5] shows (in a much more general setting) that
T is f.g. unless r « s = 1. Whether
T is f.p. in
this case remains unsettled except in a few cases indicated in the table below.
s = 1 r B 1 SL2(Pq[t]) is
s ^3
s -2
SL2(0S) is f.p. (Stuhler [27])
SL0(0Q)is not f.p.
not f.g. (Nagao) (Stuhler [2?])
r =* 2 SL3(Fq[t]) is not f.p. (Behr3 Soule [73)
r > 3 G(Pq[t]) is f.p. ( Rehmann , Soule [24]')
OCPqCt.t1"1]) is
?
f.p. (except possibly Gg) (Hurrelbrink [11]) OjpqCt.t"1]) is f.p. (Hurrelbrink [11])
? !
Rehmann has formulated a general conjecture for a semisimple group (scheme) G_ and corresponding S-arithmetic subgroups.
Here the rank r
of (3 over the different com-
pletions K of K (veS) may vary, so the crucial number is r = Z rv (= rs for a Chevalley group). The conjecture is °° vcS that T is not f.g. when r^ « 1, that r is f.g. but not f.p. when (2.5)
r^ = 2, and that
r is f.p. when
r^ £ 3-
It is not difficult to visualize how one might go
J. £. Humphreys
85
about proving that some f.g. group is f.p. be shown that such a group is not f.p.?
But how can it
In [7] Behr uses a
method somewhat like that of Stuhler [27] to deal with T = SL-dF [t])
(or other rank 2 group when -1 jl Pq2)-
Here is a very rough sketch of the method. is a discrete subgroup of the locally compact group1
T
P = SL~(k), where k is the completion of P (t) relative to a valuation for which 1/t is a prime element.
The
Bruhat-Tits theory yields an associated "building" I, a simplicial complex made up of "apartments", each in turn subdivided into "quarters".
G and hence
a quarter Q in an apartment A.
F
acts on I. Fix
Then a geometric argument
due to Soule" shows that every simplex in I is sent by §. unique simplex in Q.
But Q is in a natural way an in-
creasing union of bounded subsets Q(n), whence if
T to
I = \* I(n)
I(n) = T.Q(n). Now the "Weyl group" W of the Tits system in G is just
an affine Weyl group (the symmetric group S- extended by translations), and A is covered by W-translates of a simplex
C
c Q
whose vertices may be identified with
certain large subgroups that
F
with
T.
is generated by
P0,P-.,P2 rQ
and
of G.
It can be shown
T^ n F2 (where r± = FnP.j),
stabilizing the vertex P^.
So words in these
generators yield edge-paths in I, and relations correspond to closed paths at the vertex PQ.
Suppose
T
to be f.p.
86
Arithmetic Groups
for this set of generators.
Since I is contractible, the
paths belonging to relations can be contracted to PQ1 each contraction involving only finitely many simplices (hence all taking place in some I(n)).
To reach a contradiction,
Behr then exhibits for each n some relation whose path is not contractible in I(n).
3.
NORMAL SUBGROUPS
Another natural question to ask about an arithmetic (or S-arithmetic) group is this: groups?
Paradoxically, the answer is more complicated for
"small" groups like
(3-D
What are its normal sub-
SL2(Z) than for "big" groups like
Consider again the modular group
r = PSL2(2), cf.
[19, Ch. VIII]. It was seen in (2.1) that
T is a free
product of cyclic groups generated by elements s of order 2 and t of order 3- Note that u = st has infinite order (as the image of
f
|) ) • Much is known about normal sub30) groups of small index in T. The subgroup T2(resp.I generated by all squares (resp. cubes) is just the normal closure of t (resp. s), and has index 2 (resp. 3)-
These
are the only normal subgroups having index 2 or 3-
The
J. £. Humphreys
87
derived group and
(r,O
Tab = r/(r,O
is their intersection, having index 6, is generated by the image of u.
It can
be shown that every proper normal subgroup of finite index d other than example,
F2,r3
is_ free, of rank
l+(d/6).
For
(F,r) is free of rank 2, with generators stst
and st st.
(Thus the derived group of
p
(T,r) is free of
infinite rank and has infinite index in r.) There are other obvious normal subgroups of finite index:
T(n) = kernel of natural map
r -> PSL2(Z/n2) (n^l) =
principal congruence subgroup o_f level n.
Any subgroup
including one of these is called a congruence subgroup. The notion of level can be extended to any normal subgroup
A
(say of index d): the level of A is the least positive n for which
un e A
(so n|d).
For example,
level 6 and includes the subgroup is 72.
F(6), whose index in T
Wohlfahrt gave a nice criterion:
level n.
Then
A
(F,r) has
Let
A
is a congruence subgroup iff
have
A => F(n).
In fact, relatively few subgroups of finite index are congruence subgroups.
For example, r has infinitely many normal
subgroups of level 6, but only 4 of them include (3.2)
F(6).
Congruence subgroups can be defined in a rather
general setting.
Let K be any global field (a number field
or a function field in one variable over a finite field). Take S to be any finite nonempty set of valuations of K, including at least the archimedean ones.
Then let A be the
88
Arithmetic Groups
ring of S-integers in K, the elements at which all take nonnegative values.
v jl S
(When S consists of the archime-
dean valuations of a number field, A is just the usual ring of algebraic integers.) For a linear algebraic group G_ over K, the S-arithmetic subgroup cipal congruence subgroup
TV
TA = G(A)
has a prin-
for each nonzero ideal q of
A, defined to be the kernel of the canonical map
_G(A) _ _ •*• _ _ G(A/q).
As before, a subgroup of ^FA
containing
one of these is called a congruence subgroup.
Then it may
be asked: Is every subgroup of finite index a congruence subgroup? (It may also be asked whether
I\
has any "non-
obviousff normal subgroups of infinite index.
Though appar-
ently unrelated to the first question, this question can often be studied effectively in tandem with the congruence subgroup problem.) Serre formulated the problem in an elegant way, by considering simultaneously the group
G = G(K)
and its S-
arithmetic (resp. congruence) subgroups, cf. [3]* [14],
[25].
This leads to a short exact sequence
1 •*• C(G) -* G •*- (J •* 1, A
[10],
involving the respective completions
__
G, G
of G in the topology whose fundamental system of
neighborhoods of 1 consists of the S-arithmetic (resp. conA
gruence) subgroups.
Restrictions to the closures A
of
TA
yields the related sequence: 1 -> C(G) -> T
__
TA, TA __
+ T. +!•
J. £. Humphreys
39
The question then becomes:
Is_ C(G) trivial (and, If not,
how big is it)? In case G is simple and simply connected, G and F.
have straightforward descriptions, due to strong
approximation; e.g., FA
is just the product of the groups
G[(A ) taken over the integers A of all completions
(3.3) The case
(* = SLp
can serve as a microcosm of the
congruence subgroup problem for simple algebraic groups. In this and the following lecture I want to sketch some of the key points in Serre [25]. results: When
Consider first the "negative"
|S| = 1, C(G) is_ infinite.
This involves
three separate cases: (1) The "rational" case A = *& (cf. (3-D above). (2) The "imaginary quadratic" case, e.g., A = 2[i]. (3) The "characteristic p" case, e.g., A = FqCt] (<1
=
power of p) . In the first two cases, C(G) actually has the cardinality c of the continuum, while in the third case |C(G)| = 2C.
(According to Melfnikov [15], the structure
of C(G) in case (1) is that of a "free profinite group of countable rank" . ) (3.4) To show that C(G) is infinite in each case, Serre first replaces
T» by a suitable torsion-free subgroup T
of finite index (without loss of generality). T
can be^ the derived group of
TA
For example,
in case (1), or can be
90
Arithmetic Groups
a principal congruence subgroup in case (3). rab = r/(r,D
step is to prove that
Then a key
is infinite in each
case: (1)(2) (3)
rab
Ta
is a f.g. infinite abelian group.
is the direct sum of a f.g. group and a
vector space over (P of countably infinite dimension. Particular,
T
(In
itself cannot be f.g., cf. (2.4).)
Case (1) is classical, since torsion-free here implies free, cf. (2.1). auxiliary group F
In the other cases, Serre defines an U(F),
the direct sum of intersections of
with various maximal unipotent subgroups of G, and a
natural map
a:U(T) -*• T
. In case (2),
is already known to be f.g. number of K,
U(T)
[20].
T
and hence
rab
If h is the class
turns out to be free abelian of rank QVj
2h, while Kera has rank h, forcing
ra
to be infinite.
The proof depends on an identification of homology groups:
H^DXj,) •> H-L(Xr), where
and the orbit space (3-5)
a
X/r
with a map of
X = SL2(C)/SU2(G)
has a compactification
Xp.
It still has to be deduced that C(G) is infinite.
Cases (!) and (2) can be argued together:
Let
C«
be the
A
intersection of C(G) with the closure of C(G)
were finite,
of Bass-Milnor-Serre
Cp would be also.
T
in G.
If
Then the arguments
[3,§l6], which depend just on the
finiteness of the congruence kernel, would imply the finiteness of
H^I^Z) = Hom(rab,2),
contrary to (3-4).
J. £. Humphreys
91
In [3] it is essential that the characteristic be 0, e.g., to get the splitting of short exact sequences of finite dimensional Kr -modules, or to apply results of Lazard on p-adic groups. For case(3), one can argue that
\T\ = c,
since the
congruence topology has a countable basis of neighborhoods of 1.
On the other hand, the result of (3-4) on A
implies that
Ta
A
T
maps onto the second dual
V
of an infiA
nite dimensional vector space V over F ; here |C(G)| ;> 2C.
forcing
|V| =2
,
(Alternatively, Serre [26,11,2.?]
uses the action on the Bruhat-Tits tree to show more directly that the set of S-arithmetic subgroups of
T
has
cardinality c.)
4. (4.1) When
NORMAL SUBGROUPS (CONTINUED)
Retain the notation of (3-2): G, K, S, A, TA, G. etc. G = SLp
and
|S| = 1,
the congruence kernel C(G) is
infinite and the congruence subgroup problem has therefore a strongly negative solution.
Serrefs proof involves a
close study of the group structure of
T., G,
and of the way
TA (or its subgroup T) acts on a related topological space, but requires no delicate arithmetic information. |S| £ 2,
When
deeper arithmetic considerations enter into the
solution, which is positive or "almost" positive:
92
Arithmetic Groups
THEOREM (Serre). C(G) = y
Let
G - SL2, |S| :> 2.
Then
(the finite group of roots of unity in K)
if K
is a totally imaginary number field and S the set of all archjmedean valuations.
Otherwise
The simplest case occurs when
C(G) = 1. K = Q, S = {p,»}. (This
had been studied earlier by Ihara and Mennicke.) The assumption
|S| £ 2 crucially affects the
structure of the group U of units of A, which has the form y x an ' . In particular, U now has elements of infinite order. The proof of Serrefs theorem involves showing that
(4.2)
C(G) lies in the center of G.
Here an essential role is
played by an auxiliary family of normal subgroups of for a nonzero ideal q of A, E
I\:
is the normal subgroup gen-
erated by "q-elementary11 matrices in
T . (It is unclear
at first whether E has finite index or not.)
Now the
proof goes in steps. (1) Any subgroup N of finite index in
r.
includes
some E . (We may assume N is normal, of index n, so q = nA
will do if char K = 0.
In characteristic p, the
choice of q is a bit more complicated and uses the fact /u 0 \ that the set of u € U with \ e N has finite index JnN.)
V° M
(2) A non-central subgroup H of G normalized by an S-arithmetic subgroup N includes E
for some q.
(H must
f. £. Humphreys
93
contain some matrix
J with
ac * 0 ,
otherwise its
Zariski closure in SL^ would be a proper, normal, but noncentral subgroup.
Now (1) yields an
E , c N,
and some
delicate manipulation of matrix entries using elements of infinite order in U yields the required (3) Set C = F /E ^ rA/Eq'
then the image of
[u
_ j
in
If
E
c H.)
u c U, m « |y|,
r
^/Ea
centralizes
C .
Y
(The proof is not easy: it requires Cebotarev density, Artin reciprocity, etc.) (4)
Let
C = lim C q. T« ^ H
acts (via inner automorph-
isms) on C , hence on C, and this action extends (via to an action of G on C.
(2))
This action is trivial, whence C
is abelian (and f.g. since T
is).
action contains ju
(3)» hence is infinite. But
-]
^
(The kernel of the
G is almost simple.) ^
(5)
A
C(G) = lim C
C(G) is central in G. (4.3)
(profinite completions), and thus (This follows from
(4).)
Now the proof shifts gears, applying the theory of A
Moore [16] to the central extension
__
1 •*• C(G) •* G •*• G •*• 1.
This theory implies that G is isomorphic to the "universal covering" of G (relative to G), with C(G) isomorphic to the relative fundamental group
•n'1(G",G).
As a result of
94
Arithmetic Groups
Moore's calculation of fundamental groups, C(G) is of the form asserted in (4.1).
Moreover, C
turns out to be finite
and cyclic, of order dividing m in the totally imaginary case but trivial otherwise; so the index of Eq in finite after all, and
C = C(G).
TA A
is
As a further byproduct of
step (2) above, we see that for any subgroup N of finite index in
I\,
the normal subgroups of N all have finite
index or lie in
{+ 1} .
For example, Nab
contrast to what can happen if
is finite (in
|s| =1).
It should be emphasized that Moore's determination of relative fundamental groups involves the whole arsenal of class field theory.
So by the time Serre concludes his
argument he has invoked a considerable amount of arithmetic in order to answer what might seem to be a straightforward group-theoretic question. (4.4)
When ^ is a Chevalley group (simple, simply con-
nected, split over K) of rank ^ 2, the solution of the congruence subgroup problem is "almost" positive in the same sense as above.
For example, take S to be the set of
archimedean valuations of a number field K. if K is totally imaginary; otherwise
Then
C(G) = 1.
C(G) =y This
situation was studied independently by Mennicke and by Bass-Lazard-Serre when (j - SL
or
Sp«
G = SLn (n^3) over Q, then for —• by Bass-Milnor-Serre [3]. Matsumoto
[14] completed the treatment of Chevalley groups by making
J. E. Humphreys
95
heavy use of the results of Moore [16].
(For a partial
exposition, see [10].) (4.5)
The congruence subgroup problem for other simple
algebraic groups over K has not yet been fully solved, but there has been substantial recent progress.
The most
likely conjecture goes as follows (for (} absolutely simple, simply connected): pletion K over S).
Let r
of K for each
be the rank of G over the comv c S,
and set
r = Z r
In case S contains non-archimedean valuations,
require G to have positive Ky-rank for each such v. is a kind of non-compactness.). when
(sum
r > 2,
(This
Then C(G)ought to be finite
as for Chevalley groups of rank £ 2.
More-
over, C(G) ought to be trivial unless K is a totally imaginary number field and S its set of archimedean valuations; in this case C(G) ought to be
y (or conceivably a
quotient of y). Here is a quick summary of some recent work in this direction. In [23] Raghunathan showed that C(G) is finite if K is a number field and G has K-rank at least 2 (while C(G) has a p-subgroup of finite index in the function field case).
He also showed that each normal subgroup of an S-
arithmetic group (when G has K-rank £ 2) is either finite and central, or else includes an S-elementary subgroup (whose index is finite in the given arithmetic group).
As
96
Arithmetic Groups
in the earlier work, it is essential to show that certain extensions are central.
Building on this work, Deocihar C93,
has gotten more precise results in the case of quasi-split groups (including Du). Bak-Rehmann [2] have made a detailed study of non-split groups of type A.
