Infinite Dimensional Harmonic Analysis: Proceedings of the Third German-Japanese Symposium University of Tubingen, Germany 15-20 September 2003

X = W//G' satisfies Assumption 3.1 (d) if and only i/dim V+ = dim V~ or dim V ± > dim U. The quotient space X is naturally identified with the determinantal variety Det m (V + ® V~) of rank m = dimU (see §1.2). Theorem 3.2. If W is one of the representations of G x G' which are listed in (l)-(6) below, then the quotient maps X = W//G' ^^- W —*—>• W//G = Y satisfy Assumptions 2.1 and 3.1. In particular, a G'-orbit O' C Y is lifted to a G-orbit O C X, and the lifting map 6 is injective. (1) Let G = GL(n,C) and V = C" the natural representation ofG. We put W = (V ® V*) ® U for the natural representation U ofG' in Table 4. The quotient space X is given in Table 4 and Y = U ® U.

208

Table 4. G' 0(m,C) Sp(2m, C) SL(m,C)

condition 2m < n 4m < n 2m < n

X Sym^VeV)

Ait m (v®v*) c»f(vev)

(2) Let G - GL(n, C) and V = C" the natural representation ofG. We put G' = G'+xG'_, and W - V®U+®V*®U- for the natural representations U± of G'± in Table 5. The quotient space X is given in Table 5 and Y= Table 5.

G'+ 0(p,C) Sp(2p,C) 0(p,C) SL(f,C)

condition G'_ X 0(9) C) p + ? < n Sym J ,(V)®Sym,(V) Sp(2g,C) 2p + 2g < n Alt2p(V)®Alt 2 ,(V*) 5p(2g,C) p + 2g < n SL(q,C) p+g
dual pair (0(p,«),Sp(2n,R)) (Sp(2p,2«),0'(2n)) none none

(3) Let G' = GL(m,C) and U = Cm the natural representation of G'. We put W = V ® (U @ U*) for the natural representation V of G in Table 6. The quotient space Y is given in Table 6 and X = Detm(V ® V"). Table 6.

G 0(n,C) 5p(2n,C)

condition 4m < n 2m < n

Y Sym(U-®tT) Alt(t/©t/*)

(4) Let G' = GL(m,C) and U = Cm the natural representation of G'. We putG = G+xG-, and W = V+ ® U ® V~ ® 17* /or the natural representations V± of G± in Table 7. The quotient space Y is given in Table 7 and X = Detm(F+ ® V~). Table 7.

G+ 0(r,C) Sp(2r,C) 0(r,C)

condition y G_ Sym(J7)®Sym(I7*) 2m < r, s 0(s,C) Alt(J7)®Alt(f/*) m < r, a 5p(2s,C) Sp(2s,C) 2m < r- l,s Sym(f/)® Alt([/*)

dual pair (Sp(2m,R),0(r,s)) (0*(2m),5p(2r,2*)) none

209

( 5 ) Let V = C" (respectively U = C") be the natural representation of G = GL(n,C) (respectivelyG' = GL(m,C ) ) and assume 2m 5 n. We put W = V €9 U*@ V* 8 U. The quotient spaces are given by

X = Det,(V* 8 V) and Y = U*63 U. The corresponding dual pair is (GL(m,C),GL(n,C ) ) . ( 6 ) Let W = CP 8 Cr @ (CP)* €9 (C")" @ (CQ)* €9 (Cr)*@ CQ8 C" be a representation of G x G' with p + q 5 r, s, where G = GL(r,C)x GL(s,C) and G' = GL(p,C) x GL(q,C ) . The quotient spaces are given by X = Detp(C' €9 (Cs)*) x Detq((Cr)*63 C") Y = CP 8 (CQ)* CB (CP)* 63 CQ.

and

The corresponding dual pair is ( U ( p ,q), U(r,s)). 4. Application to Dokovit-Sekiguchi-Zhao problem

Recently, from the view point of the invariant theory, D. Z. Dokovit, J. Sekiguchi and K. Zhao [l]are studying the SL(m, C)-action on Mm(C) = C m BC" (see (0.1)). The orbit structure of the action is also being studied by H. Ochiai. Here we resolve this action into two different ways, which fit our theory of the double fibration. 4.1. Resolution via the contmction by the action of G L ( n ,C ) .

Let G = GL(n,C) and V = C" the natural representation of G. We put W = (V@V*)@Ufor the natural representation U = C" of G' = S L ( m ,C ) . In the following, we assume that 2m 5 n. Theorem 3.2 (1) tells us that the double fibration by the affine quotient maps

X=W//G'i

'

W

*

}W//G=Y

satisfies Assumptions 2.1 and 3.1. Here we only check Assumption 3.1 (e). Lemma 4.1. For any y E Y , the fiber $ - l ( y ) intersects a closed GI-orbit, which is precisely a fiber (p-'(x) for some x E X .

Proof. We identify W = V@U@V* @U with the space of 2n x m-matrices MZ,,,". Then a non-zero w E Mz,,," generates a closed GI-orbit if and only if rank w = m. In fact, for any non-zero x E G",(U), we can prove that the

210

fiber cp-l(z) consists of a single G’-orbit, hence it is closed. Thus all nonclosed orbits are contained in the null fiber cp-’(O) which is characterized by rankw < m. If we write w = ( A ,B ) E Mn,m x Mn,m, the quotient map TJ is given by

T J ( ~=)

t~~

E M,(c)

= Y. contains a full rank

Now it is elementary to verify that any fiber +-‘(y) matrix w under the condition m 5 n.

0

It is easy to see that Y = U @ U = Mm(C) inherits the SL(m,C)action in Eq. (0.1) considered by DokoviC-Sekiguchi-Zhao [l],while X is isomorphic to G z ( V @ V * ) .Thus, an SL(m,C)-orbit 0’ c Mn(C!)is lifted to a GL(n,C)-orbit 0 = O ( 0 ‘ ) C G$(V @ V * ) .Moreover, the lifting map O : Mm(C!)/SL(m,C ) + Gf(V @ V*)/GL(n,C)

(4.1)

is injective, and preserves the closure relation and nilpotency. As an example, we examine the simplest case, i.e., the lifting of the trivial orbit. Example 4.1. Let O1 = O({O}) be the lift of the trivial nilpotent orbit C Y. Then we have @ = %//G‘. Let us show that O1 is a GL(n, C!)-spherical variety with the normal closure in X = G$(V @ V*). To prove it, let us prepare some notations. We denote the set of paxtitions of length at most m by

0’= (0)

P, = {a = (a1,...,a,) E Zrn I a1 2

2 a , 2 O},

which we identify with the subset of dominant weights of GL(m,C) as usual. The irreducible finite dimensional representation of GL(m, C!) with highest weight a is denoted by 7Lm),and its contragredient 7Lm)*has the * holds. highest weight a* = (-a,, -am-l,. . . , -a1),hence T ( ~ ~=) T:;) For a,/3 E Pm,we put a O P = ( a , O ,..., O , p * ) E Z n ,

which is a dominant weight of GL(n, C). Now let us return to the proof. We note that as a GL(n,C) x (GL(m, C!) x GL(m, C))-module,

@.[%I

21 31 N

x@

TZo

a,BEP,

(T:?’

@ TAY’).

21 1

For this, see Theorem 2.5.4 in for example (note that the action of GL(rn,C) in the second factor is the dual to that in 6 ) . Then the action of SL(rn,C ) is obtained by the restriction of the action of GL(rn,C ) x GL(rn,C ) to the diagonal subgroup ASL(m,C). Note that

C if a - /3* = kl, for some k E Z, 0 otherwise, N C if where 1, = (1,1,. . . ,1) E Z". Thus we have (T$) @ and only if ai +/3,-i+l = k (1 I: i 5 rn) for a fixed non-negative integer k. From this, we obtain the decomposition of the regular function ring @[@I of the closure of the lifted orbit O1 as a representation of GL(n,C).

k20 c r - f i * = k l ,

(44 This shows that @ is a spherical variety, hence so is O1. As for the normality, T is known to be normal, and as a quotient of the normal variety, 0 1 is also normal. 4.2. Resolution via the action of the orthogonal and

symplectic groups. We denote the natural representation of O(p,C) by V+ and that of Sp(2q, C ) by V-.We put G = G+ x G- = O(p,C ) x Sp(2q, C ) and W = (V+@ V-)@ U for the natural representation U = C" of G' = SL(rn,C ) . We assume that 2m < p,2q + 1. By Theorem 3.1, the double fibration by the affine quotient maps

X = W//G' <

q w

*

bW//G=Y

satisfies Assumptions 2.1 and 3.1. Moreover, we have

Y = Sym(U) @ Alt(U) N M,(@) with the SL(m,C)-action in Eq. (O.l), while X is isomorphic to Gz(V+@ V-). Thus, an SL(rn,C)-orbit 0' C Mn(C) is lifted to an O(p,(c) x Sp(2q, C)-orbit 0 C Gz(V+@ V-).Moreover, the lifting map

e : iw,(c)/sL(rn,c) + sg(v+@ v-)/o(v+) x sp(v-)

(4.3) is injective and it preserves the closure relation and maps nilpotent orbits to nilpotent orbits.

212

Thus, if we put p = 2q, then the orbit space M,,,(C)/SL(m, C ) has two different embeddings to the same afEne Grassmaanian cone GE(V +@ V - ) . One is treated in this subsection, and the other is explained in $4.1. References 1. Dragomir Z. DokoviC, Jiro Sekiguchi, and Kaiming Zhao. On the geometry of unimodular congruence classes of bilinear forms. Preprint, 2003. 2. Andrzej Daszkiewicz, Witold Krdkiewicz, and Tomasz Przebinda. Nilpotent orbits and complex dual pairs. J. Algebra, 190(2):518-539, 1997. 3. Andrzej Daszkiewicz, Witold Krdkiewicz, and Tomasz Przebinda. Dual pairs and Kostant-Sekiguchi correspondence. I. J. Algebra, 250(2):408-426, 2002. 4. Andrzej Daszkiewicz and Tomasz Przebinda. The oscillator correspondence of orbital integrals, for pairs of type one in the stable range. Duke Math. J., 82(1):1-20, 1996. 5. Robin Hartshorne. Algebraic geomet y . Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52. 6. Roger Howe. Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond. In The Schur lectures (1992) (Tel Aviv), pages 1-182. Bar-Ilan Univ., h a t Gan, 1995. 7. fiiedrich Knop. Uber die Glattheit von Quotientenabbildungen. Manuscripta Math., 56(4):419-427, 1986. 8. Laurent Manivel. Symmetric functions, Schubert polynomials and degeneracy loci, volume 6 of SMF/AMS Tezts and Monographs. American Mathematical Society, Providence, RI, 2001. Translated from the 1998 French original by John R. Swallow, Cours SpBcialis6s [Specialized Courses], 3. 9. Kyo Nishiyama. Multiplicity-free actions and the geometry of nilpotent orbits. Math. Ann., 318(4):777-793, 2000. 10. Kyo Nishiyama, Hiroyuki Ochiai, and Chen-bo Zhu. Theta lifting of nilpotent orbits for symmetric pairs. math.RT/Q312453,2003. 11. Kyo Nishiyama and Chen-bo Zhu. Theta lifting of unitary lowest weight modules and their associated cycles. To appear in Duke Math. J., 2003. 12. Takuya Ohta. Nilpotent orbits of &-graded lie algebra and geometry of the moment maps associated to the dual pair (U@,p), U ( T , S ) )Preprint, . 2002. 13. Takuya Ohta. On the geometric quotient which appear as the restriction of the moment map related t o dual pairs. In Proceedings of Symposium on Representation Theory 2003 (Ohnuma), pages 52-61, 2003. (Japanese). 14. Gerald W. Schwarz. Representations of simple Lie groups with a free module of covariants. Invent. Math., 50(1):1-12, 1978/79. 15. B. Vinberg and V.L. Popov. Invariant theory. In Algebraic geometry. IV, pages 123 - 278. Springer-Verlag, Berlin, 1994.