In particular, they solve the congruence
, subgroup problem for many groups
SLp(D)
and "most" groups
SL (D), n ^ 3, where D is a finite dimensional central division algebra over a global field. More recently Bak Ell has announced a more comprehensive solution of the problem for classical groups (other than DJ.) of rank at least 2.
This involves a reduction to
the cases treated in C23, and uses heavily some techniques of algebraic K-theory.
(Cf. his monograph, K-theory of
forms, Ann, of Math. Studies 98 (1981).) Independently, Prasad and Raghunathan [21] have made considerable progress on the congruence subgroup problem and the related "metaplectic" conjecture. One final remark:
It is known that a non-split simple,
simply connected group G of positive K-rank contains a simply connected split group of the same rank (constructed by Borel-Tits).
It is worth asking whether the respective
congruence kernels can be related directly, since the latter is known explicitly.
Such comparisons with split
or quasi-split subgroups already play a role in the work of Deodhar and Prasad-Ragunathan.
J. £. Humphreys
97
REFERENCES 1.
A. Bak, Le problSme des sous-groupes de congruence et le problSme m^taplectique pour les groupes classiques de rang > 1, CJ.R. Acad. Sci. Paris 292 (1981), 307-310.
2.
A. Bak, U. Rehmann, The congruence subgroup and metaplectlc problems for SL « of division algebras (.preprint)
3-
H. Bass, J. Milnor, J.-P. Serre, Solution of the congruence subgroup problem for SL (n £ 3) and Sp2 (n ^ 2), Inst. Hautes Etudes Sci. Publ. Math. 33 (196?), 59-137-
4.
H. Behr, Uber die endliche Definierbarkeit verallgemeinerter Einheitengruppen, II, Invent. Math. 4 (1967), 265-274.
5-
H. Behr, Endliche Erzeugbarkeit arithmetischer Gruppen uber FunktionenkSrpern, Invent. Math. 7 (1969), 1-32.
6.
H. Behr, Explizite Presentation von Chevalleygruppen uber ZZ , Math. Z. 141 (1975), 235-241.
7.
H. Behr, SL^dP Ct]) is not finitely presentable, pp. 213-224 in: Homological Group Theory, L.M.S. Lect. Note Ser. 36, Cambridge U. Press, 1979-
8.
A. Borel, Introduction aux groupes arithmetiques3 Hermann, Paris, 1969.
9-
V.V. Deodhar, On central extensions of rational points of algebraic groups, Amer. j\ Math. 100 (1978), 303-386.
10.
J.E. Humphreys, Arithmetic Groupsa Lect. Notes in Math. 789, Springer, Berlin, 1980.
11.
J. Hurrelbrink, Endlich prasentierte arithmetische Gruppen und Kp liber Laurent-Polynomringen, Math. Ann. 225 (1977), 123-129.
12.
G.A. Margulis, Discrete groups of motions of manifolds of nonpositive curvature, Amer. Math. Soc. Transl. (Ser. 2) 109 (1977), 33-45 [Russian original appears in proceedings of 1974 Intl. Congr. Math., Vancouver]
Arithmetic Groups
13.
G.A. Margulis, Arithmeticity of irreducible lattices in semi-simple groups of rank greater than 1 [Russian], appendix to Russian translation of [22], Mir, Moscow, 1977-
14.
H. Matsumoto, Sur les sous-groupes arithmetiques des groupes semi-simples deployes, Ann. Sci. Ecole Norm. Sup. 2 (1969), 1-62.
15-
O.V. Mel'nikov, Congruence kernel of the group SL«(S), tL Soviet Math. Dokl. 17 (1976), 867-870.
16.
C.C. Moore, Group extensions^of p-adic and adelic linear groups, Inst. Hautes ftudes Sci. Publ. Math. 35 (1969), 5-70.
17.
G.D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86 (1980), 171-276.
18.
G.D. Mostow, Existence of nonarithmetic monodromy groups, Proc. Natl. Acad. Sci. USA 78 (1981), 5948-5950.
19-
M. Newman, Integral Matricesa Academic Press, New York, 1972.
20.
O.T. O'Meara, On the finite generation of linear groups over Hasse domains, J. Reine Angew. Math. 217 (1965), 79-108.
21.
G. Prasad, M.S. Raghunathan (to appear).
22.
M.S. Raghunathan, Discrete Subgroups of Lie Groups 3 Springer, Berlin, 1972.
23-
M.S. Raghunathan, On the congruence subgroup problem, Inst. Hautes Etudes Sci. Publ. Math. 46 (1976), 107-161.
24.
U. Rehmann, S. Soule, Finitely presented groups of matrices, pp. 164-169 in: Algebraic K-Theory (Evanston 1976), Lect. Notes in Math. 551, Springer, Berlin, 1976.
25.
J.-P. Serre, Le probleme des groupes de congruence pour SL2, Ann, of Math. 92 (1970), 489-527-
26.
J.-P. Serre, Arbres, amalgames, SL2, Asterisque 46 (1977)
J. E. Humphreys
99
27. U. Stuhler, Zur Frage der endlichen Prasentierbarkeit gewisser arithmetischer Gruppen im Funktionenkorperfall, Math. Ann. 224 (1976), 217-232. 28. U. Stuhler, Homological properties of certain arithmetic groups in the function field case, Invent. Math. 57 (1980), 263-281. 29-
R.G. Swan, Generators and relations for certain special linear groups, Adv. in Math. 6 (1971), 1-77.
30. J. Tits, Travaux de Margulis sur les sous-groupes discrets de groupes de Lie, Sjm. Bourbaki 1975/76, Exp. 482, Lect. Notes in Math.~5^"7, Springer, Berlin, 1977-
Modular Representations of Algebraic Groups BRIAN J.PARSHALL
These lectures provide an introduction to the modular representation theory of semisimple algebraic groups. Sections 1 and 2 assume only a basic acquaintance with the theory of algebraic groups and with the standard language of representation theory.
Later sections, however, employ
the theory of group schemes, and so make more demands on the reader.
Nevertheless, it should become clear that the
study of positive characteristic phenomena is ideally suited to the approach defined by these techniques. Many important topics as well as many proofs are omitted or only barely sketched due to lack of time. The reader may consult the papers in the bibliography for further information. It is a great pleasure to thank Warren Wong and Alex Hahn for inviting me to visit Notre Dame and for making my stay there so pleasant. 101
102
Representations of Algebraic Groups
1.
ELEMENTARY THEORY
Throughout we will fix an algebraically closed field k. Unless explicitly stated to the contrary, we assume that
k has positive characteristic
(1.1) RATIONAL MODULES. group defined over
p.
Let G be an affine algebraic
k. A finite dimensional kG-module
V
is said to be rational if the associated homomorphism Pv: G -»• GL(V) is a morphism of algebraic groups. rational G-module we mean a kG-module
V
By a
which is a union
of rational finite dimensional submodules in the above sense. ¥e let M«
be the category whose objects are the
rational G-modules and whose morphisms are the G-module homomorphisms. (1.2)
EXAMPLES/REMARKS.
(a) MQ
is an abelian category
which is closed under the formation of tensor products9 direct limits, and duals of finite dimensional modules. Further,
it is easy to see that
MQ
possesses enough
infective objects. (b) The coordinate ring
k[G] of G is a rational
G-module relative to the left translation action of
G:
(g»f)(x) = f(xg), f 6 k[G], g,x e G. We may also view k[G] as a rational right G-module by using the right translation action of
G:
(f-g)(x) = f(gx).
(c) k[G] has a well-known commutative Hopf algePor the most part, all modules are taken to be left modules.
B. J. Parshall
103
bra structure, with comultiplication A: k[G] •»• k[G] 8 k[G], counit MQ
e: k[G] •> k, and antipode
r\: k[G] •*• k[G],
is isomorphic to the category of comodules for
Then k[G]
(see [14; 1.1] for more details). (d)
Let Q be the Lie algebra of
Lie algebra of all k-derivations the identity (18 D)A = AD.
Then
D
G, that is, the of
3
k[G]
satisfying
becomes a rational
G-module, called the adjoint module, as follows.
Take
g.D, g e G, D e 3, to be the derivation defined by (g.D)(f) = g-(D(g"1-f)).
It is easy to check that g.D
satisfies the required identity to be an element of (1.3) NOTATION.
We now take up the case when
(connected) semisimple algebraic group. we will assume that
G
G
Q.
is a
For simplicity,
is simply connected.
We list
below some of the standard notation that we will use throughout: B = T.U
Fixed Borel subgroup with maximal torus T, unipotent radical U
B" = T.U"
Opposite Borel subgroup
$
Root system of
$
Positive roots defined by B
U = {ou ,... ,otp}
Fundamental roots in
W
Weyl group
w
Long word in
<,>
W-invariant, symmetric, positive definite bilinear form on E = Z$ 8 E
T
in G
$
W
104
Representations of Algebraic Groups
av
coroot 2a/ , a e $
A
Weight -lattice in E spanned by the fundamental dominant weights (i)-^,...,^- (where ^,aY> = 6..»
I £ i,j £ A) A
Dominant weights
p
o^ +...+ Co.
A* r
r X e A+ satisfying 0 <_ <X,o?> j
£
Partial order on A given by X <_ y iff y-X is a sum of positive roots
X •»• X
Opposition involution on defined by X* = -wQ(X)
Pj T = Lu T.Uj T9 JS I
Parabolic subgroup containing B with Levi factor Lj having J as fundamental roots
If V
E
is a rational G-module and X e A, let
V^ = {v 6 V|t.v = X(t)v, t 6 T} denote the X-weight space for the action of T on V. ^
If V^
/ 0, we say that
is a weight of T in V. We can now state the
following basic result, due to Chevalley [12]: (1.4)
THEOREM.
Let G be a semisimple, simply
connected algebraic group over k. (a) Let V
be an irreducible rational G-module.
Among the weights of T in V mal weight of V.
X
there is a unique maxi-
(relative to <_ ), called the high weight
Necessarily9 X e A . Any two irreducible rational
B. J. Parshall
105
G-modules with the same high weights are G-isomorphic. (b) Conversely, given
X 6 A4", there exists an
irreducible rational G-module, denoted
S(X), of high
weight X. Let us make several comments concerning the proof of this result.
The first part (a) is relatively straight-
forward and parallels closely the proof of the corresponding result for complex Lie algebras. are several ways to construct
As for (b), there
S(X), all of them important.
The original method of Chevalley involves getting
S(X)
from an appropriate linear system on the projective variety
G/B.
This is closely related to the theory of
induced modules taken up in Section 2 below. one can obtain
S(X)
Secondly,
from the corresponding complex
irreducible module by reduction
mod p
(see [44; §12]).
Thirdly, there is a direct argument of Borel which uses only the normality of
k[G]
and an elementary result of
Chevalley (see [27; 31.4] for details). (1.5)
THE PROBENIUS MORPHISM.
The next basic result in
the modular representation theory is the tensor product theorem.
To explain this result, it is convenient to
assume that the semisimple group is defined and split over the prime field to arrange this.) where G
k [G]
kQ - GP(p).
(It is always possible
Thus, in particular,
k[G] - k0CGl ® k*
is the k -algebra of regular functions on
defined over
k . We consider the Probenius morphism
106
Representations of Algebraic Groups
a: G •*• G
defined by means of its comorphism
a*: k[G] -*• k[G] (f 8 c + fp ® c, f e kQ[G], c 6 k).2 Now if V
is a rational G-module, we can, for a positive
r, form a new rational G-module v^r' as follows. fT\ As a k-vector space Vx ' = V, but the effect of applying integer
g e G to v 6 V^r' is now ar(g).v. Thus, py(r) = p,,oar. We say that
V^r' is obtained from
V
by twisting by ar.
We may assume that
T is defined and split over k . Then (T\ r clearly the weights of T in V ' are the p -multiples of the weights of T in V. (1.6) TENSOR PRODUCT THEOREM.
Thus, S(X)(r) a S(prX). Let G be a semisimple,
simply connected algebraic group defined and split over k . For X 6 A"1", let X = XQ + pX-j^ +...+ prXr
be its p-adic
expansion (so that X. 6 A-,, cf. (1.3)).
Then
S(X) s S(XQ) 8 S(X1)(1) ®...8 S(Xr)^r). This result was originally proved in general by Steinberg.
We will briefly sketch an argument from [15]-
It is given in two steps: 1) Let V(g) be the restricted enveloping algebra of the Lie algebra
g
of G.
The adjoint action of G
on Q induces a rational action of V(g)
by its Jacobson radical
G
on the quotient of
J. Furthermore, since G
is connected, it must fix elementwise the finitely many central primitive idempotents of V(g)/J. Given a For example, if G = SL (k), then a is the endomorphism of G which raises each matrix entry of g 6 G to the pth power
B.). Parshall
107
restricted irreducible g-module
S, it corresponds to a
central primitive idempotent of
V(g)/J.
It follows,
from the simple connectivity of
G, that
S extends
(uniquely) to an irreducible rational G-module. 2) S-^
If
S
is an irreducible rational G-module, let
be an irreducible g-submodule.
Horn (S-^S) 8 S^ -> S
given by
s 6 S-j^. Here Horn (S-^S)
We have a G-isomorphism
$ 8 s •*• <|>(s), 6 Horn (S-^S),
is given its natural structure
as a rational G-module, using 1) above. S(X), X 6 A^, we see easily that
Applying this to
S(X) is Q-irreducible.
Now, for arbitrary X 6 A , write X = XQ + p9 XQ e A-, and 9 6 A . Clearly, factor of S(XQ) 8 S(9K module of
where
S(X) is a G-composition
, whence any irreducible
S(X) is g-isomorphic to
S(XQ).
g-sub-
A bookkeeping
argument involving high weights establishes that S(X) = S(XQ) 8 S(6)^ by induction on
. The proof can now be completed
r.
As a bonus of the above argument, we get the result of Curtis that the S(X), X 6 A-,, are exactly the distinct irreducible restricted ^-modules. (1.7)
FORMAL CHARACTERS.
rational G-module
Given a finite dimensional
V, we can form its formal character
ch V =
dim Y
eX
in the integral group algebra Z[A] of
A (so e .ey =
108
Representations of Algebraic Groups
One is then led to ask for an explicit formula for ch S(X) as well as for dim S(X).
Both of these problems were com-
pletely solved in the case of
k = ffi in a series of famous
papers written by Weyl in the 1920Ts; for example, Weylf s character formula states that I det(w) ew(X+pV I det(w) ew(p). At present weW weW no such formula is known in positive characteristic, ch S(X) -
although a relevant conjecture has been formulated Section 5).
(cf.
The answer is not even known in general for
G = SLn(k)!!
2.
INDUCED MODULES
In this section, we will study the process of obtaining rational modules for an algebraic group from those of a closed subgroup. (2.1)
BASIC DEFINITIONS.
an affine algebraic group
Let H be a closed subgroup of G
over
be the quotient morphism, where G/H of right cosets
Hg.
k.
Let
-rr: G •*• G/H
denotes the variety
Given a rational H-module
W, we
will construct a sheaf
L,, on
G/H.
is finite dimensional.
For
an open subvariety of
G/H, set morphisms
U
First suppose
rCU,!) = Mapdr^OOjW), the space of all f : IT (U) •»• W
(of varieties) such that
f(hg) - h.f(g), h e H, g e iT^U).
In general, set
L.T » llm L,TI , the direct limit taken over all finite W fit * dimensional submodules
W1
of
W.
We call
L
the
W
B. J. Parshall
109
sheaf induced from
W.
It is clear that
a module for the structure sheaf
^G/H
L... is actually of
G/H, and in
fact one can show it is quasi-coherent (and coherent if dim W < «).3
The space w|
= MapH(G,W)
of global sections of L.,
carries a natural G-module structure by setting (g.f)(gf) = f(gfg), f e W|G, g,g' e G.
Since MapH(G,W)
is clearly a G-submodule of Homk_al (k[W],k[G]) = k[G] 8 W, it follows that wIG is in fact rational for
W
finite
dimensional, hence rational in general by (1.2). is clear that the map
Also, it
Ev(W) : wlG -* W (f •> f(l)) is H-
equivariant, and it follows directly that given any rational G-module
V
and H-module homomorphism
there exists a unique G-module homomorphism such that
Ev(W)o§ = $.
$: V •*• ¥,
$: V -*• w|
This fact (Frobenius reciprocity)
merely expresses the fact that I , which is called the induction functor, is the right adjoint to the restriction functor
MQ •* MJJ (V -*• V!H).
At times we will denote |G
by |G
when
(2.2)
EXAMPLES AND ELEMENTARY PROPERTIES.
H
needs to be mentioned. We give below
an number of examples involving induced modules. (2.2.1) TENSOR IDENTITY. G
then W|
G
If V 6 Ob(MQ), W 6 Ob
® V = (W 8 V|)| . H
To establish this, we can
quasi-coherence is straightforward if TT is locally trivial in the Zariski topology. If W is finite dimensional, one can show in general that LTT is locally free, w cf. [18],
110
Representations of Algebraic Groups
reduce to the case in which
V
is finite dimensional.
K 6 ObCMg), we have HomG(K,w|G 0 V) = HomG(K 8 V*,W|G)
For
= HomH(K 8 V*)|H,¥) s HomH(K|H,W 8 V|H) s HomG(K, (W ® V|R) |G), whence the conclusion. (2.2.2)
We have k[H]|G s k[G];
COORDINATE RINGS.
pi
n
k|
= k[G/H].
Here
k
in
k|
denotes the one-dimensional
trivial H-module. (2.2.3)
PARABOLIC INDUCTION.
parabolic subgroup of G. follows that W|
Suppose that
Since
G/H
= T(G/E9L^) is finite dimensional for any
the case in which H = B
X on
is a
is complete, it
finite dimensional rational H-module
and
H
G
W.
Let us consider
is semisimple, simply connected,
is a Borel subgroup.
For a rational character
B, it will be convenient to let
X
also denote the
one-dimensional rational B-module defined by X. H
and
G G
0 iff W|
Now for
arbitrary, it follows immediately that Ev(W) = = 0, for any rational H-module
lar, if X|
W.
In particu-
/ 0, we see the image of the dual map Ev(X)*
is a B-stable line in
(X|G)*
implies that
Conversely, if X e A , the existence
X 6 A .
of weight
-X.
Hence, -X|G ^
of a nonzero B-module homomorphism S(X)* -> S(X)*/[U ,S(X)*] = -X
guarantees that-X|G ^ 0.
Below are some basic properties of a)
-X|
for X 6 A .
-X|G has the irreducible module S(X)* = S(X*)
as its socle (= maximal completely reducible submodule).
B. J. Parshall
.
Ill
<«!
Hence , -X | is indecomposable. This is immediate from reciprocity. b)
-X|
has a unique B-stable line. line
It has weight
X* and X* is the maximal weight in -X | (with respect to
If -X| G-submodule
has B-stable line isomorphic to y, then the D
it generates is a cyclic
U~-module.
follows easily from the Bruhat decomposition of a) it follows that X* £ y.
G.
This Prom
Now (2.2.1) above and
/*!
reciprocity imply that -y| isomorphic to X.
also has a B-stable line,
Thus, X* = y and
D
is the socle of
-X| . The desired result follows. rt
c)
The formal character ch -X|
is given by Weylfs
formula (1.7) (with X* in place of X). Hence, dim -X | G is given by Weyl's dimension formula. This is more difficult, cf. (3-3-*) below. It follows that ch -X|G = where mx(y)=[-X|G: S(y)] as a composition factor of
7 , m, (y)ch S(y), M X*^y e A"*" A
is the multiplicity of
S(y)
-X| . Over the years there r%
has been intense interest in the structure of -X | (or equivalently in the duals (-X| )*, popularly known as Weyl modules) . In particular, an explicit determination of the matrix [m, (y)] (or its inverse) would essentially A solve the above problem of calculating ch S(X), cf. (1.7).
112
Representations of Algebraic Groups
(2.2.4) SMITH'S THEOREM [43]. L
U
Fix a parabolic subgroup
P = Pj
and let P~" = J- J
be
group. _
Fix
be the LdT-socle of the space
S(X) and let
V
the
opposite parabolic sub-
S(X) J of U~-fixed points. We make several observations: j a) V is irreducible as an Lj-module. This is clear since
B~
has a unique fixed line in
S(X), and, a fortiori, in V. b)
Now view
act trivially.
V
Then
as a rational P-module by making Ev(V) maps
(V|p) J
Uj
injectively into
V.
In fact, if
f 6 ker Ev(V), then
f(l) - 0.
Thus, for
g e P, u e U~, f(gu) = (u.f)(g) = f(g) = g.f(l) = 0, so f = 0 on the dense subset
Pj.Uj of
G.
Hence, f = 0.
(Notice b) holds for an arbitrary rational P-module.) c)
Since -X* is the "low" weight in
V, we get a non-
zero B-module homomorphism
V •*• -X* which lifts to a non-
zero P-module homomorphism
V •> -^*IB3 necessarily an
inclusion by a). homomorphism since
S(X)
Similarly, there is a nonzero P-module
S(X) •*• -X*|B, whose image is contained in V is a cyclic B-module, generated by a nonzero
-X*-weight vector. d)
Thus, by c), S(X) £ V|£.
an Lj-submodule of
V
by b).
It follows therefore that Lj-module.
Hence, S(X)UJ s (V|p)UJ,
S(X)
J
is an irreducible
B. J. Parshall
(2.2.5)
113
ITERATED INDUCTION.
If P1,...,Pn is any sequence
of parabolic subgroups containing B and if V is a P ... P P P P rational B-module, let VJ 1'""' n denote v| 1|B| 2 - . - l B l n, the result of successively restricting to to P.. i
Then -
P
l'"Pn = G'
[16], w
«0
V|
1 * " * "* n
= V|
The reader wil1
G
B
then inducing
as Pjj-modules —— - provided that "
find a proof of this fact in
Let us merely point out an application.
Let
= S
Q ---P S Q be a reduced expression for the long word *! N w . Setting P. = P/g j, we have P.,...PN = G. Now if V is a rational B-module which extends to a rational P.module, we have from (2.2.1) and the isomorphism kCP./B] = k x P, that V|DD = V. Thus, we get the following extension
theorem [16]: A rational B-module extends to ci rational G-module iff ±t extends to a, rational P-module for each minimal parabolic subgroup P SB.
3.
INFINITESIMAL METHODS
The theory of group schemes aims to rehabilitate in positive characteristic the classical algebraic group-Lie algebra correspondence.
Below we will indicate several
applications of this point of view.
First, we introduce
some preliminary terminology, mostly taken from [20]. Let MJE denote the category of k-functors: object in
M,E
is a functor from the category
commutative k-algebras to the category
E
an M^
of sets.
of The
114
Representations of Algebraic Groups
category
Sch/k
to the k-scheme
of k-schemes embeds naturally in X
we associate the functor (still denoted
X) R + X(R) = Homk(Spec R,X), R 6 Ob(Mk). between
Sch/k
————
and
M, E •—"J£
M.E •*• M^E
Intermediate
is the full subcategory M?P«
sheaves in the fppf topology on morphism
M,E:
M, E
of
•-""tC"~~
The inclusion
admits a left adjoint
*>: M.E •»• KE
(X •*• X) commuting with finite projective limits, called sheafification.
The point behind the introduction of
here is roughly to enlarge
Sch/k
MkE
suitably in order to
facilitate many natural constructions (especially of a group-theoretic nature). Next, recall that a k-group to the category
Gr
of groups.
affine k-group scheme)
G
G
An affine k-group (or
k[G]).
An affine algebraic
in the classical sense defines an affine k-group,
still denoted R 6 Ob(Mk). G(R)
is a functor from M.
is a k-group which is represen-
table (by its coordinate ring group
G
G, by setting
G(R) = Homk_al (k[G],R),
The Hopf algebra structure on
k[G]
with a group structure in a well-known way.
endows The
standard notions of rational representations, etc., all apply in the more general setting of affine k-groups. We cannot enter into further details of the above here.
The reader may wish to consult [20], especially
Ch. II,§1, and Ch. II,§§1,2,3, for more details. formalism involving
The
MfcE. will enter only in a technical
way below which the reader may wish to ignore at first
B.}. Parshall
115
reading. (3-D
PROBENIUS KERNELS.
Assume
G
Let
G
be an affine k-group.
is defined over the prime field
a: G •*• G
k , and let
be the Probenius morphism (1.5).
a closed subgroup scheme of
G.
Now let
H
For a positive integer r,
we define the rth infinitesimal thickening of
H, denoted
HG , by means of the pull-back diagram
Thus,
HG
is a closed subgroup scheme of
H = {e}, the trivial subgroup, we denote G^ r
G. HG
a)
Let
G = SL . Then
G
When by just
and we call it the rth — Probenius kernel of
EXAMPLES,
G.
is given by
Q P CR> = {[ajj] e SLn(R) |a£ = «±J). b)
Assume
Hopf algebra
G
is connected.
k[G-,]* of
of G.
For r = 1, the dual
k[Gn] is isomorphic to the
restricted enveloping algebra
V(g) of the Lie algebra
In particular,
M_ is isomorphic to the category "*! of restricted g-modules. c)
Let
G
be a semisimple, simply connected alge-
braic group defined and split over see that
*\f
G/Gp = G
for all
sheafification of the k-group
be
r.
k . It is easy to
Here
G/Gr
f\f
G/Gr
is the
defined by
116
Representations of Algebraic Groups
(G/G r )(R) = G ( R ) / G r ( R ) , R e Ob(^). Gr/Gg = Gr_g for r > s
Similarly,
(cf. [23; 3-6]).
It is then
easy to extend the argument of (1.6) to see that the map X + S(X)|Q , X e A*, is a bijection between A* and r representatives from the distinct isomorphism classes of irreducible rational G -modules.
(In a different form,
this result is due to Humphreys [26] for r > 1.) More generally, for an arbitrary closed subgroup scheme H of <\, ^ G, one has that HG /G = H/H x G , and the irreducible rational
HG -modules have the form
V 8 ¥ where
V
is
*\j an irreducible rational H/H xGr Gr -module and W = S(X)|'tIHG « r +
for some X e Ar . (3-2) MACKEY THEORY. affine k-groups.
Let
f : H •* G
be a morphism of
Given a rational G-module
V, we can
regard it as a rational H-module by means of the morphism f . Thus, we obtain a restriction functor
f*: IYL -> MH
which, as in Section 2, admits a right adjoint, called induction,
f^: MJJ -»• Mg
(see [l4;1.2] for more details).
The functor f^ is left exact, and, as usual, we let Rnfs: MH -*• MQ denote its nth right derived functor. no confusion results, we often write
¥| G
or
W| G for
Now fix scheme of
G.
for f*(V) and
f*(W).
f : H -> G Let
V|H
When
and let
i: L •* G
L
be a closed subgroup
be the inclusion morphism.
Consider the morphism (*) L x H + G
(in Sch/k) given
B. J. Parshall
117
by the composition of in
G.
Let
j: L x
i x f
with the multiplication map
H •»• H
be the natural inclusion mor-
phism of affine k-groups and let
fT: L x
morphism obtained by restricting
f
to
H •*• L
L x
H.
be the Now we
can state (3-2.1)
THEOREM [18].
With the above notations, assume in 'b
addition that the morphism (*) is an epimorphism in M^E.. Then we have a natural isomorphism of functors Rnj'*of'* s f*oRrilt: ML + M,
n >_ 0.
We remark that the morphism (*) is an epimorphism in case If
G H
H n L
is a reduced algebraic group and
is a closed subgroup scheme of for H x
L, we can express the
G
(3-2.1) as
H
V| |H
a V|HnL!
Now assume that k, and that
H
and
has an open orbit x
G L
Q
V 6 Ob(MH), let V by making
Hx
n = 0
case of
for V 6 Ob(ML) !!.
are closed subgroups such that
^ G/H.
in
f
Choose in
L
f e Q(k) and let
G(k).
Hx = x~ Hx, be the stabilizer of x
and if we write
is reduced and of finite type over
be a representative for
where
G
G(k) = L(k)f(H(k)),
Let x
in
L L.
= Hx x
L,
For
denote the rational H-module obtained
act on V
through the morphism Hx + H (hx-»-h),
Now we have (3.2.2) THEOREM [18],
Assume that
G/H - ft has codimen-
118
Representations of Algebraic Groups
sion >_ 2.
Then for a rational H-module
V, we have
X
In addition to the proofs of (3-2.1) and (3-2.2), [18] gives a version of (3-2.2) valid for the higher derived functors of induction. (3-3)
APPLICATIONS.
Let
G
be a semisimple, simply
connected algebraic group, defined and split over a)
PARABOLIC INDUCTION THEOREM.
Let
two standard parabolic subgroups such that One verifies that the P^-orbit K
8
PJT
k .
and
PvA be
J* U K = n.
of wO P T/P, in G/PT d d J
is open and that codim(G/P- - fl) >_ 2. Therefore, we_ w P obtain from (3-2.2) that V|G[p = V °|p npwo| K for V 6 Ob(Mp ) b) integer
[18].
THE STEINBERG MODULES.
For each positive
r, define the rth Steinberg module
S((pr-l)p).
St(r) to be
Below we sketch a proof that
St(r) = -(pr-l)plS-
The argument is given in several
steps: 1)
dim St(r) = prN, N = 1$ + I. This follows from
the r = 1 case (in view of (1.6)) where one argues directly, cf. [18] for details. 2)
Since St(r) is G -irreducible (3-D, reciprocity G gives an inclusion 1: St(r)L —*- -(pr-l)p|Rr of G r r modules. 3) Applying (3-2.1) for L = Bp, H - U~, we obtain,
B. J. Parshall
119
using (2.2.2) and the fact that Br x u~ is the r G trivial group, that -(pr-l)p |Br | a k[lT], so has r r rN dimension equal to p . Thus, i is an isomorphism by 1) above. G = -(pr-l)p| r , so r r BG r r j: StCr)!-,^ i BGr = -(p -l)p|r 'B BG
4) Similarly,
-(pr-l)p|B r!Q
there is an isomorphism
which factors as St(r)|BQ £ -(pr-l)p |B|BQ 5 -(pr-l)p|B r. r r By (2.2.3b), a is an injection on its B-socle, so a is an injection.
It follows that B is an isomorphism, which
clearly proves our claim. c)
BUNDLE COHOMOLOGY. Consider the following commu-
tative diagram, the square being a pull-back:
Here a,af,b
are the natural inclusion morphisms.
(3.2.1), we have that (since
a^
V 6 Ob(EB).
By
Rnb*oc* = ar*oRna* = ar*oRnbxoai
is exact [l4;4.2]) for all n >_ 0. If we apply the above to
Let
V(r) ® -(pr-l)p,
using b) above and the fact (2.2.1) that the exact functor - 8 St(r) commutes with induction, we obtain immediately that
120
Representations of Algebraic Groups
(3.3-1)
R n a*(V (r) ® -(p r -l)p) = R n a*(V) ( r ) 8 St(r), n >_ 0.
(This argument was first discovered by E. Cline and reproduced in [18] in detail.) Using the easily derived fact that
Rnax(V) = Hn(G/B,Lv)
for example)
(sheaf cohomology, cf . [19]
we obtain from (3-3.1) the following impor-
tant result discovered independently by H. Andersen [5] and W. Haboush [253: (3-3-2)
THEOREM.