Infinite Dimensional Harmonic Analysis I11 (pp. 213-232) Eds. H. Heyer et al. @ 2005 World Scientific Publishing Co.

ADMISSIBLE WHITE NOISE OPERATORS AND THEIR QUANTUM WHITE NOISE DERIVATIVES

UN CIG JI* Department of Mathematics Chungbuk National University Cheongju 361 -763 Korea E-Mail: uncigji @cbucc.chungbuk.ac.kr

NOBUAKI OBATA+ Graduate School of Information Sciences Tohoku University Sendai 980-8579 Japan E-Mail: [email protected]. ac.j p

An operator on a Fock space is considered as a non-linear and non-commutative function of annihilation and creation operators at points { a t , a ; ; t E T}. The derivatives with respect to at and a ; , called respectively the annihilation- and creation-derivatives, are formulated within the framework of quantum white noise theory. We prove the differentiability of an admissible white noise operator and give explicit formulae for the derivatives in terms of integral kernel operators. The qwn-derivative is a non-commutative counterpart of the Gross derivative in stochastic analysis. Mathematics Subject Classifications: 46325 60H40 81825 Key words: Fock space, admissible white noise operator, Gross derivative, annihilation-derivative, creation-derivative, quantum white noise

1. Introduction

In this paper we focus on an operator on the (Boson) Fock space r ( H ) with H = L 2(T ,u),where T is a topological space equipped with a a-finite Bore1 measure v. A fundamental role is played by the annihilation and *Work supported in part by grant (No. R05-2002-000-00142-0) from the Basic Research Program of the Korea Science & Engineering Foundation. tWork supported in part by the Grant-in-Aid for Scientific Research (No. 15340039) from Japan Society for Promotion of Sciences.

214

creation operators at a point, denoted by at and a;, in many questions arising from quantum stochastic analysis, infinite dimensional harmonic analysis, quantum field theory and so on. The pair {at, a; ; t E T} is sometimes called a quantum white noise field on T . However, they are not well defined only within the framework of the Fock space. In literatures we find two formulations. One is to consider at and a; as unboundedoperator-valued distribution by smearing t; the other is to formulate them as continuous operators on a certain Gelfand triple ( E ) c r ( H ) c ( E ) * without smearing t. The second approach is along with the classical Hida calculus and has been considerably studied under the name of quantum white noise theory [4, 8, 171. We recall that every white noise operator admits a Fock expansion, i.e., E E ,C((E),( E ) * )is decomposed into an infinite sum of integral kernel operators:

where

x at, - . . a t m v ( d s l ) .. . v ( d s l ) v ( d t l ) ..v ( d t , ) ,

(1.2)

and 6 1 ,is~a kernel distribution. One may accept (1.2) as a “polynomial” in at and a;, hence (1.1) as a function of them: E = E(at,at ; t E T). Thus we are naturally led to a kind of functional derivatives:

These motivated us to study the annihilation- and creation-derivatives. The main purpose of this paper is to formulate the annihilation- and creation-derivatives (together called qwn-derivatives), and to study gwndifferentiability of a white noise operator. In fact, a general white noise operator is not qwn-differentiableand there is difficulty in formulating (1.3) in a direct manner. Our strategy is to establish an operator version of Gross derivative (the derivative along H ) of an admissible white noise function (see e.g., [l]). The paper is organized as follows. In Section 2 we prepare some notation in white noise theory. We introduce in Section 3 the spaces of admissible (test and generalized) functions as

( E ) c B c r(H) c B* c (E)*

215

and discuss an admissible white noise operator, i.e., a continuous operator from B into G*. In Section 4 we give the definition of the qwnderivatives OF2 for a white noise operator E. Then we prove the qwndifferentiability of an admissible white noise operator and obtain formulae for the annihilation- and creation derivatives (Theorem 4.4). This result follows from the qwn-differentiability of an admissible integral kernel operator (Theorem 4.1) and the Fock expansion of an admissible white noise operator (Theorem 4.3). The notion equivalent to an admissible white noise function was first introduced by Lindsay-Maassen [13] and has been studied in many literatures for different purposes, e.g., Awe-Bksendal-Privault-Ub~e [l], Belavkin [2], Benth-Potthoff [3], Grothaus-Kondratiev-Streit [ 5 ] , Ji [7], LindsayParthasarathy [14]. We hope that our new idea of qwn-derivative opens a new direction in the study of Fock space operators. Further study and application will be discussed in [lo]. 2. White Noise Theory

2.1. Underlying Gelfand rriple

Let T be a topological space equipped with a a-finite Bore1 measure v. Let H = L2(T,v) be the (complex) Hilbert space of L2-functions and the norm is denoted by 1. l o . Let A be a selfadjoint operator (densely defined) in H satisfying the conditions (Al)-(A4) below. ( A l ) inf Spec (A)

> 1 and A-'

is of Hilbert-Schmidt type.

Then there exist a sequence

and an orthonormal basis { e j ) g o of H such that Aej = Xjej. For p E R we define M

C~yl

t E H. A ~= ~ I ~(t,ej>12, j=O Now let p 2 0. We put E, = {t E H ; I I p < m} and define E-, to be the completion of H with respect to I - I-p. Thus we obtain a chain of Hilbert spaces { E p; p E R} and consider their limit spaces: It:I

=I

<

S A ( T )= E = proj limE,, P+M

Sfi(T) = E* = indlim E - p . ,+a

216

These are mutually dual spaces. Note also that SA(T)becomes a countably Hilbert nuclear space. Identifying H with its dual space, we obtain a complex Gelfand triple:

E = SA(T)C H = L2(T,v) c E* = Si(T). As usual, we understand that SA(T) and Si(T)are spaces of test functions and generalized functions (or distributions) on T,respectively. For white noise theory Si(T)must contain delta functions. But this is not automatic and we need further assumptions: (A2) For each function ( E SA(T) there exists a unique continuous function c o n T such that ( ( t ) = r(t) for v-a.e. t E T. Thus SA(T)is regarded as a space of continuous functions on T and we do not use the exclusive symbol The uniqueness in (A2) is equivalent to that any continuous function on T which is zero v-a.e. is identically zero.

c.

(A3) For each t E T the evaluation map bt : { t)( ( t ) ,( E SA(T), is a continuous linear functional, i.e., St E Si(T). (A4) The map t t)St E Si(T), t E T,is continuous with respect to the strong dual topology of Si(T). See [17] for more discussion on these assumptions. The canonical C-bilinear form on E* x E is denoted by (., .). In other words, we set

We also write .) for the canonical C-bilinear form on H. Let J be the conjugate operator defined by (a,

M

00

It then follows that

The real parts of E, H, E* are subspaces invariant under the action of J and are denoted by ER, H R and E&, respectively. Then we obtain a real Gelfand triple:

ER C HR C E&.

(2.1)

217

(These are real vector spaces but not necessarily spaces of R-valued functions.) R e m a r k 2.1. A prototype of our consideration is the case where T = R with Lebesgue measure v ( d t ) = dt and

A = l + t - -dt2 d= 2

(t + -i ) * ( t + - $ ) + 2 .

In this case SA(T)coincides with the space of rapidly decreasing functions, which is commonly denoted by S(R). Recall also that

ej(t)= (t/;;2jj!)-"'~j(t)e-t'/2,

j = 0,1,2,-.-

,

where H j is the Hermite polynomial of degree j , constitute an orthonorma1 basis of L2(R) and Aej = ( 2 j 2)ej. This prototype is suitable for stochastic processes, where R plays a role of the time axis. Our general framework allows to take T to be a manifold (space-time), a discrete space or even a finite set.

+

2.2. Hida-Kubo-Takenaka Space

Let Ep be the Hilbert space defined in $2.1, where p E R. We consider the (Boson) Fock space:

c 00

I'(EP) =

{

A

9 = ( f n ) Z o; fn E E p , II 9 1; =

n! I

fn

;1 <

n=O

I

7

which is essentially a direct sum of symmetric tensor powers of Ep and the weight factor n! is for convention. Having obtained a chain of Fock spaces {F(Ep) ; p E R}, we set

(E) = projlimI'(E,), P+W

( E ) *= indlimI'(E-,). P-+W

Then we obtain a complex Gelfand triple:

( E ) c W)c (El*, which is referred to as the Hida-Kubo-Takenaka space [ll]. By definition the topology of (E) is defined by the norms M

n=O

218

On the other hand, for each 9 E (E)* there exists p 2 0 such that 9 E I?(EPp).In this case, we have M

9 = (Fn). n=O

The canonical C-bilinear form on (E)*x (E) takes the form: 03

((9, 4)) =

C n!

( ~ n fn) , 3

9= (

~ n E)

( E ) * , 4 = (fn) E

(~1-

n=O

Here we recall two important elements of ( E ) * . (a) White noise. By assumption (A3),

wt =(o,&,o, ...),

t E T,

belongs to (E)*and is called a white noise. According as T represents time or space, the family {Wt ; t E 2') c ( E ) *is called the white noise process or white noise field on T . (b) Exponential vector: For x E E* an ezponential vector (or a coherent vector) is defined by n!

Obviously, q ! ~E~ ( E ) * . Moreover, ~$tbelongs to ( E ) (resp. I?(Ep))if and only if [ belongs to E (resp. E p ) . In particular, 4 0 is called the vacuum vector. 2.3. White Noise Operators

In general, a continuous operator from (E) into (E)* is called a white noise operator. The space of all white noise operators is denoted by C ( ( E ) ,( E ) * ) and is equipped with the bounded convergence topology. It is noted that C ( ( E ) (, E ) )is a subspace of L ( ( E ) (, E ) * ) . For each t E R the annihilation operator at is uniquely specified by the action on exponential vectors as follows: at45 = W 4 t l

t E E.

It is well known that at E C ( ( E ) (, E ) ) .The creation operator is by definition the adjoint a; E L ( ( E ) *(, E ) * ) We . see from (2.2) that the composition . . .a:,at, . . .atmis well defined and belongs to C ( ( E ) (, E ) * ) .

219

Let I , m 2 0 be integers and

Kl,m E

KP,m = sn+l

0

C(Eam,(E@')*). We define

(In 8 Kl,m),

(2.3)

where In : E@" + E@nis the identity and Sn+l : E@(n+')+ EQ(n+l)the symmetrizing operator. An integral kernel operator El,m(Kl,m) is defined by the action 9 = (fn) H ( g n ) given by Sn+l

It is known that

El,m(Kl,m)

=

(n

+ m ) ! Kzmfn+mr n!

n

2 0.

E L ( ( E ) ,( E ) * ) .

Remark 2.2. We have C(E@",(E@')*) E (E@('+"))*by the kernel theorem. For Kl,m € L(E@m, (E@')*) let I E ~ be , ~the corresponding element, i.e., (Kl,m,

q@' 8 Pm) = ( ~ l , r n t @ P1) ~ , >

t , q E E.

We then easily understand that (1.2) in Introduction is a descriptive expression for El,m(Kl,m).In many literatures the notation E l , r n ( ~ l , r nis) used for El,m(Kl,m),see e.g., [17].

For a white noise operator symbol are defined by

-

A

%d

= ((+,

4,))

E E C ( ( E ) (, E ) * )the symbol and the Wick

>

E(t977) = W

E , 9,))

e-(t?

t,v E E ,

respectively. A white noise operator is uniquely specified by the symbol or by the Wick symbol. For an integral kernel operator we have El,m(Kl,rnY(t,q)= ( K l , m P m , ~

~ ~ ) e ( E * q ) ,

E',m(Kl,mY(t, 7)) = (Kl,mtBm, 7"). 2.4. Gaussian Realization

Based on the real Gelfand triple (2.1) we define a Gaussian measure p by its characteristic function:

The celebrated Wiener-It6 decomposition theorem says that L2(E;, p ) is unitarily isomorphic to r ( H ) through the correspondence:

220

Taking (2.2) into account, we regard (E) as a subspace of L 2 ( E * , p ) .In this sense an element of ( E )is called a test white noise function and, accordingly, an element of (E)*is called a generalized white noise function.

3. Admissible White Noise Operators 3.1. Admissible White Noise Functions For p E R we set o(I

ni 4 1;

=

C n!ezpnifni%

4 = (fn)E

w).

n=O

For p 2 0 we define 4, = {d = (fn) E r ( H ); 1 1 d 1, < co} and 4-* to be Then {G, ; p E R} form a the completion of r ( H ) with respect to 1 1 . I-,. chain of Hilbert spaces satisfying

6 = projlimGp c 4, c 4 0 = r ( H ) c 4-, c G* = indlimG-,. P-+W

,--too

Note that 4 is a countable Hilbert space but not necessarily a nuclear space (4 is nuclear if and only if H is finite dimensional), and that 9 and B* are mutually dual spaces.

Lemma 3.1. For any pairp, q satisfying 0

1 1 d Illp L I1 d llq

and

p

5 -qlog 11 A-' llop we have

*

I1 lLq 5 Ill 9 II-,

where 4 E r ( H ) and @ E ( E ) * .(The norms can be in a usual way.)

00,

9

which is understood

The proof is immediate from the definition of the norms. Then, we have

(E)c B c r(H) c G* c p)*.

(3.1)

An element in 4 (resp. G*)is called an admissible test (resp. generalized) function. The canonical C-bilinear form on G* x 4 is denoted by ((., .)) too. 3.2. Admissible White Noise Operators

We note from the inclusion relations (3.1) that C(G,P*) is regarded as a subspace of L ( ( E ) ,( E ) * ) .A white noise operator belonging to the former space is called admissible. For an admissible operator we can find a pair of real numbers p 2 q such that Z E L(GP,Gq).

22 1

Proposition 3.1. Let Kl,m E ,C(Emm,( E @ ' ) * ) .Then the integral kernel operator El,m(Kl,m) is admissible if and only if K1,m E ,C(H@'", H@'). In that case, for an arbitrary q E R and r > 0 we have

Ill %m(Kl,mM Il, 5 C II K1,m llop Ill 9 Illq+r

>

where

-and 11

Kl,m

llop

stands f o r the Hilbert space operator norm. In particular,

=l,m(Kl,m) E L(Gq+r, Gq).

Proof. If Zl,m(Kl,m) is admissible, there exist a pair of real numbers p 2 q and a constant C 2 0 such that

1 1 %,m(Kl,m)$Il, 5 C Ill 4 Illp > 4 E Gp* Taking a particular q5 = (0,.. . ,0, f m , 0,.. . ), one obtains easily

which shows that K1,m E L ( H B m ,H@'). Conversely, suppose K1,m E L(H@", H@'). For q E R and q5 = ( f n ) E G we have by definition

Note from (2.3) that bounded by

11 KZm [lop 5 11 Kl,m Ilop. Then for any r > 0, (3.2) is 00

n=O

By an elementary calculus (see e.g., 117 Section 4.11) we have sup n2o

+

~

+

( n I ) ! ( n m)! -2rn n! n! e

er/2

'+m

5 e'llmm (7) .

Combining (3.3) and (3.4), we obtain the desired estimate.

(3.4) 0

222

3.3. Admissible White Noise Opemtors with Supporter

Let U C T be a Borel set with v ( U ) > 0. Then, starting from the Hilbert space L2(U,v)we obtain the spaces of admissible functions which are denoted by

G ( U ) c G p ( U ) c I V 2 ( U , v ) )= GO(U) c G-,(V)

c G*(U).

We identify G p ( U )with a closed subspace of Gp = Gp(T)through the natural inclusion L2(U,v ) L) L2(T,v). An element in Gp(V)is called an admissible white noise function supported by U. A description of the inclusion Gp(U)c)Gp(T)is given in terms of tensor product decomposition. We first recall the following fact whose proof is standard and is omitted.

Lemma 3.2. Let T = UIU U, U U Urn be a partition into a disjoint union of Borel subsets u p t o null sets. Then, the correspondence

4t

tEL2(W,

++4EtUl @ * * * . 4 E t U m ,

gives rise t o a unitary isomorphism

G,(T) for

Gp(V1)

€3

*

- - @4 Gp(Um)

all p E R.

Now let T = U U V be a partition, where v ( U ) > 0 and v(V) > 0 without loss of generality. It follows from Lemma 3.2 that

G P W (3.5) and 9 I-+ 9 €3 4 0 tv gives the canonical inclusion Gp(U)L) Gp( T ). With each continuous operator E E L(Gp(U),Gq(U)), where we assume GAT)

G P ( W @4

2 q without loss of generality, we associate an admissible white noise operator E €3 I according to the factorization (3.5), where I is the identity operator on Gp(V).Summing up, for a Borel set U C T and a pair of real numbers p 2 q we have inclusions:

p

L(GP(U),G,(W

c L(GP(T),G,(T)) c L ( ( E ) (El*). ,

An operator in L(Gp(U),G,(U))is called an admissible white noise operator supported by U. Whenever no confusion occurs we use the same symbol E for E @4 I. The concept of an admissible white noise operator with support is useful in the study of conditional expectation and quantum martingale, see [7].

223

4. Quantum White Noise Derivatives 4.1. l h n s l a t i o n Opemtor

Since each 4 E ( E ) is a continuous function on E;L, for any translation Ted defined naturally by

” E E;t.

TC4(”) = 4(” + C),

C E E;L the (4.1)

It is known [17]that Tcq5 E ( E ) and Tc E ,C((E),(E)). However, (4.1) is not applicable to a generalized function. By the Wiener-Its-Segal isomorphism, for 4 = ( f n ) E ( E ) we have

where Gm is the right m-contraction of symmetric tensor products. It is then natural to define the translation operator by extending the right hand side of (4.2). Namely, given C E E* (hereafter we allow a complex C) and 9 = (F,) E ( E ) * ,we define Tc@by the right hand side of (4.2) with replacing fn+m by Fn+,, whenever well defined as an element in ( E ) * .

Proposition 4.1. Let C E H and 9 E Gp with s o m e p E R. Then,for any it holds that Tc9 E Gq and Tc E ,C(Gp, Gq). q < p - log Proof. Let p , q E R. By definition,

Applying the Schwartz inequality, we have

00

0

0

.

224

Given p E R, we choose q E R such that 2e-2(P-q) Then we come to

which means that Tc9 E

< 1, i.e., q < p-log&

8, and Tc E L(Gp,Gq).

4.2. Gross Derivative

+

Modelled after abstract Wiener space theory, we say that E ( E ) * is Gross differentiable if for any C E H the translation Tee@is defined for small I E ~ < €0 and if

converges in (E)*with respect to the weak topology. Dc9 is called the Gross derivative of 9 in the direction C. Proposition 4.2. Every 9 = (F,) E G* is Gross differentiableand Dg9 = ( ( n+ l)C&F,+l)F=o. Moreover, Dc is a continuous linear operator on G* equipped with the strong dual topology.

+

Proof. Let C E H , 9 = (8’‘) E Gp and set \E = ( ( T I l)CGIFn+l):=o. We first note that \E E Bq for any q < p. In fact, by direct computation we obtain

Next we show that

P = Dc9.It follows from (4.2) that

Applying a similar estimate as in the proof of Proposition 4.1, we obtain

where q < p - log&. Thus we have shown that (4.3) converges in norm and the desired assertion follows. The last assertion follows from (4.4) and 0 general theory of locally convex spaces.

A Gross differentiable function 9 E ( E ) * is called pointwisely Gross differentiable if there exists a weakly measurable function t I-$ \Et E ( E ) *

225

such that the function t and

( m a ,9)) =

/

T

I+

11 9 t [ I p

belongs to H = L2(T,v) for some p E R

c ( t ) W t , 9))

c E H,

4%

9 E (E).

(4.5)

In that case we write 9 t = D t 9 . Note that D t 9 is determined for almost all t E T . The pointwise Gross derivative plays a basic role in stochastic analysis and similar derivatives have been introduced by many authors in different contexts, see e.g., [6,12,15,16]. Proposition 4.3. [l: Lemma 3.101 Every 9 E S* is pointwisely Gross differentiable and D t 9 = ( ( n+ l)Fn+l(t, for 9 = (F,). Moreover, if 9 = (F,) E S p and q < p - log then D t 9 E S, for v-a.e. t E T.

a,

Proof. Our proof is different from the one in [l]. Consider a function t I+ 9 t = ( ( n l)Fn+l(t, which is defined for almost all t E T by Fubini theorem. We note that

.))rz0,

+

S,11

qt

1;

v(dt)=

S,2

n!e2,”(n

+ 1)21Fn+l(t,*)I:

v(dt)

n=O

00

=

C(n+ l)e-2Pe-2(P-,)n(n + ~ ) ! e ~ p ( ~I F,+~I; +l) n=O

5 e-2pCp”-p1 1 9 Il l2p < 00. Then 9t E S, for almost all t E T. Since II *t

(4.6) IIqA0

I II q t I I,Ao I II *t II,

by Lemma 3.1, we see from (4.6) that the function t I+ 11 9t ll,Ao belongs to H = L 2 ( T , v ) . Finally, (4.5) follows from Proposition 4.2 with direct computation. Thus D t 9 = 9ffor almost all t E T. 0

Corollary 4.1. For 9 E ( E ) we have Dtq5 = atqi It is shown by norm estimates that

)

n

90 9=

(C k=O

Fk&,,-k

p* is closed under the Wick product:

03 7

@=

n=O

(Fn),

* = (Gn).

Then the next result is straightforward. Proposition 4.4. For 9, 9 E

G* we have

D c ( 9 0 9)= ( D c 9 ) o 9

+9

D t ( 9 o *) = ( D t 9 )o 9

+ 9 o (DtQ),

0

(Dcq),

c E H, for almost all t E

T.

226

4.3. Annihilation- and Creation-Derivatives

Let Z E L ( ( E ) ,( E ) * ) It . is proved that for any Q E E there exists 9, E (E)* uniquely specified by

40)

((*?I,=

((5*4q, 4d) ((4-7, 4d)

.t E E.

9

By using the Wick product we may write

9, = (E*+,)o&,. Now assume that 9, is Gross differentiable for all Q E E. This assumption is equivalent to that so is 3*&, since 9, o 4, = E*&. If, in addition, ( ( D c ~ ,$0) , is the Wick symbol of some operator in C ( ( E ) (, E ) * )for any C E H , denoted by DYE, i.e.,

(((D;Z)4b 4,))e-(c9q) = ((DC+,, 4th & Q E E, (4.7) then E is said to be differentiable in annihilation parts and DFZ is called the annihilation-derivative of Z with C E H . Similarly, the creation-derivative D l Z E C ( ( E ) (, E ) * )is defined by

WpMC, 4v))e-(Et9)= ((DC% 4,)L

t,QE E ,

where

*c = ( q t )

04-C.

We say that E E C ( ( E ) ,( E ) " )is qwn-differentiable if DFZ E L ( ( E ) ,( E ) * ) exists for all C E H . The derivatives DFE are regarded as non-commutative extension of the Gross derivative. Let us study the qwn-derivatives of an integral kernel operator. We need notation. For Kl+1,, E C(HGm,H6(l+l))and E H we define C*Ki+l,,,, E C(HGm,H6') by

{ (C * K l + l , m ) E @ m ,Pi}= (Kl+l,mPrn,Pi€3 c} > E , rl, c E H. Similarly, for C ( H G m ,H6") by

E L(H&("+l), H G r )and ( E H we define Kl,m+l* I E

((Kl,m+l*

C)E@'",

7 1 y = {Kl,m+1Prn €3

c, Q@).

Theorem 4.1. A n admissible integral kernel operator is qwn-differentiable. Moreover, for any Kl,m E C ( H G m ,H c z ) and C E H we have

D,Si,m(Ki,m) = mZ,m-i(Ki,m DfSl,,(Kl,,)

= G-l,m(C

*%?a).