For a rational B-module
V, we have an
isomorphism of rational G-modules: Hn(G/B,Ly)(r) 0 St(r) s Hn(G/B,Lv(r) ^ ^(pr.1)p), n^ This theorem has several important consequences. (3-3-3)
COROLLARY.
Let X e A+.
for all positive integers
Then
Hn(G/B,L_x) = 0
n.
This result, due to G. Kempf [36], follows easily from (3-3-2) using the ampleness of the line bundle
L_ ,
cf. [5] for more details. (3. 3.4)
COROLLARY. For X 6 A+,
ch -X|G is given by
Weylfs formula (1.7) (with X* in place of X). This is a well-known consequence of (3-3-3), cf. for the argument. (3-3.5)
COROLLARY. For
V 6 Ob(Mg), there is an injec-
B. J. Parshall
tion
121
Hn(G/B,Lv)(r) -»• Hn(G/B,Lv-(r)) of rational G-modules
for all n ^ 0, r ^ 0.
This result, which had been conjectured earlier by Cline, Scott, and the author, was proved by Andersen in [5]. It follows easily from (3-3.2).
4.
RATIONAL COHOMOLOGY
The rational cohomology of affine algebraic groups was first studied by Hochschild in the early 1960fs.
It plays
an important role in the representation theory, and in this section we will survey some of the work done to date. (4.1) Let
BASIC DEFINITIONS. FQ
Let
G
be an affine k-group.
be the functor from the category
M«
to the cate-
gory of k-vector spaces which assigns to each V B Ob(M~) G 4 the space V of G-fixed points. It is trivial to see that MQ
possesses enough injectives, so we can speak of
the nth right derived functor V 6 Ob(MQ), write
RnFQ
of
FQ, n >_ 0.
For
Hn(G,V) - RnFQ(V), the nth rational co-
homology group of_ G
with coefficients in
V.
Similarly,
we can define the rational ExtG-groups, Ext^(V,-) = RnHomG(V,-), V 6 ObCMg). Now let let
W
H
be a closed subgroup scheme of
be a fixed rational G-module.
have that
HomH(W,-) = Koj, where
G
and
By reciprocity, we
J = |H
is induction
More precisely, if we view V as a k[G]-comodule with structure map Ay: V + k[G] 8 V, then VG = {v 6V|Ay(v) =
1 3 v}.
122
Representations of Algebraic Groups
from MJJ to Mg J
and
K = Hora
Q(w>-)-
takes injective objects in
Ifc
is immediate that
MIT to injective objects in
MQ, so there is a Grothendieck spectral sequence (4.1.1)
E^
= Ext^(W,RtJ(V)) «>Ext|+t(W,V)
for all rational H-modules
V.
(4.2)
In this section
SOME BASIC RESULTS.
G
will be a
fixed semisimple, simply connected algebraic group, defined and split over
k .
(4.2.1) TRANSFER THEOREM [19]. module.
Let V
be a rational G-
Then the restriction map on cohomology induces a Hn(G,V) •* Hn(B,V), n >_ 0, of cohomo-
natural isomorphism logy groups.
To prove this result, we will use the spectral sequence (4.1.1) with
H = B.
It is not hard to see that
H (G/B,LZ) for every rational B-module
Z.
Z - V e Ob(MQ), we obtain from (2.2.1) that V 8 RtJ(0^), where B-module.
£
Taking
W
Now if RtJ(V) =
denotes the one-dimensional trivial
It follows from (3-3-3) that
for t > 0.
RtJ(Z) a
HG/B^) = 0
to be the trivial G-module, our
spectral sequence collapses to give the desired result . (4.2.2) REMARKS: V e Ob(MG) and Extg(V,-X),
a)
The same argument shows that for
X 6 A+, we have that ExtQ(V3-x|G) =
n ^ 0. See [19]-
B. J. Parshall
b)
123
One can also show for
r
a positive integer and
V e ObfM^) that there is an isomorphism Hn(G,V) •*• Hn(BGp,V), n ^ 0. (4.2.3) THEOREM [193- Let that -X
V 6 ObCMg), X 6 A+.
Suppose
is not strictly greater (in the partial order >^
(1.3)) than any weight G
Ext£(V,-X| ) =0,
n
of
T
in V.
Then
n > 0.
For the proof see [19;3-2].
This result has the
following important consequence (also taken from [19]): (4.2.4)
COROLLARY.
Let
a) -X|G 8 -y|G = 0 for all positive b)
-X| 8 -y
(4.2.5) REMARK. that
X,y 6 A+.
is G-acyclic, i.e., Hn(G,-X|G 8 -y|G) n. is B-acyclic.
Recent work of Wang Jain-pain [45] shows
-X|G 8 -y|G
(X,y 6 A+)
has a G-filtration with
sections isomorphic to induced modules at least as long as
p
-co|
is sufficiently large.
(o> 6 A ), It follows
from (4.2.4) therefore that an arbitrary tensor product -X-J0 8...® -*n!G
U±
6 A"1")
is G-acyclic.
Using the
parabolic induction theorem (3.3a)a D. Vella has determined (unpublished as yet) conditions which guarantee that -X| , X e A , has an L-filtration with sections isomorphic to induced modules -o>| where
L
(co a dominant weight for
L),
is a Levi factor of a suitable parabolic sub-
124
Representations of Algebraic Groups
group of
In particular, this means -X|G is L-acyclic.
G.
The above results have an interesting (though formal) application, observed first in [42] (and motivated by similar results in characteristic 0), to the representation theory.
For V,W
finite dimensional rational B-modules,
we define X(V,W) =
I 1-0
(-1)1
dim ExtJ(V.W). B
(It is not hard to see that the Ext-groups here are finite dimensional and vanish for i
sufficiently large, cf .
+
[19].) Now for X,u 6 A , (4.2.4) implies that X(y*,-X|G) - 6, (Kronecker delta). A ,11 ch -X|G -
I x(y*,-X|G) ch -y|G, X eA+ additivity of X, we get that (4.2.6) (4.3)
ch S(X) -
FURTHER RESULTS.
Thus, and so, by the
I . X(U*,S(X)) ch -y|G. Let
G
be a semisimple, simply
connected algebraic group, defined and split over kQ. (4.3-D
GENERIC COHOMOLOGY . For q - pd, let
denote the subgroup of points.
G
G(q)
consisting of GF(q) -rational
Then, given a rational G-module
V, we can con-
sider the classical discrete cohomology groups
Hn(G(q),V).
The "generic cohomology" arises from the stability of these groups.
More precisely:
B. J. Parshall
125
(4.3-1.1) THEOREM [19]. rational G-module.
Let
V
be a finite dimensional
For a fixed
n, the cohomology groups
Hn(G(q),V) achieve a stable value
Hnen(G,V) as
d -» «> Hn(G,V^r')
which is given in terms of rational cohomology by for r » 0.
Besides the proof of this result, [19] contains arithmetic conditions on G v
r
and
n
d
n
which guarantee that (r)
H!Lv^ gsn > ) ~ H (G(q),V) = H (G,V ). The above result has also been extended to include the twisted groups [9]. We next introduce a variation on the Kostant partition function
P.
Namely, for n « 0,1,...,
P (X) denote the number of ways sum of
n
positive roots.
P = PO + P-, + ...
X
and X e A,
let
can be written as a
Thus, we have that
. In the following result, we assume
is of simple type. (4.3.1.2)
THEOREM [22], [23].
condition <X+p,a^> £ p, where root in
$*.
Assume also that
Coxeter number of $.
Let X 6 A* aQ
satisfy the
is the maximal short
p >^ 2h, where
h
is the
Then for 0 <_ n < (p-2h+2)/4
and
all r >_ 1,
0
n odd
det(w)p (w(X+p)-p), n=2m is even. In view of (4.3.1.1), the above formula calculates
G
126
Representations of Algebraic Groups
also the generic cohomology of (4.3.2)
S(X) in a range of degrees.
INFINITESIMAL COHOMOLOGY.
Let
G
be as above,
and consider the relationship between the cohomology of and that of its Probenius kernels (4.3.2.1)
THEOREM [14].
Let V
G
G
(3.1).
be a finite dimensional
rational G-module. a)
The natural restriction map
is an isomorphism for each b)
Hn(G,V) •> llm Hn(Br,V)
n.
The natural restriction map
H^G^V) + llm Hn(Gr,V)
is an isomorphism for n <_ 2, and an injection for all In b) above, it turns out that for r »
0, at least so long as
n.
H^G/V) = H1(Gr,V)
p 7* 2
or
G
has no com-
ponent of type
C . In general, stability does not hold
in a) or in b)
(for n = 2).
J6
We also would like to mention the following result. (4.3.2.2) THEOREM [8].
Assume that
p ^ 2
or that
G
has no component of type
C0Xf. Then for X e A_, r we have that Extp (S(X),S(X)) = 0. r Finally, we remark that [23] contains a calculation (in a range) of the cohomology
H*(Gr,k).
This in fact
plays an important role in the proof of (4.3.1.2).
B. J. Parshall
127
5.
LUSZTIG'S CONJECTURE
Throughout this section,
G will be a fixed simple,
simply connected algebraic group over k. (5-1) THE AFFINE WEYL GROUP. Let E = Z$ 8 B
be the
Euclidean space associated to the root system $ of G (1.3).
Let sex ,n (a 6 $, n e Z) be the reflection of E about the hyperplane Ha = {x 6 E|<x,av> = np}, and let ¥Qa (the "affine Weyl group") be the subgroup of Aut(E) generated by these s . If a denotes the maximal short root in $+, and if S = {sa
-J9 then Q,. . ., sa Q5, s 1* jl o9 (¥&,S) is a Coxeter system. Clearly, ¥ is a semidirect
product
¥.T, where
T
the translations of E
is the normal subgroup generated by by p-multiples of roots.
In
addition to the usual action of ¥0a on E, there is the "dot" action defined by wX = w(X+p) - p. For u,X e E, write p t X provided that p = s <_ X ^
for some a e $
are
and integer
n.
«X
¥e say that y and
strongly p-linked (resp. p-linked) provided there
is a sequence p= yQ *^
t...* un
= X
(resp. X = w-p,
for some w 6 ¥_). a (5.1.1) THEOREM [3].
Let
X 6 A
and let
composition factor of Hn(G/B,L"~A *) some non-negative integer w e ¥.
n.
S(y) be a G-
(cf. (3-3c)) for
Suppose that w-X = X1 6 A ,
Then p is strongly p-linked to
Xf.
It follows easily from (5.1.1) and (2.1) that the
128
Representations of Algebraic Groups
high weights of the G-composition factors of an indecomposable rational G-module are p-linked. (5.2)
THE GENERIC HECKE ALGEBRA. Let A = %tq*9q.'^
be
the ring of integral Laurent polynomials in the indeterminate
q^. Given the Coxeter system (¥_,S) of (5.1) (or a any Coxeter system for that matter) we define its generic
Hecke algebra
_______
H
to be the free A-algebra with basis
T•yy ,
w 6 Wa Q, and multiplication defined by =T W ww' (TO +1)(TD -q) =0,
if
*<ww f ) = £(w> s e S.
(Here fc denotes the usual length function on ated to
+
*(wf)
Wa .) Associ-
H , Kazhdan and Lusztig [3^] have constructed for
each pair
w,z e W , with w < z in the Bruhat order on a -— W , a polynomial P (q). These can be described in terms of certain symmetric elements in H : Namely, the involution a •* a
of
A
defined by
q^
= q"^
extends to an involu-
tion of the Hecke algebra by putting I a._ T_ = Z a__ T" n . W WT W •nr—.L Then there exists for each w e WQa such that 5
a unique element
Cw
w= C w
c = y (. w
where and
Py TT TT 6 Z[q] 9w w,w = ^
P
y <_w
has degree at most
U(w)-Jl(y)-l)/2
B. J. Parshall
129
The reader will find details of the above in [34], as well as equivalent descriptions of the Kazhdan-Lusztig polynomials
P
y >w .
Part of the motivation behind these poly-
nomials in general lies in the role they play in measuring the failure of Poincare duality in the closures of the cells BwB/B
(w e W)
In fact,
P
in
y»" (q)
G/B
(the so-called Schubert varieties).
turns out to be (at least for
W
and
k =ffi)the Poincare polynomial for a geometric cohomology (due to Goreski, MacPherson, and Deligne) of these Schubert varieties. (5-3)
THE CONJECTURE.
for i = 0, !,•••,A.
Choose
X e A
(The existence of
p > h, the Coxeter number of
X
-p< <X+p,ou> <0
requires that
e A , w e Wa. S(u*) is a G-composition factor
According to (5.1.1), if
§.) Let
such that
wX
/!
of -(wX)| , then y = yX, <w(X+p),av> < p(p-h+2). (5-3.1)
some y 6 W . Now assume that 'a* Then the conjecture is that
ch S((w-X)«) - E ( - i )
w
y p
(i)ch -(yX)[G, y ,w
where on the right side the summation is over all satisfying
y £ w
y*X e A .
In view of (4.2.4) above, we could also express f T__ w(l) s y jW For further results
(5-3-1) in terms of an identity involving the and certain rational Ext-groups. along these lines see [42], [7].
Although it is not yet clear where a proof of this conjecture might lie, it is tempting to speculate that
130
Representations of Algebraic Groups
some connection between rational cohomology and a geometric cohomology which involves the
P
f
y * ™s
might do the trick.
This was, in fact, essentially the case in the KazhdanLusztig conjecture [3^] concerning Verma modules in characteristic 0 (proved independently by Brylinski and Kashiwara [11] and by Beilinson and Bernstein [10]), where (very roughly speaking) the connection between the algebraic cohomology (in the category 0 of
BGG) and the
geometric cohomology (alluded to above in (5-2)) was obtained via homological properties of the sheaf of algebras of differential operators on
G/B.
B.J.Parshall
131
REFERENCES 1.
H. Andersen, Cohomology of line bundles on G/B, Ann. Sci. ficole Norm. SUP. 12 (1979), 85-100.
2.
H. Andersen, The first cohomology of a line bundle on G/B, Invent. Math. 51 (1979), 287-296.
3.
H. Andersen, The strong linkage principle, J. Reine Angew. Math. 315 (1980), 53-59. '
4.
H. Andersen, On the structure of Weyl modules, Math. Z. 170 (1980), 1-14.
5.
H. Andersen, The Frobenius morphism on the cohomology of homogeneous vector bundles on G/B, Ann. of Math. 112 (1980), 113-121.
6.
H. Andersen, On the structure of the cohomology of line bundles on G/B, J. Algebra 71 (1981), 245-258.
7.
H. Andersen, An inversion formula for the KazhdanLusztig polynomials for affine Weyl groups, to appear.
8.
H. Andersen, Extensions of modules for algebraic groups, to appear.
9.
G. Avrunin, Generic cohomology for twisted groups, Trans. Amer. Math. Soc. 268 (1981), 247-253-
10.
A. Beilinson and J. Bernstein, Localisation de gmodules, C.R. Acad. Sc. Paris 292 (1981), 15-18.~~
11.
J. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems. Invent. Math. 64 (1981), 387-410.
12.
C. Chevalley, Seminaire sur^la classification des groupes de Lie alge"briques, Ecole Norm. Sup. (1956-58), Paris.
13-
E. Cline, B. Parshall, and L. Scott, Induced modules and affine quotients, Math. Ann. 230 (1977), 1-14.
14.
E. Cline, B. Parshall, and L. Scott, Cohomology, hyperalgebras, and representations, J. Algebra 63 (1980), 98-123.
15.