* C),

(4.8) (4.9)

227

Proof. For simplicity we set Z = Zl,m(Kl,m).It follows from (4.7) and Proposition 4.4 that D;Z E L ( ( E ) ,( E ) * )is characterized by

(((qws,

47))e-(c97)

= (((Q=*47>0 4-77 4s))

+ ((=*4, 0 ( W - o ) , 4s))

*

(4.10)

The right hand side being equal to

(4.10) is equivalent to

As is verified by direct computation, DcZ*4,, = (h,) is given by

where n = 0,1,2,. .. (the second term vanishes for n = 0). Then the first term of the right hand side of (4.11) becomes

Therefore (4.11) becomes

from which we see that Z admits the annihilation derivative and (4.8) holds. A similar argument can be applied to (4.9). 0

228

4.4. Fock Expansion of an Admissible Opemtor

We assemble some general results on an admissible white noise operator to discuss its qwn-differentiability in the next subsection. As a special case of [9] we obtain

Lemma 4.1. Let p , q E R. For each L[,m E C ( H S m , H g 1 )there exists a unique operator Il,m(Ll,m) E L ( q ,Gq) such that

Il,m(Ll,mjiJ,7 ) = (Ll,mPm,

In this case, II Il,m(Ll,m)]lop I

rn 11 L1,m

5 , E~E -

9

110p-

Theorem 4.2. Let p , q E R. For any E E L($,,Gq) there ezists a unique family of operators Ll,m E C ( H G m , H g ' I ) ,Z,m 2 0 , such that 00

5=

16,m(LI,m),

(4.12)

l,m=O

where the series converges weakly in the sense that m

( ( ~ 4q)) , =

C

((1l,m(~l,rn)4, +))

3

4E~

p ,

1c1 E 9-q.

l,m=O

The expression (4.12) is called the chaotic expansion of 5. In fact, Ll,,,, is obtained by the formula: 1

Ll,m = -I?EIm, l!m!

(4.13)

where I, E L ( H Q m ,4)defined by ImFm = (0,. . . ,0,F,, 0,. . .). On the other hand, the Fock expansion of Il,m(Ll,m)is easily computed:

Inserting (4.14) into the chaotic expansion (4.12), we obtain the Fock expansion: 1Am

n!

l,m=O

(4.15)

Theorem 4.3. Let Z be an admissible white noise operator and let

c M

5=

l,m=O

Zl,m(Kl,rn)

(4.16)

229

be the Fock expansion. Then for all 1, m 2 0 we have Kl,mE L(HNm,l?@‘). Moreover, if E E L(Gp,Gq) for some p,q E R, then (4.16) converges in L ( G q - s + r , Gq-s) for any r > 0 and s > 0 satisfying (4.17) Proof. Given Z E L(G,G*), we define Ll,m as in Theorem 4.2. Comparing (4.15) and (4.16), we obtain

(4.18) from which the first assertion is obvious. We shall prove the convergence. Suppose that E E L(Gp,Gq) with some p , q E R and denote by 1 1 E 1 1 the operator norm. It follows easily from (4.13) that epm-d 1, m 2 0. II Llm llop 5 Ill 2 Ill 7

m

Then (4.18) becomes

Applying the Schwartz inequality, we see that the last quantity is bounded by

Thus, (4.19) becomes (4.20) Now let r , s > 0. Applying Proposition 3.1 and a simple inequality nn enn!, we obtain

5

Ill %n(Kl,m)+ lllq-s Hence for any r, s > 0 satisfying (4.17), the Fock expansion (4.16) converges 0 in L ( G q - s + r , G q - s ) .

230

4.5. Q WN-Derivatives of an Admissible Operator

Theorem 4.4. Eve y admissible white noise operator is qwn-differentiable. More precisely, i f the Fock e q a n s i o n of Z E L(B,G*) is given as in (4.16), then for any E H we have

<

l,m=O

l,m=O

where the right hand sides converges in the same manner as mentioned in Theorem 4.3. Moreover, DF is a continuous linear operator o n L(B,G*). Proof. Each El,m(Kl,m)is admissible by Theorem 4.3 and hence qwndifferentiable by Theorem 4.1. Then, it is sufficient to show the onvergence of the right hand sides of D t E . Suppose that Z E L(Bp,Gq). Applying Proposition 3.1 and (4.20),we obtain easily

1 1 m S , m - l ( K , m * C M lllq-s

1 1 11 I C 10 1 1 4

'

I!q-S+T

This estimate is almost the same as (4.21) and the series 00

C

1 1 mS,m-l

(Kl,m

* 04 I I I ~ - ~

l,m=O

converges whenever (4.17) is satisfied. In this case we have

IH (DT'I4

1IIq-S

'

I C IH HII C IOHI 4 !lq-S+T

7

with some constant C = C(p,q, r , s), which proves that DC is a continuous linear operator on L(B,G*). The argument for D;' is similar. 4.6. Pointwise Q WN-Derivatives

A qwn-differentiable operator E

E L ( ( E ) (E)*) , is called pointwisely qwndifferentiable if there exists a measurable map t I+ D:E E L ( ( E ) (, E ) * ) such that

((P;%,4 7 ) ) =

/T

( ( ( D ? W b hJ)C(t)v(dt),

c E H , 6 77 E E.

The following examples support the intuitive idea (1.3) in Introduction.

231

Example 4.1. For f E E* define KOJE L ( E , C ) and K1,o E L ( C , E * )by

K0,l :

< * (f,<) ,

K1,O : c

* cf,

respectively. The integral kernel operators

axe respectively called annihilation and creation operators associated with f . If f E H , both A f and A; are pointwisely qwn-differentiable and

D r A f = f ( t ) l , D t A f = 0;

DFAj = 0 , D:Aj = f ( t ) I .

Example 4.2. The number operator and the Gross Laplacian axe defined by

N = %,1(1)=

/

a f a t v(dt), AG = Z0,2(1)=

T

/

a:

v(dt),

T

respectively. Then we have

DFN = a;,

D:N = at,

DFAG = 2at,

D,+AG= 0.

References 1. K. Aase, B. Bksendal, N. Privault and J. Ub0e: White noise generalizations of the Clark-Haussmann-Ocone theorem with application to mathematical finance, Finance Stochast. 4 (2000), 465496. 2. V. P. Belavkm: A quantum nonadapted It0 formula and stochastic analysis in Fock scale, J. Funct. Anal. 102 (1991), 414-447. 3. F. E. Benth and J. PotthoE O n the martingale property for generalized stochastic processes, Stoch. Stoch. Rep. 58 (1996), 349-367. 4. D. M. Chung, U. C. Ji and N. Obata: Quantum stochastic analysis via white noise operators in weighted Fock space, Rev. Math. Phys. 14 (2002), 241-272. 5. M. Grothaus, Yu. G. Kondratiev and L. Streit: Complex Gaussian analysis and the Bargmann-Segal space, Methods of Funct. Anal. and Topology 3 (1997), 46-64. 6. T. Hida: “Analysis of Brownian Functionals,” Carleton Math. Lect. Notes, no. 13, Carleton University, Ottawa, 1975. 7. U. C. Ji: Stochastic integral representation theorem for quantum semimartingales, J. Func. Anal. 201 (2003), 1-29. 8. U. C. Ji and N. Obata: Quantum white noise calculus, in “Non-Commutativity, Infinite-Dimensionality and Probability at the Crossroads (N. Obata, T. Matsui and A. Hora, Eds.),” pp. 143-191, World Scientific, 2002. 9. U. C. Ji and N. Obata: A role of Bargmann-Segal spaces in characterization and expansion of operators o n Fock space, J. Math. SOC.Japan 56 (2004), in press.

232 10. U. C. Ji and N. Obata: Annihilation-derivative, creation-derivative and representation of quantum martingales, preprint, 2003. 11. I. Kubo and S. Taken&: Calculus on Gaussian white noise I, Proc. Japan Acad. 56A (1980), 376-380. 12. H.-H. Kuo: “White Noise Distribution Theory,” CRC Press, 1996. 13. J. M. Lindsay and H. Maassen: An integral kernel approach to noise, in “Quantum Probability and Applications I11 (L. Accardi and W. von Waldenfels Eds.).” Lecture Notes in Math. Vol. 1303, pp. 192-208, Springer-Verlag, 1988. 14. J. M. Lindsay and K. R. Parthasarathy: Cohomology of power sets with applications in quantum probability, Commun. Math. Phys. 124 (1989), 337-364. 15. P. Malliavin: “Stochastic Analysis,” Springer-Verlag, 1997. 16. D. Nualart: “The Malliavin Calculus and Related Topics,” Springer-Verlag, New York, 1995. 17. N. Obata: “White Noise Calculus and Fock Space,” Lect. Notes in Math. Vol. 1577, Springer-Verlag, 1994.

Infinite Dimensional Harmonic Analysis I11 (pp. 233-247) Eds. H. Heyer et al. @ 2005 World Scientific Publishing Co.

PDE APPROACH TO INVARIANT AND GIBBS MEASURES WITH APPLICATIONS

MICHAEL ROCKNER Fakultat fur Mathematik Universitat Bielefeld 0-3361 5 Bielefeld (Germany) Email: [email protected]. de

In this paper we give a pedagogical account of the PDE approach to invariant and Gibbs measures in finite and infinite dimensions. As an application we describe some recent new results on the classical problem whether “invariance implies Gibbsian” and illustrate how they apply to a wellstudied lattice model from statistical mechanics with non-compact single spin spaces.

AMS 1991 Subject classification: Primary: 58G32,58B99, 82B21. Secondary: 53C21, 53C80,58G03,60560,82B05. Keywords and phrases: invariant measures, diffusions on manifolds, elliptic equations for measures, Lyapunov functions, Gibbs distributions, logarithmic gradients To appear in conference proceedings.

1. Introduction This paper is an extended version of my talk given at the symposium in Tiibingen. The main purpose is to give a concise pedagogical account on the PDE (= partial differential equation) approach to invariant and Gibbs measures, developed in a series of papers during the last few years by S. Albeverio, V.I. Bogachev, Y.G. Kondratiev, T. Pasurek, F.Y. Wang and the author (cf. e.g. AKR97a AKR976, AKRTOO BRWOl BROl AKPROI), and de-

,

,

7

7

scribe a recent new application to the classical problem whether any invariant measure is Gibbsian (cf. BRWo2). The latter will be done in more detail, but the main result will be only precisely formulated, without recalling the proof from BRWo2. Instead, we shall present an application to an intensively studied model from statistical mechanics (see e.g. B H K 8 z ) .

234

2. The PDE-approach to invariant and Gibbs measures 2.1. The finite dimensional case

Consider the d-dimensional Euclidean space Rd (or more general a ddimensional manifold M d ) and a Bore1 measurable function E : Rd + (-m, +00] such that e-E E L1(Rd,dx) where dx denotes Lebesgue measure. For a better understanding of the following one should think of E ( z ) as the "energy" of the "configuration7' x = (xi)l
(2) p satisfies the following first order PDE:

aip = -&E (where

ai :=

(3) Setting

*

p

V 15 i 5 d

& and & p denotes the distributional derivative of p).

:= ( & ) l < i < d ,

&

d

:= &E and

LZ := c(8: - &&), then p i=l

symmetrizes LZ with domain C,"(Rd), i.e.

s

LZUv dp =

s

u LZVdp V U , v

E

C,"(Rd).

(4) p is Lz-infinitesimally invariant, i.e.

/

LZU dp = 0 V u E C,"(Rd),

or shortly,

L>p = 0 (so p satisfies a second order PDE).

Then (1) H (2) @ (3) + (4). Under our present smoothness assumptions, i.e. E E C 1 ( R d )the , im(3) are pretty much obvious. For (2) + (1) a plications (1) + (2) regularity result to ensure the existence of a sufficiently regular RadonNikodym derivative with respect to dx for any p satisfying (2) is necessary. For details on the latter in a much more general case, namely where merely e- 3 E E Hk:(IRd, dx) (= local Sobolev space of order 1 in L~o,(Rd, dx)) and e - i E > 0 dx-a.e. is assumed, we refer e.g. to Proposition 1.5 and its

235

proof in A R Z g 3 (see also Remark 2.1 below). For (3) =+ (2) we refer e.g. to Lemma 1 in BR03. The equivalence of (1) and (2) is of great importance since as we shall see in the next section it generalizes to infinite dimensions, so gives the possibility to study Gibbs measures by PDE-methods. Concerning the relation between (3) and (4) by choosing v, E Ci(Rd) with v, = 1 on a ball of radius n and letting n 4 00 we obviously deduce (4). So, we have “Gibbsian =+ infinitesimally invariance”. The converse in infinite dimension is a famous conjecture of Gibbs (originally formulated for so-called Hamiltonian dynamics). Under our present regularity assumption on E , i.e. E E C’(Rd),in this finite dimensional case the converse is also true. This follows from two highly non-trivial general results from Corollary 2.3 (see in particular also Remark 2.4.(i)) in Sta99 and Theorem 3.1 in BRSoo (see also Theorem 4.1 in B R S 0 2 ) which imply that the PDE in (4) has a unique solution, which since p in (1) is a solution, is therefore Gibbsian. If one relaxes the assumptions on E the situation becomes much more complicated. We summarize this in the following remark.

Remark 2.1. For E : Rd -+ mentioned condition

(-00,00]

e - i E E Hk:(Rd,dx)

and

let us only assume the already

e - i E > 0 dx-a.e.

(2.1)

Setting p := e-E (E H k t ( R d ,dx)) we can reformulate (2) as follows:

(2)

p satisfies the following first order PDE

for some (Borel) dx-version

(9)of 9.

9

Here we set as usual := 0 on { p = 0). In this case as mentioned above always p << dz, so the superfix “J’ can be dropped a posteriori. Note that in general q! L:,,(Rd,dx), so even if E E L:,,(Rd,dx), in general # & E , since the right hand side is always a Schwartz distribution. Then we still have

9

%

(1) =+ (2)

@

(3) =+ (4).

(9)and (3), (a) are just conditions (3), (4) respectively, augmented by the condition (9)-L t c ( R d , p ) 1, 5 i 5 d. Since Here we take

Zi :=

E

236

by Theorem 1 in BKR97 any solution of L*p = 0 is absolutely continuous with respect to dx, we can again drop a posteriori. Under condition (2.1) the PDE in (2) can have infinitely many solutions (cf. Example 6.1 in BR95 and also Remark 3.6(ii) in AKR97b) even for d = 1. So, (2) i+ (1). If, however, in addition, to (2.1) also E L:,,(Rd,dx), then p from (1) is the unique solution of the PDE in (27 (cf. Theorem 6.2 in BR95 and Theorem l.S(ii) in A B R 9 9 ) . In addition, = &E, 15 i 5 d , “N”

9

(cf. Lemma 6.4 in B R 9 5 ) . On the other hand, even if E L:,,(Rd,dx), it appears to be unknown whether + (3). But also no counterexample seems to be known. For conditions so that (4) + (3) in the present situaton we refer to Theorem l.S(i) in A B R g g for the most general result we are aware of that does not require higher integrability of If one

(a)

y.

assumes that

E Lt:E(Rd,dx),e

> 0 , then by the same results from

(a)

mentioned above the PDE in has a unique solution (among probability measures), so + ( l ) , hence (1) @ (2) H (3) @ in this case. Stagg, BRSoo, BRSo2

(a)

(a)

If we replace Rd by a suitable manifold, then there are easy examples even for smooth 2 where (4) + (3).

Example 2.1. (cf. Remark 2.5(ii) BRWo2) Let M be a connected complete Riemannian manifold with infinite volume measure A M , such that there exists a positive harmonic function h, integrable with respect to the volume measure AM (cf. GhzL83,Ls84 for existence). Choose the vector field 2 to be identically equal to zero, so LZ = A (= Laplacian on M ) . Define

Then for u E C i ( M )

237

since A h = 0. But for u , v E C i ( M )

s

L Z U Vd p =

/

u Av h dXM

+

/

uv A h d X M

+2

s

u ( V v ,V h ) d X M

= / u L ~ v d p + 2 / u ( V v , V h )d X M

since h cannot be constant because X M ( M )= 03. Therefore, p is infinitesimally invariant for L z , but not symmetrizing, hence not Gibbsian. In infinite dimensions much less is known about when or not (4) implies (3). We have included the quite detailed discussion in Remark 2.1 since it displays a typical characteristics in comparing finite and infinite dimensional analysis. Difficulties in the analysis of PDE or differential operators in infinite dimensions are reflected in part in finite dimensions if the coefficients become singular. The last remark of this subsection concerns the relation between “infinitesimal invariance” and “invariance”. We refer to Subsect. 2.5 in BRSoo and Sections 3 and 4 in BRSo2 for more details.

(a)

Remark 2.2. Assume again that (2.1) holds and that p is as in (cf. Remark 2.1). Set L := L z . Suppose there exists a closed extension ( ,b ’ , D ( i p ) ) of the operator ( L , C i ( R W don ) ) L 1 ( R d , p )which generates a Co-semigroup (T:)t>o = (et e w )t20 on L1(Rd,p).If p is the special measure in ( l ) ,then it follows by Corollary 2.3 and Remark 2.4.(ii) in that ( i p , D ( i p ) )must be the closure of ( L ,Ci(Wd)) and this closure really generates a Co-semigroup on L1(Rd,p ) . In this case a simple consideration implies that

T f p=

Jf

dp

for all t > 0, f E L 1 ( R d , p ) ,

(a),

(2.2)

For a general p satisfying however, the i.e. p is (T:)t>o-invariant. mere existence of (T,”)t?o= ( e e w ) t l o for a suitable closed extension 2 . of L is unknown and, if it exists, it might not be the closure of L. So, it is unclear whether p will be its invariant measure in the sense of (2.2). On the other hand, if (T,”)t>oexists and satisfies (2.2), by differentiating at t = 0, we deduce from (2.2) that p satisfies L*p = 0. So, even in finite

238

dimensions “infinitesimally invariance” seems to be a more general notion than “invariance”. 2.2. The infinite dimensional case

Consider now the d-dimensional lattice RZm,m E N,instead of Rd. (Again we could also consider a product Mi of finite dimensional Rieman-

n

iEZd

nian manifolds, cf. BRW02 for details). We are going to restrict the class of “energy”-functionals E : R”” -+ (--oo,oo] a bit, with applications to statistical mechanics in mind. So, let ~~

~ ( x:= )

C

uA(xA),

x

= (xi)icWZm

(2.3)

ACZ”

I4<m

where /A1 denotes the cardinality of A, X A := ( x i ) i E A , and UA : RA + (-00, 001 is Borel-measurable. Of course, (2.3) is purely informal, because this sum almost never converges. U A ( X is ) called the “potential of the configuration x in A” and as before E ( x ) is the LLenergy of the configuration z = ( x i ) i E p m ” . Similarly as in finite dimensions one then defines corresponding Gibbs measures p on R”” (equipped with the product of the Bore1 a-algebras on R) as

(1)’ p is a Gibbs measure (with energy E ) , i.e.

with

dxi

:= Lebesgue measure on

R1.

Of course, also (1)’is purely informal, since an infinite product of Lebesgue measures does not exist in a suitable sense and, as said before, E ( x ) is not well-defined. But it turns out (and has been well-known for many years) that the expression in the right hand side of the equality in (1)’ can be given sense. In general, however, the correspondingly defined measure is not unique. One reason is e.g. that if one defines the right hand side as a limit along a “localizing sequence” A, /” Z”, n + 00, with A, finite, the limit might depend on the sequence. Furthermore, “localizing” implies that one has to fix ‘Lboundaryconditions” outside every A,, and different choices of these might also lead t o different limits. A precise definition of a Gibbs measure taking all these issues into account requires the notion of a “local specification” and then a Gibbs measure can be defined by

239

determining its conditional probabilities on R" for each finite A C Z" by this specification fixing the configuration outside A, i.e. the measure satisfies the DLR (= Dobrushin-Lanford-Ruelle) equations. We refer e.g. to the exposition in Geo88 for details, since we do not need this below, since this rigorous version of (1)'is as in finite dimensions equivalent to the following, infinite dimensional analogue (2)' of (2) which is rigorous. This equivalence has been proved in in various frameworks: AKR97a,AKR97b1AKRT00

(2)' p satisfies the following first order PDE (in infinitely many variables):

Precise assumptions for "DLR-version of (1)'

(2)"' to hold are e.g.

c Z" and for some R > 0 and diamh > R

for all finite A

UA = 0 if

("finite range interaction"). To define the distributional derivative 13ip in (2)' we need a test function space. As usual in infinite dimensions we take for e E N U {co}

FC; := { u : RZn

I

R 3 finite A c Z" and g

E C;(RA)

such that ~ ( I c = ) g ( X A ) for all

IC E

(2.6)

ItZm}.

Then a probability measure p on R"" satisfies the PDE in (2)' if

where

za :=

&uA,

iEZm.

(2.8)

A:iEA ACZm, A finite

Note that by the finite range condition the sum in (2.8) has only finitely many non-zero summands, so Zi is well-definied. As in the preceding subsection we now consider two further assertions about a probability measure p on R"":

240

(3)’ Setting Z := (Zi)iEzrn,Zi as in (2.8)’ and L z := then p symmetrizes LZ with domain 3C;, i.e.

cz“=,Ca? - Zit$),

(4)’ p is Lz-infinitesimally invariant, i.e.

/Lzudp=O

VUE~C?,

or shortly,

L>p = 0 (so p satisfies a second order PDE in infinitely many variables). We emphasize that since u (and v) are in 3Cz the sum in the definition of LZU has again only finitely many non-zero summands. So, all is welldefined. In A K R g 7 a , AKR97b, AKRToo also the equivalence (2)’ ($ (3)’ has been proved, and obviously (3)’ + (4)’by taking v 3 0. So, altogether as in finite dimensions we have: DLR-version of (1)’($ (2)’ @ (3)’ +-(4)’. However, as mentioned in the previous subsection the implication “(4)’+ (3)”’ even under stronger smoothness assumptions on the UA is a major problem in this infinite dimensional case. In the next section we shall present a result giving a sufficient condition for this to hold.

Remark 2.3. (i) Let p satisfy (2)’ (H (3)’ @ DLR-version of (1)’). In this infinite dimensional case ( as in finite dimensions, cf. Remark 2.2) again even the mere existence of (T[)t20= (ett”)tgo as a Co-semigroup on L1(RZrn,p)for a suitable closed extension LP of L is not clear in general and only known under quite stringent assumptions. However, if ( T [ )-t >=~ (etep)t>o - exists and the analogue of (2.2) holds for p, then as in finite dimensions always L*p = 0 by taking $ It=o. So, “invariance” implies “infinitesimally invariance”, but the converse is unlike in finite dimensions in fact known to be wrong in general (cf. Chap. 5b in Ebe99 for counter examples). Concerning the question whether (4)’ + (3)’’ our result in the next section is therefore more general than just stating it for invariant measures, since we prove it for a larger class. In particular, it generalizes the classical well-known

241

results in e.g. H S 8 1 , Fri82 since it holds also for infinite products of manifolds (cf. B R W o 2 ) . (ii) To be precise we mention that in (2)', (3)', (4)' above one has to assume, in addition, each time that Z i E L2(Rzm,p) V i E Z", in order to have that (Lz,FC;) is an operator on L2(Rz",p) and that the Gibbs measure in (1)'defined through the DLR-quations is ternpered in a suitable sense. We suppressed this point above since in applications, the square integrability of Zi is automatic (cf. Section 4 below on applications). 3. Infinitesimally invariance implies Gibbsian

Consider the situation described in Subsection 2.2, so UA, Z = ( Z i ) i E z m , L z are as defined there. Assumptions on the potentials UA: Let UA, A c Z", IAl < 00, satisfy assumption (2.5) for some (fixed) R > 0, and in addition:

where

Remark 3.1. We note that obviously for Ic E N Ek(z)

=Ek(xAk),

E IWZm 9

and for i E Z"

SO,

zi(x) = z i ( x i + A l ) ,

x E Rzm,

zi(x)= Z i ( x A k + l ) ,

2 E

if i E h k ,

Rzm.