E. Cline, B. Parshall, and L. Scott, On the tensor product theorem, J. Algebra 63 (1980), 264-267.
16.
E. Cline, B. Parshall, and L. Scott, Induced modules and extensions of representations, Invent. Math. 47 (1978), 41-51.
132
Representations of Algebraic Groups
17-
E. Cline, B. Parshall, and L. Scott, Induced modules and extensions of representations II, J. London Math. __ (1979), "-3-43Soc. 20
18.
E. Cline, B. Parshall, and L. Scott, imprimitivity theory, to appear.
19-
E. Cline, B. Parshall, L. Scott, and W. van der Kallen, Rational and generic cohomology, Invent. Math. 39 (1977), 143-163.
20.
M. Demazure and P. Gabriel, Groupes algebriques 1^, North-Holland, Amsterdam (1970).
21.
S. Donkin, The blocks of a semisimple algebraic group, J. Algebra 67 (1980), 36-57.
22.
E. Friedlander and B. Parshall, On the cohomology of Chevalley groups, Bull. Amer. Math. Soc. (1982), to appear.
23-
E. Priedlander and B. Parshall, On the cohomology of algebraic and related finite groups, to appear.
24.
W. Haboush, Reductive groups are geometrically reductive: a proof of the Mumford conjecture, Ann, of Math. 102 (1975), 67-83.
25.
W. Haboush, A short proof of the Kempf vanishing theorem, Invent. Math. 56 (1980), 109-112.
26.
J. Humphreys, On the hyperalgebra of a semisimple algebraic group, in Contributions to algebra; a collection of papers dedicated to Ellis Kolchin, 203-210, Academic Press, New Yorlc (1975).
27.
J. Humphreys, Linear algebraic groups, SpringerVerlag, New York.(19757^
28.
J. Humphreys, Modular representations of finite groups of Lie type, in Finite Simple Groups II, 259-290. London, Academic Press (1980).
29.
J. Jantzen, Zur Charakterformel gewisser Darstellungen halbeinfacher Gruppen und Lie-Algebren, Math. £. (1974), 127-149-
30.
J. Jantzen, Darstellungen halbeinfacher Gruppen und kontravariante Formen, J. Reine Angew. Math. 290 (1977), 117-141.
A Mackey
B. J.ParshaU
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31. J. Jantzen, Uber das Dekompositionsverhalten gewisser modularer Darstellungen halbeinfacher Gruppen und ihrer Lie-Algebren, J. Algebra 49 (1977), 441-46932. J. Jantzen, Weyl modules for groups of Lie type, in Finite Simple Groups II, 291-300, London, Academic 33-
J. Jantzen, Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne, J. Heine Angew.' Math. 317 (1980), 157-199.
34. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. 35. D. Kazhdan and G. Lusztig, Schubert varieties and Poincare duality, in: Geometry of the Laplace Operator, Proc. Symp. Pure Math. 36 (1980), 181P203. « 36. G. Kempf, Linear systems on homogeneous spaces, Ann. of Math. 103 (1976), 557-591. 37-
G. Kempf, The Grothendieck-Cousin complex of an induced representation, Advances in Math. 29 (1978), 310396.
38. G. Kempf, Representations of algebraic groups in positive characteristic, Ann. Sci. IScole. Norm. Sup. 14 (1981), 61-76. 39. G. Lusztig, Some problems in the representation theory of finite Chevalley groups, in: The Santa Cruz Conference on Finite Groups, Proc. Symp. Pure Math. 37
U9yo), JI^STT:
40. G. Lusztig, Hecke algebras and Jantzen1s generic decomposition patterns, Advances in Math. 37 (1980), 121-164. 41. B. Parshall and L. Scott, An imprimitivity theorem for algebraic groups, Indag. Math. 42 (1980), 39-47. 42. L. Scott, Representations in characteristic p, Proc. of Symposia in Pure Math. 35, Amer. Math. Soc. (1980), 319-33143.
S. Smith, Irreducible modules and parabolic subgroups, J. Algebra 75 (1982), 286-289.
44. R. Steinberg, Lectures on Chevalley groups, Yale U. lecture notes (1968).
134
Representations of Algebraic Groups
45.
Wang Jain-pain, Sheaf cohomology on G/B and tensor products of Weyl modules, 'J. Algebra9 to appear.
46.
W. Wong, Irreducible modular representations of finite Chevalley groups, J. Algebra 20 (1972), 355-36?.
Abstract Homomorphisms of Big Subgroups of Algebraic Groups BORIS WEISFEILER
This is a survey of results centered around the problem of describing automorphisms, isomorphisms, or homomorphisms between subgroups of algebraic groups.
An attempt was made
to follow the lectures rather closely. However some material was added to give a broader view and other material deleted in an attempt to stay within allotted space.
I
apologize to all the people whose results are bypassed or not fully represented. It is a pleasure to acknowledge here my gratitude to the members of the Mathematics Department at Notre Dame for their invitation and warm hospitality and, also, to Alex Hahn, Don James, Alex Lubotzky, and Ross Staffeldt for many fruitful discussions and helpful suggestions. NOTATION Most of our notation and terminology is standard and is explained at appropriate places; we record it here for the 135
136
Homomorphisms of Algebraic Groups
benefit of a reader who does not want to search for these explanations,
k
always denotes a field; it will be
infinite unless otherwise specified; its characteristic is "char k"; k field of R
is its algebraic closure;
p-adic numbers and Z
0
is the
is its ring of integers.
de]
K/k
kotes Weil's restriction of scalars from a ring
K 2. k
has
to
^or
^S
a
functor
(R K/k V)(L) - V(L 0 K) .
V
If
from rings to sets one
G is a group then N Q (A)
K.
(resp. Z~(A))
is the normalizer (resp. centralizer ) in G
of a subset A; if ZG(A)
G
is an algebraic group then
can be considered as algebraic groups.
commutator of subgroups
A
and
B
of
G.
or Lie group G, Lie G is its Lie algebra. M.Q
is its Haar measure.
For
NQ(A) and
[A,B] is the
For an algebraic For a Lie group G,
SLn(k), S0(n,l) etc., PSLn(k) ,
PSO(n,l) etc., denote the quotients of the former groups by their centers.
A subset X of a topological space is locally
closed if it is open in its closure and it is relatively compact if its closure is compact. 1.
MOTIVATION
The problem we are concerned with belongs to the following broad class: given an object having several structures, how much information is lost if some of the structures are forgotten? and what is the minimum informatipn required to recover the complete information? algebraic group
G
Ours is the case of an
which we consider as a functor
B. Weisfeiler
137
G: (fields) •* (groups).
Take the value
G(k)
of
G
at a
fixed field k.
Does it determine the functor? More
precisely:
a: G(k) -»• G'(k'') is an abstract group
If
homomorphism, does it come from a homomorphism of functors? If it does one goes still further and asks: what are subgroups H of G(k) and group horaomorphisms H -»• G'(k') that come from a map of functors? (1.1) A classical example of a problem of the same broad class is the Fundamental Theorem of Projective Geometry (PTPG for short). n > 1,
Consider the protective space
as an algebraic variety over
k.
Fn
It has a
distinguished family C of subvarieties: curves of degree 1 (called lines). functor
Now consider an algebraic variety V as a
V: k -»• V(k)
from fields to sets.
Then we have a
functor 3Pn: (fields) -> (sets) and 3Pn(k) has, for any field k, a distinguished family of subsets: sets L(k) for L € C, L defined over k. Classically,
UPn(k)
1-dimensional vector subspaces of k
is the set of and distinguished
subsets, still called lines, are the sets of one-dimensional subspaces of kn kn+1.
contained in a 2-dimensional subspace of
Let a: Fn(k)
-*!>m(k'), n,m > 2, be a bijection which,
together with its inverse, carries lines into lines. the PTPG says that if
m,n > 2, then m = n
exist a field isomorphism 31 1 1
11 1 1
p: k " " -•> k' " " with
Then
and there
and a bijective map
p(ax+by) = cp(a)p(x) + q>(b)p(y) for
138
Homomorphisms of Algebraic Groups
a,b € k, x,y S kn+1, x € kn
, x £ 0.
such that
o(kx) = k'p(x)
for
(Recall that the above properties of
p
are called (p-semilinearlty . ) We can reformulate this result in a more invariant form. (1.2)
Assign (see [BT,l-7]) to an affine algebraic variety
V defined over
k
and to a field homomorphism
the algebraic variety the ring
k'l^V]
^V
defined over
k'
by taking for
of regular functions the algebra
k[V] ® k' (cp is used to identify k as a subfield of k' ) . k For general V we cover V by open affine pieces V - UV±,
each defined over
The natural injection notation
k,
and set ^V
V(k) -> ^(k')
is denoted
V(cp), the value of the functor V on the map (p,
is also used in the literature). (1.3)
Now the PTPG takes the form:
Let
n,m > 2, be a bisection such that both lines into lines. cp: k •* k'
a
and
-*pm(k')> a
carry
Then there exist a field isomorphism
and a k' -isomorphism
varieties such that (1.4)
a: Pn(k)
$: ?Pn -*• Pm
of algebraic
a = "p o cp° .
Generalizations and interpretations of the FTPG are
closely related to different advances in and approaches to the problem of abstract homomorphisms of subgroups of algebraic groups.
The reasons for this are many.
But the
fact itself is not surprising at all if one remembers that the first result in the area, a description by 0. Schreier
B. Weisfeiler
139
and B.L. van der Waerden (Abh. Math. Sem. Univ. Hamburg 6(1928), 303-322) of the automorphisms of the projective special linear groups for any automorphism
PSL (k) was based on the FTPG: a
of PSL (k) with
there exist a field automorphism A € GLn(k)
such that
A cp (S*)""3^"1; PSL (k), if
S
here
S,
and a matrix
A cp (S)A""1
is the image of
is the transpose of
S = (s..).
q>: k -»• k
a(S) is either S
n £ 3,
or
S € SLn(k)
and
in
In the more abstract language (1.2) the
claim of this theorem is: There exist a field automorphism cp: k -> k
and an isomorphism
groups over
k
such that
£: ^PSL
-*• PSL
a(h) = pi o cj>°(h)
of algebraic
for
h € PSLn(k)
Since 1928 when this result was discovered the area has been developed by many a mathematician of renown: E. Cartan, H. Freudenthal, J. Dieudonne, Hua Lo-keng, Wan Zhe-xian, I. Reiner, C. Rickart, O.T. O'Meara, A. Hahn, D. James, B. McDonald, G. Mostow, A. Borel, J. Tits, G. Prasad, G. Margulis, M. Raghunathan and many others. attempt here to give a historical survey.
We will not
Instead we will
outline the major achievements in the subject and their interrelation. The ideal goal of the theory would be to obtain a theorem which includes all known results. goal is, of course, unrealistic.
This
But it is good to keep
it in mind for orientation and proper perspective. For this same purpose we discuss occasionally results from
140
Homomorphisms of Algebraic Groups
adjacent areas. 2.
Let
EVIDENT RESTRICTIONS
G and G' be connected algebraic groups defined
over fields k and k'. Let H be a subgroup of G(k) and a: H -*• G'(k') a group homomorphism. (2.1)
Suppose that G = G" is the group
e3L
(so that
<E8LQ(k) is the additive group of k). Let B be a basis of k
over its prime field.
Then any map
B -* k gives rise to
a homomorphism
GaQ(k) -»• a
G to be
commutative. Neither do we want it to have commutative quotients. This restricts us to the case when G coincides with its own commutator subgroup.
In particular, G is a
linear algebraic group. Actually we often restrict ourselves to semi-simple groups. (2.2) Next, if we want to recover G from a homomorphism a: H -> G'(k'), where H is a subgroup of G(k), then H should be Zariski-dense in G. This condition can sometimes be replaced by additional assumptions on a, H, and k' (say, k and k' are finite, H = G(k), and a is an isomorphism). assumption.
But in general, it is a very reasonable
B. Weisfeiler
141
(2.3) By a theorem of J. Tits (J. Algebra 20(1972), 250270, Theorems 3a^(vi)) G(k)
contains Zariski-dense free
subgroups H if either char k = 0 over its prime field. into
G'
For a free
or H
k
is not algebraic
no good map of
G
can of course be recovered from a general
homomorphism
H -*• G'(k'). This tells us that
well embedded in
G.
H
must be
To specify the meaning of this "well
embedded" is one of the problems of the theory. Examples of relatively small but well-embedded subgroups are (1)
full (of transvections or of rotations) subgroups of
O.T. O'Meara and (2) lattices in semi-simple groups over local fields (see §6 below).
In a number of cases it has
been shown that groups of type (1) and (2) are actually arithmetic (see (6.7) below and also [Va], [Se], and [VW]). 3.
ISOTROPIC SEMI-SIMPLE GROUPS OVER FIELDS.
(3-1) This case can be considered as a paradigm because the situation here is well-understood and can be completely described.
Let
G
be a connected semi-simple algebraic
group defined over k (or, shorter, k-group).
G is called
absolutely almost simple if all normal subgroups of G(k) are finite central; G is almost k-simple if only finite central algebraic normal subgroups of G are defined over k; G is k-isotropic if G contains a proper parabolic ksubgroup; G+(k) is the normal subgroup of G(k) generated by U(k) where U is the unipotent radical of a minimal parabolic
142
Homomorphisms of Algebraic Groups
k-subgroup. k-group.
For example,
SL (k) is a simply connected
Its subgroup U is the group of upper triangular
matrices with all diagonal entries equal to 1. Since SL (k) is generated by transvections, it is generated by the conjugates of U(k). Thus (SL^Ck) - SLn(k). Another example is SO (f), the special orthogonal group of a non-degenerate (and of defect at most 1 if char k - 2) quadratic form
f =
over k if a... € k
Z a..x.x.. l
This group is defined
1 < i, j < n.
It is k-isotropic
iff f represents zero non-trivially over k.
In this latter
(S0n(f))+(k) = S0£(f,k), the spinorial kernel group.
case
The following theorem is a generalization of a result previously known for "most" groups. (3-2) THEOREM.
(J. Tits, Ann. Math. 80(1964), 313-329)
Let G be an almost k-simple and isotropic over k algebraic k-group. (i) If H 2. G+(k) is a subgroup of G(k) then any normal subgroup of H is either central or contains G (k). (ii) One has G(k) = Z(k)'G+(k) where
Z
is the
centralizer of a maximal split k-torus of G. (3.3) To see the relevance of this Theorem to our problem, take a homomorphism
a: H -> Q
Then by (3-2(1)) either Ker a Ker a
contains
of
H
into a group Q.
is finite and central or
G (k). In other words,
monomorphism or an almost trivial map.
a is an almost Let us look at the
B. Weisfetter
143
latter case first.
The problem is to understand
G~(k) = G(k)/G+(k).
This consists of two different
contributions . (3-1*)
Let
TT: G -»• G
easily seen that G(k)/Tr(G (k))
be the universal cover of
Tt(G(k)) £ G+(k) . Therefore
as a factor-group.
G.
It is
G"(k) has
This latter can be
easily bounded using the exact sequence of Galois cohomology associated with the short exact sequence 1 •*• Ker TT -»• (* -»• G -* ->• 1.
Namely, we get the exact sequence
-> H°(k,G) -> H°(k,G) 4 H^kjKer TT) or •> G(k) -> G(k) ^ H1(k,Ker TT). Thus G(k)/Tt(G(k))
with a subgroup of
1 -> H°(k,Kerrr) -> 1 -> (Ker Tt)(k) ->
6 identifies
H^CkjKer TT). This latter
is a commutative periodic group of exponent equal to the exponent of (the finite algebraic group) example if
G = S0n(f), n > 3,
char k £ 2
then
6
then
Ker TT . For
G = Spinn(f) and if
is the spinor norm homomorphism.