Now we can formulate one of the main results from

BRW02.

Theorem 3.1. Let p be a probability measure on R"" such that

242 (2)

zi E L2(R"",p)

(ii) L > p (iii) & E k Set f o r k

v i E Z", i.e. p is Lz-infinitesimally invariant, E L2(R"", p ) 'd i E A k , k E N.

= 0,

E

N

-

where Ep[. I a ( A k ) ] denotes conditional expectation of p with respect t o the X A ~ x , E RZm. If there exist a-algebra u(hk) generated by the map x c k E [ ~ ; , c of l) (O,CO) such that (3.3)

then p is Lz-symmetrizing, i.e. Gibbsian. Instead of giving an account of the proof of Theorem 3.1 we refer to BRW02 and shall rather discuss an application in the next section. We only mention here that the crucial quantities LIE, Ic E N,in (3.2) exactly capture how strongly the PDE's in (3)' and (4)' are coupled with respect to the one dimensional coordinates of x = ( x i ) i E Z m or "how much" p differs from a product measure on R"". This will become, particularly, clear in the applications below. (3.3) just says that DE should not grow too fast with

k. 4. Application

In this section we shall apply the above, in particular, Theorem 3.1, t o a well-studied model from statistical mechanics (cf. e.g. B H K 8 2 and the references therein). This is a lattice system over Z" with a two-body interaction of finite range R > 0, i.e. in the frame from the previous section we have for finite A c Z":

UA = 0, If A

=

unless d i a m h 5 R and IAl 5 2.

{i} we set

v, := U,,}, and if A = { z , j } , i

# j , we set wi,j

:= U { i , j ) .

(4.1)

243

So, in particular, Wi,j = Wj,i. Assumptions on V,, Wi,j:

V, E C'(R),

(4.2)

W2,j E Cl(rW2).

There exist K , K o E (0, co),p E (m,co) such that K > 12Ko(1+R)2+p, and C E (0, m), a E [ 2 , co),such that for all i, j E Z m and all s, t E R (wi,j(s,t)J I Ko(1

+ ISIQ f ItI"),

+

J & W i , j ( S , t )5 JK

o(~ sv,'(s) 2 KISI"

-

+ Itla-'), c. (4.3)

It can be easily shown that (4.2), (4.3) imply conditions (2.5) and (3.1). Let us first calculate the corresponding Zi and &Ek. We have for i E Z",k E N,in this situation

C

z ~ ( Z )= - ~ l ( Z i )-

a1Wi,j(Zi7Zj)

(4.4)

j€Z"

li-jls R &Ek(Z) = -K'(Zi)-

& W i , j ( Z i , Zj),

if i E h k .

(4.5)

jE,Ak

li-3llR

By Examples 6.12 and 4.6 BRol there exist probability measures p on R"" such that the following properties hold:

(A) L>p

= 0.

(B) ("temperedness of p7')

Furthermore, for all such p and every such that

T

E

(0,co) there exists M , E (0,co)

in particular (by (4.2), (4.3))

zi, &Ek for i E

Z" and k

E

W

E L2(R"",p)

is such that a E Ak.

244

Now let us calculate the crucial quantities D:, k E W, from the conditions of Theorem 3.1. Fix k E N and i E A k , then since aiEk is a ( A k ) measurable we can use Jensen's inequality to obtain

J

'

jCZrn\Ak

li-jlgz

where we used (4.4), (4.5) in the last step. By ( 4 . 3 ) and Holder's inequality the latter is bounded by

5 3(2R + 1 ) 2 m ~( i1 + 2 ~ 2 , - 2 ) =: C ( R ,m,KO, a ) where we used (4.7) in the second step. So, for any such p as above and kEN

D: I C(R,m,Ko,Q) I h k \Ak-ll

=: C k .

Note that the latter is of order k m - l , so

where "M" means "equal up to a constant". So, Theorem 3.1 applies if and only if m 5 2 . Hence we have

Corollary 4.1. For the one- and two-dimensional lattices Z1 and Z2 for the above model we have: infinitesimal invariance implies Gibbsian. Concluding Remark 4.2. (i) Let ZLX denote the set of all probability measures on R"" satisfying conditions (A), (B) above. By similar techniques one can prove (cf. B R W 0 2 ) : if one p E ZLX satisfies the logarithmic Sobolev inequality, then #ZLX = 1 , i.e. we have uniqueness in this case. This extends results known for compact spin spaces (i.e. lattices of type Mi,

n

iEZm

245

compact manifolds, cf. SZ92a 1 SZ92c S.Z92b Sz95 Zeg92 7 Zeg96 ) with M~ for a particular case with to t h e non-compact case (see also Mi = R as above). (ii) There are examples on R”” where “infinitesimal invariance + Gibbsian” for all lattice dimensions m. We refer to BRW02 for details.

Acknowledgement

It is a pleasure for t h e author to t h a n k Professor H. Heyer for a very pleasa n t and stimulating conference in Tubingen and for t h e financial support through t h e DFG. Financial support of t h e BiBoS-Research Centre a n d t h e German Science Foundation (DFG) through t h e DFG-Research G r o u p “Spectral Analysis, Asymptotic Distributions and Stochastic Dynamics” is also gratefully acknowledged.

References ABR99. AKPR04.

AKR97a. AKR97b.

AKRTOO.

ARZ93.

BHK82.

BKR97.

BR95.

S. Albeverio, V. I. Bogachev, and M. Rockner, On uniqueness of invariant measures for finite and infinite dimensional diffusions, Comm. Pure Appl. Math. 52 (1999), 325-362. S. Albeverio, Y . G. Kondratiev, T. Pasurek, and M. Rockner, Euclidean Gibbs measures on loop lattices: existence and a priori estimates, Ann. Prob. 32 (2004), no. l A , 153-190. -, Ergodicity of L2-semigroups and extremality of Gibbs states, J. F’unct. Anal. 144 (1997), 394-423. S. Albeverio, Y . G. Kondratiev, and M. Rockner, Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states, J. Funct. Anal. 149 (1997), 415-469. S. Albeverio, Y . G. Kondratiev, M. Rockner, and T. V. Tsikalenko, A-priori estimates on symmetrizing measures and their applications to Gibbs states, J. F‘unct. Anal. 171 (2000), 366-400. S. Albeverio, M. Rockner, and T.-S. Zhang, Markov uniqueness for a class of infinite dimensional Dirichlet operators, Stochastic Processes and Optimal Control (H. J. Engelbert et al., eds.), Stochastic Monographs, vol. 7, Gordon & Breach, 1993, pp. 1-26, J. Bellissard and R. Hpregh-Krohn, Compactness and the mazimal Gibbs states for random Gibbs fields on a lattice, Comm. Math. Phys. 84 (1982), 297-327. V. I. Bogachev, N. V. Krylov, and M. Rockner, Elliptic regularity and essential self-adjointness of Dirichlet operators on Rd, Ann. Scuola Norm. Sup. Pisa C1. Sci., Serie IV XXIV (1997), no. 3, 451-461. V. I. Bogachev and M. Rockner, Regularity of invariant measures on finite and infinite dimensional spaces and applications, J. Funct. Anal. 133 (1995), 168-223.

246 BRO1. BR03. BRSOO.

BRSO2.

BRWOl.

BRWO2. Chu83. Ebe99.

Fri82. Geo88. HS81. LS84.

Sta99.

SZ92a.

SZ92b. sz92c. sz95. YOSO 1.

Zeg92.

-,

Elliptic equations for measures o n infinite dimensional spaces and applications, Prob. Th. Rel. Fields 120 (2001), 445-496. -, O n LP-uniqueness of symmetric diffusion operators o n Riemannian manifolds, Matem. Sbornik. 194(7) (2003), 969-978. V. I. Bogachev, M. Rockner, and W. Stannat, Uniqueness of invariant measures and essential m-dissipativity for diffusion operators o n L1, Infinite dimensional Stochastic Analysis (P. Clement et al., eds.), Royal Netherlands Academy of Arts and Sciences, Amsterdam, 2000, pp. 39-54. ___ , Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions, Matem. Sbornik. 193:7 (2002), 3-36, (Russian version), 945-976 (English version). V. I. Bogachev, M. Rockner, and F.-Y. Wang, Elliptic equations f o r invariant measures o n finite and infinite dimensional manifolds, J. Math. Pures Appl. 80 (2001), 177-221. ___ , Invariance implies Gibbsian: some new results, BiBoSPreprint 02-12-106 (2002), to appear in Comm. Math. Phys., 18 pp. L. 0. Chung, Existence of harmonic L1-functions in complete Riemannian manifolds, Proc. Amer. Math. SOC.88 (1983), 531-532. A. Eberle, Uniqueness and non-uniqueness of semigroups generated by singular diffusion operators, Lecture Notes in Math., vol. 1718, Springer, Berlin, 1999. J. Fritz, Stationary measures of stochastic gradient systems, infinite lattice models, Z. Wahr. verw. Geb. 59 (1982), 479-490. H.-0. Georgii, Gibbs measures and phase transitions, de Gruyter, 1988. R. Holley and D. W. Stroock, Diffusions o n a n infinite dimensional torus, J. Funct. Anal. 42 (1981), 29-63. P.Li and R. Schoen, Lp and mean value properties of subharmonic functions o n Riemannian manifolds, Acta Mathematica 153 (1984), 279-301. W. Stannat, (Nonsymmetric) Dirichlet operators o n L1: Existence, uniqueness and associated Markov processes, Ann, Scuola Norm. Sup. Pisa C1. Sci., Serie IV 28 (1999), 99-140. D. W. Stroock and B. Zegarlinski, T h e equivalence of the logarithmic Sobolev inequality and the Dobrushin-Shlosman mixing condition, Comm. Math. Phys. 144 (1992), no. 2, 303-323. -, T h e logarithmic Sobolev inequality for continuous spin systems on a lattice, J. Funct. Anal. 104 (1992), no. 2, 299-326. ~, The logarithmic Sobolev inequality for discrete spin systems o n a lattice, Comm. Math. Phys. 149 (1992), no. 1, 175-193. -, O n the ergodic properties of Glauber dynamics, J. Stat. Phys. 81 (1995), no. 5-6, 1007-1019. N. Yoshida, T h e equivalence of the log-Sobolev inequality and a maxing contition for unbounded spin systems o n the lattice, Ann. Inst. H. Poincare Probab. Statist. 37 (2001), no. 2, 223-243. B. Zegarlinski, Dobrushin uniqueness theorem and logarithmic Sobolev

247 inequalities, J. Funct. Anal. 105 (1992), no. 1, 77-111.

Zeg96.

-, T h e strong decay t o equilibrium for the stochastic dynamics of unbounded spin systems o n a lattice, Comrn. Math. Phys. 175 (1996), no. 2, 401-432.

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Infinite Dimensional Harmonic Analysis I11 (pp. 249-264) Eds. H. Heyer et al. @ 2005 World Scientific Publishing Co.