See J.-P. Serre, Cohomologie Galoisienne, (Lecture Notes in Math., vol. 5, Springer, Berlin; 1964) for details and additional examples. (3.5) Another contribution, G~(k) « G(k)/S+(k) is much more interesting. where
It comes from
P - [Zg(T) ,Zg(T) ]
and unexpected result that
P(k)/(P(k) n G+(k))
(see (3-2)(ii)). (f~(k)
It is a recent
can be nontrivial.
To
describe an example of such a G consider a central division p algebra D over k with dii%D = N < °°. Then there exists an
144
Homomorphisms of Algebraic Groups
(automatically simply connected) algebraic k-group G such that
G(k) ~ SLn(D)a n > 2, and G(k) ~ SLnN(k) . Set SKZ(D) »
D /[D*,D*] where D
is the group of elements of reduced
norm 1 in D. The Dieudonne determinant establishes the isomorphism G~(k) ~ SK,(D).
Examples of D such that
SK-^D) # {1} were constructed by V. Platonov and V. Yanchevsky, and somewhat later by P. Draxl.
The simplest
example I know of a D with SK-^D) / 1 is described in P. Draxl's lecture notes [Dr , §24], Moreover there are fields k such that SK.,(D)
runs over all commutative finite
groups when D runs over all division algebras over k5and for every countable commutative torsion group C of bounded period there is a field k and a central division algebra D over k such that SK^D) = C.
Nevertheless
SK-^D) is
always trivial if k is a local (T. Nakayama and Y. Matsushima, 19^3) or a global (S.S. Wang, 19^9) field. of these results that the equality
It was because
SK.,(D) = 1
was expected
to hold always. We refer the reader to [Dr], [DK], [T5H and references therein for more information.
We conclude by
mentioning that V. Voskresensky [Vo] gave an interesting algebro-geometric interpretation of SK-^. (3.6) To proceed we need certain notions. homomorphism
p: G -*• G'
is an isogeny if
An algebraic p
is an
epimorphism with finite (schematic) kernel; an isogeny p is special if its differential is not identically zero.
B. Weisfefler
145
If char k - p the Frohenius map Fr(s
Fr: SLn -> SLn
) SB Cs?.) is anon-special isogeny.
(see e.g. [D]) map SOp +, -* Sp2
given by
The standard
in characteristic 2
is a special isogeny; for pairs (G,G') a non-trivial (.i.e., not an isomorphism) special isogeny exists only when char k < 3: if char k * 2 then CG,G') must have type CBn,Cn),(Cn,Bn),0^,F4) and if char k - 3 then only the pair (G2,G2) can be involved. See [BT, §3]
for more details.
Using these notions the situation when
a
is an "almost
monomorphism" (see (3-3) above) is completely described by (3-7) THEOREM (A. Borel and J. Tits [BT]). Let
G
be an
absolutely almost simple algebraic group defined over an (infinite) field k.
Let
k'
be another field and let
G'
be an absolutely almost simple algebraic group defined over k'.
Let
let
a: H -»• G'(k') be a homomorphism of abstract groups
such that
H
be a subgroup of
G(k) containing
a(G (k)) is Zariski dense.
G (k) and
Assume that either
G is simply connected or G' is adjoint.
Then there exist
a field homomorphism q>: k -»• k', a special k'-isogeny P: ^G -»• G' that
and a homomorphism
a(h) = r(h) • p(cp°(h)) Note that
Y
Y: H -* center (G'(k'))
for all
such
h € H.
here (classically called a radial
homomorphism) is an almost trivial map. Therefore (3-*0 and (3-5) can be used to study
Y-
The ideas, as well as
important technical steps, of the proof of (3-7) are outlined
146
Homomorphisms of Algebraic Groups
on pp. 502,503 of the Introduction to [BT]
and by
R. Steinberg in his Bourbaki seminar report
[St2]
on the
paper [BT]. Theorem (3-7) was, even in the case of
SLn, a vast
technical and conceptual generalization of previously known results on homomorphisms of isotropic groups over fields.
It opened new avenues of research and set new
standards.
Among its most important achievements was a
uniform treatment of all (isotropic'semi-simple)
algebraic
groups (including exceptional ones) and all homomorphisms between them (including isomorphisms SOc ~
PSLj,~ PSOg,
PS
Pii* triality for Dh , and the non-trivial special
isogeny
S^n+l "*
Sp
2n
ln cnaracteristlc 2
)•
But
still
more remarkable was the step from isomorphisms and automorphisms to homomorphisms with dense image. (3-8) Among crucial technical tools the proof of (3-7) uses are the existence of Bruhat decomposition in H and the algebraic geometry of the target group G'.
These tools are
still available if H is replaced by a (non-algebraic) group with a BN-pair (say by an isotropic classical group over an infinite-dimensional division algebra or by the infinite P 2 P twisted groups Bp, G2» and Fii of M" Suzuki and Rsee [Stl, §11]). assumptions on
But it is difficult to see how the
G'
can be weakened unless methods
compensating for the absence of algebraic geometry are devised.
B. Weisfeiler
147
(3-9) To conclude this discussion of isotropic groups we remark that (3-7) implies (see [Tl, no. 10]) a generalization of the PTPG to irreducible Tits buildings of spherical type (the particular case of (3-7) when G = SL , G' = SLm, m,n > 3> stated in (1.1)).
a
is an isomorphism,
is equivalent to the PTPG as
Conversely, an analogue of the FTPG for
Tits buildings would, it seems, imply (3.7) in relative rank > 2. 4.
SHORT REMARKS ON O f MEARA f S METHOD
The lectures did not cover work of O.T. O'Meara and his school.
There are excellent books
[0!M1]
and
[0*M2]
f
by O Meara himself which have made his theory accessible to a wider audience than these notes can hope for. Therefore we discuss OfMearafs approach only briefly to indicate the features which make it conceptually different.
OfMearafs
method associates with a subgroup of G(k) a certain geometry. This is, roughly, a Tits geometry if G is isotropic But it is some other yet mysterious geometry if G is anisotropic.
In this latter case the geometry is constructed
from a group having no parabolic k-subgroups.
This geometry
(and the rule used to construct it from the group) are therefore very fascinating.
On th.e other hand 0!Mearafs
method permits one to probe into which groups are well embedded (in the sense of (2.3) above).
This is another
conceptually important feature of the 0fMeara"method.
148
Homomorphisms of Algebraic Groups
One class of such well-embedded subgroups are the full subgroups of
GLn(D), D
a division ring.
We recall that
a subgroup H of GLn (D) is ~""™""""~*""" full (. of transvections) if for """"""™ every hyperplane h in Dn and every vector v # 0, v € h, there exists A € H Ker(A - Idn) = h.
such that
[So], is
(4.1) THEOREM.
Let
H
are division rings.
isomorphism
that all
Let
a : H •*• H'
n = n^
(or
n
n
a(h) » F ° h « 3" "1 h € H.
(Here
h
qp : D° •*• D')
(or 3 : D° + D"n) a(h) - f « h « F"1)
(Or
3
such for
on pro-
is the protective contragredient of
(of transvections) subgroup H of
and
[Se]
that a full
GL (D), n £ 3, contains a
This is a kind of
It has been extended to appropriately
interpreted full subgroups of Chevalley groups, see [VW]. (4.3)
We remark that O'Meara's method can be (as a paper
[Hal] of A. Hahn shows) applied to groups which are not quite full.
h.)
GL (0) where 0 is a subring of D
whose field of quotients is D. "arithmeticity" result.
D^
and a q>-
n
It has been proven in [Va]
congruence subgroup of
and
be an isomorphism.
"3 is the map induced by V
D
and there exist a ring
& : D -»• D"
jective spaces and (4.2)
respectively, where
q> : D •* D'
semilinear map
and
and H" be full subgroups in
PGL ^(D^)
If n,n' £ 3 then
= Dv
A sample result of O.T. O'Meara [O'MS],
as extended in
PGL (D) and
Im(A - Idn)
Finally, we want to point out a similarity
B. Weisfeiler
149
between O'Meara's and Mostow's C6.5) results -—both impose the same conditions on both H and 1
whereas
superrigidity (.6.6) is more akin to Borel-Tits1
Margulis
Theorem (3-7): imposed on 5.
a("_H) —
only the condition of Zariski density is
a(.H).
TITS1 RESULTS ON. HOMOMORPHISMS OP-LIE GROUPS
Again, as with OfMearafs method, the author's exposition in [T3] is beyond imitation, but, on the other hand, the results and ideas are so substantial that they may not be bypassed. (5.1) THEOREM.
Let
G
be an algebraic simply connected group
defined over 3R, the reals. (2.1)). Let H = G'(B)
H
G = CG,G] (see
be a Lie group countable at infinity (.e.g.,.
where
a: G(B)° -* H
Suppose
G'
is an algebraic H-group) and let
be a group homomorphism.
Then there exist
a finite-dimensional E-algebra K, a homomorphism of rings
with dense tin the Hausdorff topology) image,
and a homomorphism of Lie groups a = p o cp°. (Here
^G
and
p: (pG(K) -*• H
such that
cp° are defined in the same
way as they were defined before for field homomorphisms.) (5.2) The following example outlines the meaning of (5.1). Let
K =B[e] where
e2 - 0
and let
orthogonal B-group of the form
2
S0n
be the special 2 x. . Thus SO^CR) x n
150
Homomorphisms of Algebraic Groups
is the usual special orthogonal group. a Lie group, an extension of V 2:ii32n(n"1)
with SOn(B)
representation. G(B)
r: B[e] -»• B
d € Der~.B
acting on V
is given by
be a derivation of B
case
via the adjoint Lie G(B) of
Lie G(B) = Ker(G(B[e]) Gir) G(B)),
ring homomorphism given by of
SO (B) by a vector group
(Actually, the Lie algebra
can be defined by
where
SO (B[e]) is also
r(a + b e ) = a . ) Let and
q>: B -*-B[e]
B[e] if
d 9* 0.
+ d(X)e.
the
The image
In this
Lie groups with dense image.
To conclude this example we
remark that dim Der~ B = °° so that "bad" homomorphisms do exist.
To construct one of them one takes a trans-
cendental element, say
t,
in B
extends the differentiation
over
d/dt from
<$, and then Q(t) to
B.
Below we reproduce Tits1 outline of the proof of
(5-3) (5-1).
We suppose for simplicity that
has no compact factors. subgroups
F.^ of
G,
(G/Rad G)(B)
Then there exist algebraic
B-
i = 1,..., d = dim G, isomorphic
to a semi-direct product Gd Q * G^ m of the additive and multiplicative groups such that Lie G = © Lie [P.,P.]. Set
G± - £Fi>Fil
we have a.(B).
and
a.:B -* H.
a
i=
a
Let
A^
IGiCR).
Since
G±(B) ~ B
be the Hausdorff closure of
It is a commutative group.
Denote by K^ the
B.Weisfeiler
151
(commutative) subring of the endomorphism ring of generated by the automorphisms
a *-»• a(x)a a(x
Ai
) for
x € P., a € A. . This gives us, for all i, homomorphisms of algebras
cp.^: B •*• K.^ with dense image, and homomorphisms *PJI of Lie groups pf.: G.(K.) •* H such that a. = pf. o q>° Consider of K
q> = wcp.^: B -»• <£K., and denote by
the closure
cp(B) in *K. . Since each K. is still a quotient of and
q^: B -*• K^
are p. : ^G.CK) -*• H Since
^0(10
factors through B •*• K •+- K. , there still such that
Lie G(B) = e,Lie G±(B)
Therefore
if the U±
Since the
are neighborhoods of e in ^(K).
We
is a homeomorphism
p. are analytic we recover an analytic map Take connected open V. £ U.
so that,
V -V £ U, and V"1 £ U for V = nv±.
a o cp°: v n cp°(G(B)) -»• H is a homomorphism and
follows that
coincides with
coincides with
cp°(G(B))
Then
py
on
is dense in
V.
p:
GCK) -*• H.
a on U n q>°(G(B))
Since
and since
connected (use the simple connectedness of G) a = p ° cp°.
Since
^GCK) it
Py is a local homomorphism. Therefore
extends to a homomorphism
that
i.
we have Lie ^GCK) = eLie^G^K).
to be so small that IIU. -»• U
Py = np. : U -* H. e € V±,
a. = p. <> cp° for all
U = nu.^ is a neighborhood of identity e in
can take U.
a
K
py
° p o cp
^GCK) is it follows
152
Homomorphisms of Algebraic Groups
Going over Tits1 proof C5-3) of C5-D the reader is
(5.4)
most impressed by the contrast between the simplicity of the ideas and power of the result.
One would expect, therefore,
that the methods of
[T3] would have wider applicability.
And indeed they do.
Tits points out in
[T3, 5-1] that
results similar to (5.1) can be obtained for homomorphisms of simply connected.Chevalley groups over fields of characteristic different from 2 into arbitrary groups.
algebraic
He mentions that in characteristic 2 there are
counter-examples. 6.
RIGIDITY, STRONG RIGIDITY, AND SUPERRIGIDITY OP LATTICES Let
B.
G
be a semi-simple algebraic group defined over
Then G(E) is a Lie group.
Let
T
be a lattice in
G(B)
(i.e., a discrete subgroup such that the volume
/ dg G/r
of
G/r
is finite; here
measure, see [Hu, (1.1)]). Then
dg
is induced by a Haar
r
is finitely generated
(see [Hu, (1.3)]).
Let
x1,...,xn be a set of generators.
Every homomorphism
a: T -* G(E) determines a point
(a(x1),...,a(xn)) € G(E)n. n
(s^) € G(E) say
for which
Let
R(T,G)
x.^1-*- s.^ determines a homomorphism,
p: r -> G(B), such that
p(r) is a lattice.
a variation of the usual definition of Hausdorff topology of and
G(B) acts on
be the set of
G(R)n
R(r,G)
R(r,G)).
(This is The
induces a topology on continuously via
R(T,G),
B. Weisfeiler
153
(Ad g)(s±) - (g s^"1). Call a € R(r,G)
locally
weakly) rigid if the orbit
a
AdG(R)(a)
of
(or
is open in
R(r,G). We recall that a lattice if the projection of GCH)°
r
r in
G(B)° is irreducible
on every simple factor of
is dense, see [Hu, (1.5)1;
compact; and torsion free if
uniform if
r
G/r
is
contains no elements of
finite order (this condition always holds for an appropriate subgroup of finite index in a lattice
r
in
embedding of
r.
(6.1) THEOREM. Let
r
G(B)°
let
r
by [R, Theorem 6.11]). idp
For
denote the identity
(A. Weil, see [R, Theorems 6.7, and 7-66]).
be a uniform irreducible lattice in Q(E)°. If
G
is adjoint semi-simple and GQR)° has no compact factors and is not isomorphic to PSL2CR) then
idr
is locally rigid.
This theorem was first established by A. Selberg [S] for G = PSL , n > 3. There he also pointed out an application: (6.2) THEOREM.
(Weak arithmeticity, see [R, Proposition
6.6]).
be as in (6.1).
Let
G
lattice in G(E)°,
If
r
is a locally rigid
then there exist a number field
a structure of an algebraic group defined over and an element
g € GCR) such that
g r g"
k
k £ B, on
£ G(k).
(6.3)
REMARK. Actually, see e.g., [M2, Lemma 1],
field
(Q(tr Ad r)
generated over
G,
the
dj by the traces of all
154
Homomorphisms of Algebraic Groups
Ady* Y € r,
can be taken as
k.