DEFORMATIONS OF CONVOLUTION SEMIGROUPS ON COMMUTATIVE HYPERGROUPS

MARGIT ROSLER Mathematisches Institut, Universitat Gottingen Bunsenstr. 3-5, 0-37073Gottingen, Germany E-mail: roeslerOuni-math.gwdg.de MICHAEL VOIT Fachbereich Mathematit, Universitat Dortmund D-44221 Dortmund, Germany E-mail: michael.uoitOmathematik.uni-dortmund.de It was recently shown by the authors that deformations of hypergroup convolutions w.r.t. positive semicharacters can be used to explain probabilistic connections between the Gelfand pairs (SL(d,C),S U ( d ) )and Hermitian matrices. We here study connections between general convolution semigroups on commutative hypergroups and their deformations. We are able to develop a satisfying theory, if the underlying positive semicharacter has some growth property. We present several examples which indicate that this growth condition holds in many interesting cases.

1. Introduction

Klyachko recently derived a connection between SU(d)-biinvariant random walks on SL(d,C) and random walks on the additive group H ~ , of o all hermitian d x d-matrices with trace 0, whose transition probabilities are invariant under conjugation by S U ( d ) . He used this connection t o transfer the recent solution of the spectral problem for sums of hermitian matrices (7, lo) t o the possible singular spectrum of products of random matrices from SL(d,C) with given singular spectra. The singular spectrum of a matrix A g SL(d,C) here means the spectrum of the positive definite matrix Klyachko’s connection between SL(d,C) and H ~ , w oas explained in a different way and extended by the authors in 18; it is shown in l8 that the commutative Banach algebra of all SU(d)-biinvariant bounded measures on SL(d,C) may be embedded into the Banach algebra of all bounded measures on the Euclidean space H ~ , in o an isometric, probabil-

m.

250

ity preserving way. The proof of this fact, which has some applications in probability theory (see 18), depends on so-called deformations of hypergroup convolutions with respect t o positive semicharacters as introduced in 20. These deformations lead to connections between random walks and convolution semigroups on different, but closely related hypergroups. This forms the motivation to investigate systematically when and how convolution semigroups of probability measures on a commutative hypergroup ( X ,*) can be transformed canonically into convolution semigroups on a deformation ( X ,0 ) of ( X ,*). In particular we show that the generators and Levy measures of the original and the deformed convolution semigroup are closely related whenever this transformation is possible. We mention that the deformation of convolution semigroups is closely related t o Doob’s h-transform, and that L6vy processes associated with a convolution semigroup and its deformation are related by a Girsanov transformation on the path space; see 21, The paper is organized as follows: In Section 2 we collect some facts on deformations and present examples. In particular we indicate how for a maximal compact subgroup H of a complex, non-compact, connected semisimple Lie group G, the double coset hypergroup G / / H may be r e garded as deformation of an orbit hypergroup. This includes the examples above. Section 3 is devoted t o deformations of convolution semigroups w.r.t. positive semicharacters ao. We show that this concept works in a satisfying way under a canonical growth condition on the convolution semigroup together with some growth condition concerning (YO. Section 4 finally contains examples where this condition on a0 is satisfied. In fact, we have no example for which this condition would not hold. 2. Deformations of commutative hypergroups We give a quick introduction. First, let us fix notations. For a locally compact Hausdorff space X , M + ( X ) denotes the space of all positive Radon measures on X , and M b ( x ) the Banach space of all bounded regular complex Bore1 measures with the total variation norm. Moreover, M 1 ( X ) c &(x) is the set of all probability measures, M , ( X ) C &(x) the set of all measures with compact support, and 8, the point measure in z E X . The spaces c(x)3 cb(x)3 C o ( x ) 3 cc(x)of continuous functions are given as usual.

Definition 2.1. A hypergroup (X, *) consists of a locally compact Hausdorff space X and a weakly continuous, probability preserving convolution

251

* on Mb(X)such that ( M b ( x )*), is a Banach algebra and * preserves compact supports. Moreover, there exists an identity e E X (such that 6, is the identity of ( M b ( x ) *)) , as well as a continuous involution z I+ Z on x that replaces the group inverse. For details we refer to and 6 . We here only deal with commutative hypergroups ( X ,*), i.e., * is commutative. In this case there exists an (up t o normalization) unique Haar measure w E M + ( X ) which is characterized by w ( f ) = w(f,) for all f E C,(X) and z E X , where we use the notation

Similar t o the dual of a locally compact abelian group, one defines

x ( X ) :={aE C ( X ): a # 0 , a(z * y) = a(z)a(y)for all z,y E X } , X * :={a E X ( X ):

a(z)=a(.> for z E X I ; 2 := X * n cb(x).

Elements of X * and 2 are called semicharacters and characters respectively. All spaces are locally compact w.r.t. the compact-uniform topology. Example 2.1. (1) Let K be a compact subgroup of a locally compact group G. Then

Mb(GIIK) := { p E Mb(G) : 6,

* p * 6,

= p for all z,y E K }

is a Banach-*-subalgebra of Mb(G) with the normalized Haar measure dlc E M 1(G)of K as identity. The double coset space G / / K := { K z K : z E G } is locally compact w.r.t. the quotient topology, and the canonical projection p : G -+ G / / K induces a probability preserving, isometric isomorphism p : ikfb(GIIK)-+ Mb(G//K) of Banach spaces by taking images of measures. The transport of the convolution on Mb(GIIK) to ikfb(G//K) via p leads t o a hypergroup structure ( G / / K ,*) with identity K and involution ( K z K ) - := K z - l K , and p even becomes a Banach-*-algebra isomorphism. If G / / K is commutative, i.e., (G,K ) is a Gelfand pair, then a K-biinvariant function cp E C(G)with cp(e) = 1 is spherical if cp(z)cp(y) = 1, f(zlcy) dlc for z, y E G. The functions a E x ( G / / K ) are in one-to-one correspondence with the spherical functions on G via a I+ a o p for the canonical projection p : G + G / / K . (2) Let (V,( . ,.)) be a finite-dimensional Euclidean vector space and K c O ( V ) a compact subgroup of the orthogonal group of V. For

252 p E Mb(V), denote the image measure of p under

k

E

K by k(p).

Then the space of K-invariant measures

M F ( v ) := { p E M b ( v ) : k ( p ) = p for all k E K } is a Banach-*-subalgebra of M b ( v ) (with the group convolution) with identity 60. The space V K := {K.z : z E V } of all K-orbits in V is again locally compact, and the canonical projection p : V + V K induces a probability preserving, isometric isomorphism p : M F ( V ) 3 M b ( V K ) of B a n d spaces and an associated orbit hypergroup structure ( V K *) , such that p becomes an isomorphism of Banach-*-algebras. The involution on V K is given by 5 = -K.z. Moreover, the continuous functions

are multiplicative on ( V K *) , for X E V I ,the complexification of V , a p if and only if K.X = K.p. It is known (see 6, that and ax V K = {ax : X E V } . h

BY

20,

positive semicharacters lead to deformed convolutions:

Proposition 2.1. Let a0 E X * be a positive semicharacter o n the commutative hypergroup ( X ,*), i.e., ao(z)> 0 for z E X . Then p

v = ao((a,'p)

* (a,lv))

( Av E

M

C

W

)

(2.2)

extends uniquely to a bilinear, associative, probability preserving, weakly continuous convolution 0 o n M b ( X ) , and (x,0 ) becomes a commutative hypergroup with the identity and involution of ( X ,*). ( X ,0 ) will be called deformation of ( X , * ) w.r.t. ao. Eq.(2.2) shows that p I+ aop is an algebra isomorphism between ( M ,( X ) ,*) and ( M c( X ), a ) which for unbounded a0 cannot be extended to M b ( x ) ; 6.Section 3. Many data of ( X ,0 ) can be expressed in terms of a0 and corresponding data of ( X ,*). For instance, if w is a Haar measure of ( X ,*), then a i w is a Haar measure of (X,.). Moreover, the mapping Ma0 : a I+ a/ao is a homeomorphism between ( X ,*)* and ( X ,.)*, and also between x ( X , *) and x ( X , 0 ) ; see 2o and 1 8 .

Remark 2.1. Deformation is transitive as follows: Let (X, 0 ) be the deformation of ( K ,*) w.r.t. ao, and let PO be a positive semicharacter on ( K ,0 ) .

253

Consider further the deformation ( K ,0 ) of ( K ,0 ) w.r.t. PO. The function a& is a positive semicharacter on ( K ,*), and ( K ,0 ) is the deformation of (K,*)w.r.t. a&,. For PO = l/ao, one obtains o = *. We next present some examples; for further examples see Section 4.

Example 2.2. Let (V,( . , . )) be an n-dimensional Euclidean vector space, K a compact subgroup of the orthogonal group O ( V ) , and ( V K , * )the associated orbit hypergroup. Fix p E V with -p E K.p, and consider the function e p ( z ) := e ( p j z ) on V and

M ! ~ ~ ( v:=) {e,p

:p E

M,(v) K-invariant}.

The multiplicativity of ep on V yields that w.r.t. the group convolution on M,(V), we have e,p * epu = e,(p * u ) . Hence, M,",K(V)is a subalgebra of Mb(V),and its norm-closure

Ma"'K(V):= M p ( V ) a Banach subalgebra. On the other hand, ao(K.z) :=

s,

e,(k.z)dk (z E V )

is a positive semicharacter on ( V K *); , see Example 2.1(2) above as well as Proposition 2.8 of 18. Proposition 2.8 of l8 also states that for the deformation ( V K , o ) of ( V K , * )w.r.t. ao, the canonical projection p : V -+ V K induces a probability preserving isometric isomorphism of Banach algebras from M l Z K ( V )onto M b ( V K , o ) . In other words, the deformed hypergroup algebra may be regarded as Banach algebra of (not longer Kbiinvariant) measures on V .

Example 2.3. It is well-known that the double coset hypergroup SL(2,C ) / / S V ( 2 ) and the orbit hypergroup (IR3)s0(3) may be identified with [0,XI[, and that the associated hypergroup structures on [0, m[ are deformations of each other; see I , 18, or 20. Here is the higher rank extension of this example:

Example 2.4. Let G be a complex, noncompact, connected semisimple Lie group with finite center and K a maximal compact subgroup. Consider the Cartan decomposition g = e p of the Lie algebra of G, and choose a maximal abelian subalgebra a c p. K acts on p via the adjoint representation as a group of orthogonal transformations w.r.t. the Killing form ( . , .) as scalar product. Let W be the Weyl group of K , which acts on a as finite reflection group; here and further on we identify a with its dual a* via the

+

254

Killing form. Fix some Weyl chamber a+ C a and the associated set C+ of positive roots. Then the closed chamber C := Q+ is a fundamental domain for the action of W on a, and C can be identified with the orbit hypergroup ( p K , *), where a K-orbit in p corresponds to its representative in C. C can also be identified with the commutative double coset hypergroup G / / K where x € C corresponds t o the double coset K(e")K. Denote the corresponding convolution by 0 . Using the known formulas for the spherical functions on G / / K and pK (see Helgason 4), we proved in l 8 that ( G / / K ,0 ) = (C,0 ) is the deformation of the orbit hypergroup ( p K , * ) = (C,*) w.r.t. the positive semicharacter a-ip (in the sense of Example 2.1(2)) with P := Cae.E+ a E a+ *

As - p E K.p, the construction in Example 2.2 shows that Mb(G1I.K) may be embedded into Mb(p) in an isometric, probability preserving way. Here are the most prominent examples (c.f. Appendix C of 9 ) . (1) T h e Ad-1-case. K = SU(d) is a maximal compact subgroup of G = SL(d,C). In the Cartan decomposition g = t p we obtain p as the additive group H: of all Hermitian d x d-matrices with trace 0, on which SU(d) acts by conjugation. Moreover, a consists of all real diagonal matrices with trace 0 and will be identified with

+

{2= ( x 1 , . . . , x d )

E

xd : cxi =o} i

on which the Weyl group acts as the symmetric group take

sd.