Weil's theorem (6.1), and therefore its corollary (6.2), were extended to certain non-uniform latticesby H. Bass, A. Borel, H. Garland, and M. Raghunathan. A. Weil were cohomological —
The methods of
it was at the time when
deformations and their relation to cohomology were intensively explored.
The proof consisted: of two steps.
One of them was to show that r
H (J,Ad) = 0
with coefficients in Lie G(B),
Ccohomology of
and the second one was
to establish that the existence of deformations implies H1(r,Ad) # 0. In the same groundbreaking paper [S], A. Selberg conjectured that any uniform irreducible lattice group
G(E)°
PSLpCR) is arithmetic.
Recall that a lattice
arithmetic if there exist over
Q
in a
assumed adjoint, semi-simple, without compact
factors and different from (6.4)
r
T
in
G(B)°
is called
an algebraic group H defined
and an epimorphism of Lie groups IT: H(B)° •* GOO °
with compact kernel such that index in both
r
and
integral matrices in
Tr(.H(22)).
r fl rr(H(Z)) is of finite (H(Z)
is the set of
^-representation of H; this definition of arithmeticity does not depend on
cp or on the choice of a Q-basis in the
representation space of
cp.) An example: let
k
be a
totally real number field, 0 the ring of integers in
k,
B. WeisfeUer
f(x) =
155
Z
a. xif, a. £ k*, a quadratic form on
kn,
G - S0n(f) and T = S0n(f,0). Let CTO = 1,^, . . . ,am be the embeddings of k
into II . Suppose that CT. a . < 0 , l < i < r < n , a.J >0 for r < i < n, j > 0, cr. and a . *J > 0 for all i and all j > 0 . Then r is an arithmetic lattice in G(B) (take
H = RWQG>
then our
choice of the a,, implies that H(IR) = G(R) x (compact group)) This lattice is uniform if f (x) = 0 has no non-zero solutions, in particular, if
[k: Q] > 1.
uniform if k = Q and f(x) = 0 for some In the late
It is not x € Qn, x ^ 0.
sixties counter-examples to the conjecture
in [S] were found in the groups S0(n,l), 3 < n < 55 see §8.
New versions of the conjecture were proposed by
A. Selberg and I. Piatet ski-Shapiro.
The latterfs version
was very general, it included lattices in products of groups G1(k.), G. semi-simple over k.^ and k^ locally compact. It was essentially I. Piatet ski-Shapiro's conjecture that was subsequently established by G. Margulis (see (6.7)). However, a breakthrough was achieved by G. Mostow whose results are summed up by (6.5) THEOREM (Strong rigidity).
Let
G and
G' be
connected adjoint algebraic semi-simple groups over IR. Suppose that
G(B)° and G'(B)° have no compact factors and
are not isomorphic to PSLp(B). Let
a: r -»• I"
bean
156
Homomorphisms of Algebraic Groups
isomorphism between irreducible lattices of GCR)° and G'CR)°
respectively.
P: G -*• G'
Then there exists an B-isomorphism
such that
a(g) = p(.g) for
g € T.
[The absence of a field homomorphism by
AutJEl = {1}
and the absence of a "radial" homomorphism
*0
Y
is ensured by the assumption that
compare with (3-7). rigidity.]
G'
is adjoint,
Note also that strong rigidity implies
Mostow's original proof of C6.5) worked only
for the uniform lattices.
But the missing pieces for an
extension to all lattices were localized and were later provided by G. Margulis, G. Prasad, and M. Raghunathan. Mostowfs results led to a number of spectacular developments which culminated in Margulis1 (6.6) THEOREM (.Superrigidity, see [M2]). connected semi-simple adjoint rkpG > 2
and
Let
G
be a
B-group such that
G(B)° has no compact factors.
Let
G'
be
a connected k-simple adjoint algebraic group defined over a local field k of characteristic 0.
Let
a: r -*• G'(k)
be a homomorphism with Zariski-dense image of an irreducible lattice (i)
r c G(B)°
G'(k).
Then either
a(r) is relatively compact in the Hausdorff
topology of (ii)
into
G'(k), or
k =B
or
(D, G' = G£ x G£
(direct product of
algebraic k-groups), pr., o a: r -»• G£(k) has relatively
B. Weisfeiler
157
compact image, and there exists a homomorphism of algebraic k-groups
p: G -* G£
such that
pr2
« a(g) = p(g)
for
g € r. This theorem implies (6.5) if a
from (6.5) we have that
rk~G > 2.
I" = a(r)
therefore (6.6)is applicable with
is Zariski-dense and
k = B
must have by the assumption of (6.5) on But then (6.6) reduces to (6.5).
Indeed, for
if rk.^ > 2. We G'
that
G£ =
{!}.
The general version of
(6.6) (see [Ml] and [T2]) implies (see [T2] and [T4]) a perfect analogue of (3-7) for homomorphisms (with Zariski dense image) of lattices
r
such as in (6.6) into k-simple
k-groups over an arbitrary (infinite) field k. To see the relevance of the different conditions and implications of (6.6) one need only look (and we will in (6.12)) at Margulis1 proof of (6.7) THEOREM (Arithmeticity theorem) as in (6.6).
Then
r
Let
G
and
r
be
is arithmetic.
This theorem was first proven by G. Margulis in special cases; in one of these cases a similar result was obtained by M. Raghunathan. The counter-examples in S0(n,l) and SU(n,l) (see §8) show that the gap between (6.5) and (6.6) can not be closed without additional (as compared with (6.6)) assumptions on G or a or both. (6.8) Before proceeding further we mention several related developments.
G. Prasad has contributed very much,
158
Homomorphisms of Algebraic Groups
especially in the non-archimedean case, to the study of lattices, see e.g. [Pr], There is an ongoing investigation, led by R. Zimmer (see [Z] and [P]), of generalizations of Margulis1 results to ergodic actions. On the other hand, Y.-T. Siu has generalized the geometric version of (6.5)-
This version considers two
compact locally symmetric spaces X and X' of non-positive sectional curvature, of dimension ^2, totally geodesic factors.
and having no global
The claim then is that
any isomorphism of fundamental groups of X and X' extends, modulo normalizing factors, to an isometry X -*• X'. Y.-T. Siu [Sil,Si2] drops the assumption that X is locally symmetric but assumes that both X and X' are compact Kahlerian. of (6.5).
He also proves in CSil] other generalizations An example, by G. Mostow and Y.-T. Siu [MS],
shows that non-locally symmetric X exist for which the conditions of CSil] are satisfied. P. Parrell and W.-C. Hsiang, e.g.
[PH],studied
topological generalizations of the geometric version of (6.5). A class of compact manifolds M, dim M = n, whose universal covers are contractible but not homeomorphic to Bn is constructed in [Da] for n > 4. The fundamental groups
rr-CM) of such
M
are generated by
B. Weisfeiler
159
"reflections". In view of (8.7). it is improbable that these ir-jCM) are isomorphic to lattices in Lie groups for
n > 30.
However, such M defy the usual techniques to establish topological rigidity. (6.9) Margulis1 proof in [M2] of C6.6) splits naturally into two parts.
The first part constructs a measurable map co
between algebraic varieties, and the second part shows that co
is essentially algebraic.
The first part was somewhat
streamlined and conceptualized by R. Zimmer, see [Z]; we follow his exposition. In the notation of (6.6), let P(resp. P') be a minimal B - (resp., k-) parabolic subgroup of G (resp., G'). Then, by a theorem of C. Moore, r
acts ergodically on
G(H)/P(H).
Then a result of
H. Purstenberg ensures the existence of a measurable r-map co: G(E)/P(1R) •+ M(.G'(k)/P'(k)) 'from
G(B)/P(H) into the
space of probability (positive of total mass 1) measures on
G'(k)/P'(k).
The orbits of
are locally closed.
G'(k) on M(G'(k)/P'(k))
This implies that
co(.G(B)/P(B)) is
(up to measure 0) contained in an orbit, say G'(k).
G'(k)/D, of
It was shown by C. Moore and R. Zimmer that D is
either compact, or the Zariski closure H' of D is a proper algebraic k-subgroup of
G'.
In the first case
a(r) is
relatively compact. In the second case we combine the natural map
G'(k)/D -+ G'(k)/H'(k)
to obtain a
measurable r-map (also denoted by co) co: G(E)/P(E) •*
co with
160
Homomorphisms of Algebraic Groups
•* G'(k)/H'(.k). Assume for tlue rest of this outline that a(r)
is not relatively compact.
(6.10) Then one must recover from G -»• G'
co a rational map
of algebraic k-varieties Cor show that
essentially a map into a point if k ^ B
or
co is
C). To
succeed one must find a link connecting objects of absolutely different nature: measurable maps and rational maps.
Let
\|r: XQR)° -> Y(.k) be a measurable map of points
of algebraic varieties X and Y defined over B and k respectively with Call t
XCR) assumed to have a measure
M-y-rational if, up to measure 0,
into a point when
k #B
of a rational k-map
or
X -*• Y
(C and when
nx-
\|r is a map
\|r is the restriction
k =E
or
C.
To establish
a link between "measurable" and "rational1^ Margulis considers a measurable (with respect to the Lebesgue measure m+n
on E
m+n
) map
almost all n
f: K
x te E
-> Y(k). Then he proves that if for
x € Rm, y € Bn m
and B
^ y
the restrictions of
f to
are rational with respect to the
Lebesgue measures on Mn [im+n-rational.
M-m+n
and IRm, then
f is
To make the above theorem applicable,
Margulis proves that for any B-split subtorus T £ G the map
cpg: Z - ZQ(1R)o(T(B)) -> G'(k)/H'(k) given by
cp (z) = cp(gz) U
of
Z
is
Hy-rational for any unipotent subgroup
and for almost all g € G
Haar measure on
U).
(where
^
is the
The above statement is vacuous if
B. Weisfeiler
161
rkr,G = 1 — hence the restriction that
increases the unipotent subgroup of
q> (where 5
r f c G > 2.
Now one
IL- \ , the restriction
q> (u) = q>(gu)) to which is S
(in
-rational
U
for almost all g, by adding new root subgroups UQcL
one
after another and applying the Pubini theorem and the theorem about maps Bm x Bn -* Y(k) to the ^g1 U(r+l)= U(r)x Ua "* G'(k)/H'(k)-
Thls
completes
the proof in the case when k ? E,
one identifies a maximal unipotent subgroup
G(B)
U
of
such that tl • P(B) is a big Bruhat cell with an
open subset of G(JR)/P(E). Thus one gets a n^-irational map
G(B) -* G'(k)/H'(k). It is not difficult to lift it
to a k-rational homomorphism
G •+ G'.
Needless to say, the complete proof is extremely intricate both conceptually and technically.
The fact
that it, together with the necessary definitions, occupies just 35 pages is due to Margulis1 very condensed style of writing and his ability to extract the "concrete essence" out of every notion and proof he uses. His results are also much more general than the versions we gave here, see [M1],[T2], (6.11) Let us outline now a derivation of (6.7) from (6.6). Since superrigidity implies local rigidity, G is defined over a number field k and r is contained, by (6.2), in
G(k).
Set H = H
(
G - Let *: T •+ H(Q) - G
the
162
Homomorphisms of Algebraic Groups
natural embedding.
If we take
then, modulo replacing ip(r)
r
k = Q(trAd D , see (6.3),
by a subgroup of finite index,
can be assumed Zariski-dense in
H.
On the other
R,,/^ we have that H ~ n UG, K-/W -a a direct product of k-groups where a runs over all
hand, by the definition of
embeddings of
k
into
a
(C and
G
is defined in (.1.2).
If we identify the k-group G as the factor a k-map G' £ G let in
K: H •* G.
Moreover
G, then we get on
r
-
ket now
be another non-trivial simple E-factor of H and
tr be the projection G'
K «t
= id
H -+ G';
since it is dense in
a = IT o \|r: r -* G'
H.
ir(r) is Zariski-dense Suppose that
extends to a rational map
Then the .diagram
H 5- G
commutes since
p: G -»• G'.
K o ^ = id . Let
ir \ / P G'
us restrict it to
Ker K £ H.
our choice of
and
G'
Then rc(Ker K) = G'
by
IT. On the other hand
p(ic(Ker K)) = p(eQ) = eQ'>
a
not extend to
and therefore, by (6.6(11)),
tr(r)
p: G ->• G'
contradiction.
is relatively compact in
G' . Hence
Thus
a
does
H(B) ~ G(B)x
x (compact group). (6.12) The final step is simple. H(Q) -*H(
Since p
r
Consider the natural map
is finitely generated
(namely, for all
p
x|r(D £ H(Z )
which do not enter
in the denominators of entries of generators of
B. Weisfeiler
163
considered as matrices}. primes p, the closure topology of
H
(^L)
By C6.6CO1 we know that for all
r/- * of tCr) in the Hausdorff
is
compact.
|r(p)/(r( x n H(Z ))| < -
Therefore
since
H(* ) is open in HtQ ).
Thus
HCZ) fl r = fl CKr) n H(Z )) is of finite index, say P P m, in r. Since H(2Z) is a lattice we have that H(Z) n r is of finite index m-vol(GCR)/H(2E))(vol GGED/r)"1 H(Z).
in
Thus we have the situation described in C6.4). 7.
QUOTIENT GROUPS OF LATTICES
The next theorem, which is a particular case of a much more general theorem due to G. Margulis [M3] and £M4], establishes a dichotomy similar to that of section (3-3)• (7-1) THEOREM.
Let
normal subgroup of
G r
and
r
be as in C6.6).
Then every
is either central or of finite index.
In view of the Arithmeticity Theorem (6.7)
one can
deduce (7-1) from a positive solution of the Congruence Subgroup Problem CCSP) in the cases where this latter has been established (see [Hu, §§3,^1 for a discussion of isotropic groups and [Kn ] for the only known solution of the CSP for anisotropic groups).
However, Margulis!s proof is
much more uniform than the other known proofs of the CSP, and it is applicable to a wider class of groups of B-rank £ 2. A more general form of the above theorem was used by Margulis [M5] to solve the CSP positively for the group D
of
elements of reduced norm 1 in a central quaternion division
164
Homomorphisms of Algebraic Groups
algebra over a number field; this extended an earlier partial result of V. Platonov and A. Rapinchuk.
The
corresponding local result was obtained by B. Pollak CPo] and, for general division algebras, by C. Riehm CRi]. (7-2) The proof of (.7.1) uses two representation-theoretic notions pertaining to a locally compact group H: Property (T) Cintroduced and used by Kazhdan, and therefore referred to as Kazhdan1s property (T)): the identity A
representation of
H
is an isolated point of the set
of all irreducible unitary representations of Amenability:
H
ajid
is amenable if for every action of
on a non-empty compact
X
probability measure on
X.
(7-3)
H,
H
H
there is an invariant (under
H)
It can be shown that an amenable group has property
(T) iff it is compact, and that solvable Lie groups are amenable.
Also we have
(7.4)
THEOREM CD. Kazhdan [Kal]).If a locally compact
group
H
H.
has property (T),then so does any lattice
Therefore
r/[F,r]
is finite for such
T
in
r.
To apply this theorem one must exhibit Cor, better, characterize) a class of groups
H
having property CT).
This is done in (7-5) THEOREM.
A semi-simple adjoint connected Lie group
H has property (T) iff it does not have or
PSU(n,l), n > 1,
as factors.
PSO(n,l)° n > 2,
B. Weisfeiler
165
Here
S0(n,l) = {A € SLn+1CB) | AJAt » J} and
SU(n,l) • {A € SLn+1(
H
= J} where
(1.5) was proven [Kal] by
which do not have simple factors of
R-rank 1 (.actually, Kazhdan fs proof was for "most" simple Lie groups of B-rank > 2; it was extended to full generality by L. Vaserstein (Punct. Anal. Appl. 2(1968), 17*0.
Then B. Kostant showed [Ko] that a simple connected
Lie group
G(B)
of B-rank 1 has property (T) iff it is not
locally isomorphic to S0(n,l), n > 2, that the family of simple Lie groups consists of n > 1,
or
SU(.n,l).
G(B), rkG
Recall
= 1,
PSO(n,l)°, n > 2; PSU(n,l), n > 1; PSp(n,l),
(the unitary group of a skew-Hermitian form of Witt
index 1 over the Hamilton quaternions H) 9 and of a group of type
Fjj. (There are the following
low-dimensional
isomorphisms: PSO(2,1)0- PSU(1,1) ~PSL2(B), PSO(3,D°2l -PSL2(C), PSO(4,1)°- PSp(l,l)» PSO(5,D°21 PSL2(E).) Finally, note that compact groups have (T) — for them A H is discrete. The following theorem follows from (7-3) and (7-1*) if
GCR) satisfies the conditions of (7-1*). In
general it follows from (7.1) • (7.6) THEOREM.
If
H
is an adjoint semi-simple connected
Lie group which is not locally isomorphic to n > 2,
or
PSU(n,l), n > 1,
PSO(n,l),
and has no compact factors,
166
Homomorphisms of Algebraic Groups
then
r/[T5r]
is finite for any irreducible lattice
r
in
H.
In the case when
r
is cocompact and
G
has no
R-factors of rank 1, (7-6) was previously proven by S. Kaneuki and T. Nagano [KN]. For uniform lattices it was proven by J. Bernstein and D. Kazhdan [BK]. The latter's proof is extendable to non-uniform lattices. (7-7)
The proof of Kaneuki and Nagano was based on
Y. Matsushima1s fundamental interpretation [Ma] of H^rjR) ~ Cr/[r,T]) R through certain representation2 2 theoretic properties of L (G/r) (the space of squareintegrable functions on
G/r). Matsushima1s results are
described in [BW, Ch. II]. (7.8)
We will now give a short sketch of Margulisf proof
of (7.1). factors
See also [T2] and PSO(n,l)° or
quotient of
r
[Z]. First, if
PSU(n5l)
G
has no
then every amenable
is finite by (7-2), (7-3), and (7.4).
The
remaining cases of rank s> 2 groups are more difficult, they are taken care of in [M4], In [M3] Margulis shows that non-amenable quotients of
r
from (7-1) are central.
do this he shows that for any non-amenable group N there exists a metrizable compact X on which
N
acts without
To
B. Weisfeiler
167
fixing any probability measure on
X.
(Note that a
compact X exists by definition of non-amenability in (7.2); the problem is to find a metrizable compact.) Suppose that N = r/I^, r; < r,
N
non-amenable. Then by a theorem of
Furstenberg (which was already used in (6.9) in the case X = G'(k)/P'(k))
there is a measurable r-map
co: G(R)/PCR) -^M(X);
P
meaning as in (6.9) and
and r
M
here have the same
acts on
X
the natural quasi-invariant measure on
via N.
Let
G(B)/P(B).
further we need a difficult and remarkable
G(E)/?CR)
H-parabolic subgroup of
G.
on
where
But then
GCR)/P(H), which implies that
T^
r^
To move
result of
Margulis which enables us to identify (M(X), co#(n.)) measurable T-space with
M- be
P
as a
is a proper
acts trivially
is central.
Of
course, the result we have used is the heart of the argument, and the assumption
rk^G > 2
is used in its
proof. 8.
LATTICES IN
In this section
PSO(n,l) k
AND
PSU(n,l)
is always a number field.
(8.1) Theorems (6.6), (6.7), (7.1), (7.4), (7.5), and (7.6) are not applicable to the simple Lie groups n > 2, and
G = PSU(n,l), n > 1.
G = PSO(n,l) ,
Recall that
(i) PSO(2,1)° - PSU(1,1)- PSL2CR);
168
Homomorphisms of Algebraic Groups
(ii) (iii)
PSO(3,1)° ~ PSO(n,l)°, n > 2,
of the group of isometries Lobachevsky space (iv)
is the connected component
of an n-dimensional
n
A;
PSU(n,l) is the connected component of the group
of holomorphic automorphisms of an n-dimensional ball Bn = {(x±) € (Dn, Z|x±|2 < 1};
it also coincides with the
connected component of the group of isometries of the Bergman metric on (8.2)
Bn.
The uniform torsion-free lattices
T
in
PSO(2,1)°
are the fundamental groups of the compact Riemann surfaces r\A . Therefore they are 2r-generator-l-relator groups 2 (where r is the genus of r\A ) . Non-uniform lattices are even worse: they contain a free subgroup of finite index. (8-3)
We assume in the sequel that
G T^ PSL2CR).
Counter-examples to (.7.6) in PSOCn3l)°a n > 3,
were
constructed by E. Vinberg for special lattices when 3 < n < 10. For arithmetic lattices
r
of the type
described in (6.3) (with r » 1) J. Millson [Mi] showed that. a congruence subgroup
A
of
r
contradicts (7.6).
The corresponding cohomological statement — —
H (AJR) * 0
was generalized in [MR]; e.g. H (A,R) * o for
1 < 1 < n.
Note that if
A/ [A, A]
is infinite, then the
same holds for all subgroups of finite index in
A.
B. Weisfeiler
169
(.8.4) Millsonfs proof of
|A/[A,A]| - -
simple topological observation.
is based on a
Namely, the fact that
An
contains totally geodesic subspaces of codimension 1 makes it possible to construct a non-trivial homology class c € Hn""1CA\An^R).
By Poincare" duality this implies that
1
0 # H CA^E) ~ (A/[A,A]) SB. Millson's ideas do not apply 7L to PSU(n,l) because the ball Bn Cwith Bergman metric! has no totally geodesic subspaces of codimension 1. (-8.5)
Counter-examples to* (7.6) in PSU(nal)a n > 2.
Consider arithmetic subgroups in PSU(n,l) constructed as in (6.3).
Namely, take an imaginary quadratic extension K of a
totally real number field k
and denote by ~~
the complex cpnjugation in (D. Then is a Hermitian form on Kn.
f = Z a^^x^, a^ € k,
if the a.^ satisfy the same
conditions a.s in C6.3) and if, moreover, r = 1 then T = PSU(f,0) is an arithmetic lattice in PSU(n,l). If k # Q D. Kazhdan has shown Csee [Ka 2] and [B¥, Ch. VIII]) that the lattices
PSU(f,0) have congruence subgroups
infinite
A/[A,A]; these
To prove that such techniques — Sp2n
A
k * ft (thus
with
are automatically uniform.
exist Kazhdan uses powerful
properties of the Weil representation of
and Matsushima1s
of a lattice.
A
A
[Ma]
interpretation of homology
G. Shimura [Sh] removed the condition A
can be non-uniform); his construction
is more explicit and gives more information.
170
Homomorphisms of Algebraic Groups
(8.6) Counter-examples t£ (6 . 6 ) and (6.7) In PSO(nal) were first constructed by P. Lanner in 1950 > but he did not know that they were counter— examples to a conjecture stated much later.
Then V. Makarov constructed infinitely many non-
arithmetic lattices in PSO(.3,1)° ~ PSL2(C). [Vi 1]
E. Vinberg
understood the nature of the known examples and
developed general techniques to construct more.
His
lattices are groups generated by reflections in totally geodesic hypersurfaces in
An.
Such groups are easily
described in terms of generators and relations.
Vinberg
distinguished two ways in which arithmeticity and superrigidity can break down. Namely, suppose that lattice in PSO(n,l)°.
M -R k G -
r
x( compact group).
Quasi-arithmetic if M(B)° ~ PSO(n,l)° By (.6.4) any arithmetic lattice is quasi-
For a quasi-arithmetic
r
the first step (6.11)
of Margulis1 proof of arithmeticity works. r
r is
G(k) and is, moreover, Zariski-dense in
Cal1
arithmetic.
is a
Let k = Q(tr Ad r) . Then, as in
(6.11), it can be assumed that G is defined over k, contained in
r
Therefore if
is not arithmetic, the second step (6.12) must break down
This means that for some prime p the matrix entries of the' y € r
contain arbitrary large powers of this prime in the
denominators. other hand:, if
This contradicts also (6.6(1)). r
On the
is not quasi-arithmetic then (6.11) does
B. Weisfeiler
171
not work. too.
This shows, as in (6.11), that (6.6(ii)) fails
Vinberg gives in CVi 1] examples of both uniform (for
n = 3
and
4) and non-uniform (for
n = 3) non-arithmetic,
quasi-arithmetic lattices; he also gives examples of both uniform (for and
n = 3*4
and
5) and non-uniform (for
4) non-quasi-arithmetic lattices.
n = 3
A recent result
CVi 2] of Vinberg, extending his own work, and also that of V. Nikulin, and proving a conjecture of his, is (8.7) THEOREM.
There are neither uniform nor arithmetic
lattices generated by reflections in PSO(n,l)°, n > 30. Thus the methods used to construct non-arithmetic lattices in PSO(n,l)° do not work for
n > 30.
An optimist
can consider this as an indication that all lattices in PSO(n,l)° are arithmetic for large n. (8.8)
Counter-examples to (6.6) and (6.7) in PSU(n.l).
n > 23
are much more difficult to construct.
The walls
of a fundamental domain have codimension 1 but can not be totally geodesic (compare (.8.4) and (8.6)).
G. Mostow
developed [Mo 2] a very deep theory of certain polyhedra in Bn. B
The walls of his polyhedra are surfaces of points in
equidistant from two given points (in the Bergman
metric).
These polyhedra are used to construct and study
uniform non-arithmetic subgroups in SU(n,l), n = 2. Mostowfs subgroups in [Mo 2] are uniform and contained in the group of integral points of an appropriate algebraic group.
Thus
172
Homomorphisms of Algebraic Groups
they are not quasi-arithmetic,
Mostow's polyhedra were
also used to construct interesting Kahler manifolds see [MS] and (6.8). Using completely different ideas going back to E. Picard, Mostow constructed new both uniform and non-uniform non-quasi-arithmetic
lattices in PSU(2,1); these new groups
2
are monodromy groups of certain branched covers by B . The lattices mentioned above are all in PSU(2,1).
However,
G. Mostow has an example of a non-uniform lattice in PSU(3,1) as well. (8.9) Counter- examples t£ a generalization ojT the strong rigidity theorem (6.5). in
PSO(n,l)°, n > 3,
By (6.5) isomorphisms of lattices
and
PSU(n,l), n > 2,
extended to isomorphisms of Lie groups.
can be
Mostow, using
a generators and relations description of lattices, constructed in [Mo 2] T^
and
T2
of
two uniform arithmetic lattices
PSU(.2,1) and an epimorphism a: T^ -*• T2
with infinite kernel.
Clearly such
a
can not be
extended to an automorphism of PSU(2,1). Similarly, if one looks at Vinberg's examples [Vi 1], one can find many pairs of lattices PSO(3*1)°
A-, and
such that there exist epimorphisms
with infinite kernels. epimorphisms
A«
in
a: A-, •*• A2
Moreover, there are sequences of
-* A^ -*• A. ., -+...-* A., infinite to the left
B. WeisfeHer
173
and all having infinite kernels. notation of
For example, in the
A. = I1'. or r m are fixed integers, r > 1,
[Vi 1] , one can take
A. = r~. where r and m r m m > 4. I was not able to find in [Vi 2]
such examples
in PSO(n,l)°, n > 4. (8.10) The attention given at present by topologists to 3-manifolds has led to many results and conjectures about lattices in
PSO(3A)0 = PSL2((C).
One should consult
work of W. Thurston (but I can not give an exact reference). • On the other hand, lattices in PSL2(
F. Grunewald and J. Schwermer
showed in [GS] that arithmetic lattices in PSL2(
r
in
PSLp(^)
has a subgroup
such that the normal closure in
r
A
of finite index
of any element
x € A, is a normal subgroup of infinite index.
x # 1,
He also
mentions that M. Gromov has proved similar results for lattices in PSO(n,l)°, n > 3(8.11) The notion of an FQ-group can be useful, as was pointed out to me by A. Lubotzky, in explaining the examples described in (8.3), (8.5), (8.9). Indeed, if a lattice r is an FQ-group, it or its subgroup of finite index can be
174
Homomorphisms of Algebraic Groups
mapped onto any finitely generated group. This would mean that the information about the commensurability class of
r
(and by (6.5) not all
r
in
PSO(n,l)°
or
PSU(n,l)
are commensurable) is contained in the kernel N of a homomorphism onto a free group and in the extension of N by a free group.
It would be then interesting to find out what
this information is and how it is carried. (We say here that two groups are commensurable if they have isomorphic subgroups of finite index.) 9.
CONCLUDING REMARKS
(9.1) Recently V. Petechuk [Pe] concluded investigation of automorphisms of
SLn(A), GLn(A), and En(A), A
commutative ring: if
n > 4
an arbitrary
they are "standard"; for n = 3
see Petechuk, Math. Notes 31(1982). The proof is based on a description of the normal subgroup structure of E (A). In a rough outline it goes as follows: normal subgroups correspond to ideals; therefore an automorphism of Spec A; for p € Spec A
a
induces a bisection
it induces isomorphisms
En(A/p) ~ En(A/a*Cp)) which are standard and can be glued together to give a "standard" automorphism(.9-2) A. Hahn [Ha2] has found a new setting —Morita equivalence —
for the isomorphism theory of the classical
groups over rings. (9-3) D. James [J] and myself [We] have obtained results about homomorphisms (possibly with infinite kernels) of
a*
B. Weisfetter
175
anisotropic algebraic groups. (9.4) Reference [JWW] contains a list of problems pertaining to the present survey. SUMMARY TABLE FOR §6,7,8 LATTICES IN ADJOINT SEMI-SIMPLE LIE GROUPS
In this table G Is an adjoint semi-simple connected Lie group and r Is an Irreducible lattice In 0 ^xGroup G
S0(n,l)°, n > 3
Property^.
SU(n,l),
n > 2
see
PSp(n,l) , n >2
see
( 7 - 5 )Fl| of rank 1 (7.5)
rk R G>2 'yes", if G does not have PSO(n,l) or PSU(n,l) as factors; 'no" otherwise, see (7.5)
Kazhdan ' s property (T) , o, see (7-5) see (7.2)
yes, see C7.5)
r/[r,n
finite, see (.7.4) and (7.6)
finite, see (7.4) and (7.6)
There are both amenable, see (8.3), C8.5), and non-amenable , see (8.9), infinite quotients Some lattices have other lattices as non-trivial quotients, see (8.9), (8.10) see also (8.11)
every finite quotient is not amenable see (7.2) and (7.3)
every quotient is finite, see (7.1)
Strong rigidity, see (6.5)
yes, see (6.5)
yes, see (6.5)
yes, see (6.6)
Superrigidity, see (6.6)
no, -see (8.6). (8.8), but..., see (8.7)
Quotients
an be Infinite, see 8.3), (8.5).
Quasl-arlthmsti in PSO(n,l) non-arithmetic uniform (n • 3,4) lattices, non-uniform (n • 3) see (8.6} see (8.6), (8.7) non-quasiarithmetic lattices see (8.6)
in PSO(n,l) uniform (n - 3,4,5) non-uniform (n • 3,4) seeja^i^iS..!! in PSU(n,l) uniform (n - 2) non-uniform (n • 2,3) see (8.8)
unknown
yes, see (6.6)
unknown
no, see (6.7)
unknown
no, see (6.7)
176
Homomorphisms of Algebraic Groups
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