We thus

C : = { x = ( Z 1 , . . . , x d ) E R d x: l 2 X z > . . . > % d r c x i = o } . i

+

+

Then in particular, p = (d - 1 , d - 3 , . . .,-d 3, -d 1). (2) T h e Bd-case. For d 1 2 consider G = S 0 ( 2 d + 1,C) with maximal compact subgroup K = S0(2d 1).Here a may be identified with Rd, and we may choose

+

c={ x E Rd : Z 1 > 2'2 > > x d 2 0 ) * * *

with the Weyl group W N s d K Z,! and p = (2d - 1,2d - 3 , . . . ,1). ( 3 ) T h e Cd-case. For d 2 3 let G = Sp(d,C) with the maximal compact subgroup K = Sp(2d 1). Here, a = Rd with C and W as in the Bd-case. We have p = (2d, 2d - l , , .. ,2). The preceding results on hypergroup deformations imply that the hypergroups

+

255

+

+

+

S p ( d , C ) / / S p ( 2 d 1 ) and S 0 ( 2 d l , C ) / / S 0 ( 2 d 1) are (UP t o isomorphism) deformations of each other; see also . (4) The Dd-case. For d 2 4 let G = S 0 ( 2 d , C) with maximal compact subgroup K = S O ( 2 d ) . In this case a = Rd and we may take

c = {z E Rd : 2 1 2 2 2 2 '

'

2 Zd-1 1 I.dl}

with p = (2d - 2,2d - 4,.. . ,2,0).

3. Deformation of convolution semigroups We now always assume that a0 is a positive semicharacter on a a-compact, second countable commutative hypergroup ( X ,*) and that ( X ,0 ) is the associated deformation. We show how under a natural growth condition, convolution semigroups on ( X ,*) can be deformed into convolution semigroups on ( X ,.). To describe this condition, we introduce the spaces

M2;(x) M:o(x)

Mbl+(X) : aop E Mb,+(X)}, := M ' ( X ) n M2b+(X) := { p E

as well as the transformation

Lemma 3.1. Let p, v E M b ! + ( X ) .'Then p * v E M::(X) if and only if p, u E ( X ) . Moreover, i f one of these conditions holds then

Ra0(P * v ) = Rcq ( P ) RaO ( v ) Proof. If p, u have compact support, then the lemma is clear by Eq. (2.2). In the general case, choose compacta (K,),>l - in X with X = U,K, and K,+1 3 K, for n E N. Put pn := p l ~ and , v, := ~ I K , . As the p, * u, have compact support, we have

Monotone convergence implies that

where one term is finite if and only if so is the other one. This proves the first part of the lemma. Moreover, if these terms are finite, then the same monotone convergence argument shows that for all f E cb(x)with f 2 0,

256

This implies R,, ( p * Y) = R,, ( p ) 0 R,,

(Y).

Remark 3.1. Notice that the mapping R,, : M A o ( X ) + M 1 ( X ) is not (weakly or vaguely) continuous whenever QO is unbounded. In fact, choose (zn)n21C X with ao(z,) + 00 and a0(zn)2 1. Then the measures pn := (1- ( Y O ( Z ~ ) - ~ ) ~~, o ( z ~ ) - ' 6 ~ ,tend , to 6, while

+

does not tend t o 6, = ROO(6,). We now investigate convolution semigroups.

Definition 3.1. A family (pt)t>o - C M1(X) is called a convolution semigroup on ( X , * ) , if po = 6,, if p8+t = p8 * pt for s , t 2 0, and if the mapping [0, 00[+ M1( X ) , t I+ pt is weakly continuous. It is well-known (see Rentzsch 12) that each convolution semigroup (pt)t>O - admits a LCvy measure r] E M + ( X \ {e}) which is characterized by 1

1

f dpt

for

f E C c ( X ) with e

# supp f.

(pt)tlo is called Gaussian, if r] = 0 which is equivalent to saying that for all neighborhoods U of e E X , 1imt-o i p t ( X \ U )= 0.

We next study under which conditions convolution semigroups on (X, *) can be deformed w.r.t. QO. We here need the following condition on (YO.

Definition 3.2. A positive semicharacter a0 on ( X ,*) is called exponential if there exists a neighborhood U of e E X and a constant C > 0 such that for all z, y E X with y E z * U ,ao(y)/ao(z) 5 C. We conjecture that positive semicharacters are always exponential. Unfortunately we are not able t o prove this. However, we present at least some criteria and examples in Section 4 below. The following theorem is motivated by 5 , 19, where a variant for the group case is studied.

Theorem 3.1. Let a0 be an exponential positive semicharacter on (X, *) with a0 2 1. Then the following statements are equivalent for a convolution semigroup (&>o - on ( X ,*) with Lkwy measure r]. (1) pt E M A o ( X )holds for some t > 0. (2) pt E MA,(X) holds for all t 2 0, the mapping cp : [O,oo[+]O,oo[ given by q(t) = J a 0 dpt is continuous and multiplicative, and (R,,(pt))tLo is a convolution semigroup on ( X ,0 ) .

257

(3) For any neighborhood U of e E X , s x , r r adq ~ < 00.

If one and hence all of these statements hold, then aoq is the L&vy measure of the convolution semigroup (Rao(pt))t>O o n ( X ,0 ) . In particular, Gaussian semigroups on ( X , * ) always lead t o Gaussian semigroups on ( X ,0 ) .

+

Proof. (1) (2): Lemma 3.1 implies that cp 2 1 is well-defined and multiplicative. To check continuity, we observe that the multiplicativity implies that for N E N and 0 5 s 5 1/N, c p ( ~ > ~ c p( l s N ) = cp(1) and hence cp(s) 5 cp(l)llN+ 1 for N + 00. Therefore, cp is continuous at t = 0 and hence, as a multiplicative function, on [0,00[. Using Lemma 3.1 and the fact that the mapping [0,00[+ M 1 ( X ) ,t C ) Ra,(pt) = p(t)-'aOp is vaguely and hence weakly continuous, we conclude that (Rao(pt))t>o is a convolution semigroup on ( X ,0 ) . (3): The measure p := l{ao>z}qE M b l + ( X )is the Levy mea(2) sure of the Poisson semigroup (vt := e-IIPllt exp(tp))t>O,e x p denoting the exponential function on the Banach algebra ( M b ( X ) ,*,.' Moreover, it is easy t o see that q - p is the Ldvy measure of a further convolution semigroup (ijt)t>o with pt = ut * fit for t 2 0. Lemma 3.1 shows that vt E M k o ( X )for t 2 0. As obviously p 5 (ellPllt/t)vt for t > 0, we obtain p E M:;(X) and thus (3). Furthermore, for f E C c ( X )with e $! supp f ,

-

Hence, aoq is the Levy measure of the semigroup (Rao(pt))t>O on ( X ,0)n The proof of (3) =+ (1) is more involved. Recapitulate that for a convolution semigroup (pt)t>o - on ( X ,*), the translation operators Tt(f ) := p; * f (t 2 0 ) form a strongly continuous, positive contraction semigroup on L 1 ( X , w ) ,w being the Haar measure of ( X , * ) ; see [BH]. Let A be its infinitesimal generator with the dense domain DA C L 1 ( X , w ) . We have:

Lemma 3.2. Let a0 be a positive semicharacter and (pt)t>o - a convolution ~ q satisfies Stao2z}a0 dq < 00. semigroup o n ( X ,*) whose L & Jmeasure Then f o r each neighborhood U of e E X there exists f E C c ( X )n DA with f 2 0 , f = f * # O , s u p p f c U ,a n d J l A f l a o d w < o o . Proof. Let U be a compact neighborhood of e E X with U - = U . Then by 12, there exists f E D A with f dw = 1, f 2 0, f = f*, and supp f c U.

sx

258

Let x # U*U and y E U.Then f (x*y) = 0, which means that the translate fz given by fi(y) := f (x * y) satisfies fi = 0 on U ,and hence 1

A f (4= t+O lim -t( P t

s

* f d e ) - f z ( e ) >= f (x * Y) drl(Y).

Consequently, by Fubini’s theorem,

lAfl-aodw

+

/

x\u*u

lAfl.aodw

Now

(1) in the theorem now follows from Lemma 3.2 and the follow(3) ing result.

Lemma 3.3. Let a0 be an exponential positive semicharacter with a. 2 1, and (pt)t>o a convolution semigroup on ( X , *) with generator A. Assume that for each neighborhood U of e E X there exists f E C c ( X )n D A with f 2 0 , f = f*#O,suppf c U a n d J l A f l a o d w < o o . T h e n f o r a l l t 2 0 , JaodClt < 00. Proof. Let U be a neighborhood of e E X and C1 > 0 a constant with clao(x) 5 ao(z) for x E X and z E U*x. Let f E c c ( X ) n D A with f >_ 0, f = f * # 0 , SUM, f c U and [Aflaodw < 00. Then for all m E N,the functions am:= a0 A rn E C I ( X )also satisfy clam(%) I a,(z) for x E X , z E U * x. Hence, there is a constant C2 > 0 depending on f such that for all m E N a n d x E X ,

s

am(x) I c 2

/

am(x * Y)f (Y)

WY)

= c2 . a m

* f (.I.

(3.1)

Moreover, as a. 2 1, we have for all m E N and x, y E X,

a m ( x * y) I m A ao(x * Y) = m A (ao(x)ao(y)) I am(x)am(y). (3.2)

259

Define hm(t):= J(pt * f ) .am dw = J a m * f d p t . As f E DA and Af E L1 ( X ,w ) holds, we obtain $pt * f = pt * Af and hence

Therefore, by (3.2) and (3.1),

This yields h m ( t ) I hm(0) etc for t 2 0 and some constant C 2 0 independent of rn. Hence, again by (3.1),

for all m E N. This yields the claim

J(YO

dpt

< 00 for t 2 0.

Notice that the growth condition on a0 was needed above only for the preceding lemma. Theorem 3.1 therefore admits the following variant. Theorem 3.2. Let (YO be a positive semicharacter and (pt)t>o c M 1 ( X ) a Poisson semigroup o n ( X ,*), which means that pt = e-tllPl&xp(tp) f o r all t 2 0 and some p E M b * + ( X ) .Then p is the L6vy measure of (pt)t>o, - and the statements (1)-(3) of Theorem 3.1 are equivalent.

Proof. It suffices t o check (3) =+- (1). However, if R := Jcto d p < 00, then for all n 2 0 , J (YO dp(") = R" and hence J (YO dpt < 00 for all t 2 0.n Remark 3.2. Let a0 be an exponential positive semicharacter and (pt)t>o C MAo( X ) a convolution semigroup on ( X ,*). Then the convolution operators (Tt)t>Oon C o ( X )with Tt f := pt * f form a Feller semigroup. Its generator A with 1 (2 E x,f E D ( 4 ) A m ) = t+O lim -(p; * f(.) - f (.)I t admits a II.Il,-dense domain D ( A ) in C o ( X ) ; see 13. Now consider the generator Ano of the Feller semigroup on C o ( X ) which is associated with the renormalized convolution semigroup (Rno(pt))tro on (X, 0 ) . Using the notation above, we have

260

Theorem 3.1(2) shows that cp(t)= ect for some c E E.,and 1 lim -(l/cp(t)- 1) = -c. t+O

t

Hence

Therefore

A"" = M l/ao O A O Mao - c

(3.3)

at least on D(AQo)n C c ( X ) ,where M , denotes the multiplication operator with g E C(K). The same holds for other function spaces like LP(X,w). 4. Exponential positive semicharacters

It seems reasonable t o conjecture that positive semicharacters are always exponential. Unfortunately we are not able to prove this. Here are, at least, some criteria and several examples: Lemma 4.1. (1) If ( X ,*) is discrete, then (YO is always exponential. (2) Let ( Y O , ( ~ 1be exponential positive semicharacters o n ( X ,*), and let (X,.) be the deformation of ( X , * ) w.r.t. ( Y O . Then (YI/(YO is an

exponential positive semicharacter o n ( X ,0 ) . Proof. Part (1) is clear by taking U = { e } . For the proof of (2) choose U1 of e and constants CO, C1 associated with (YO, a1 neighborhoods UO, respectively. For U := UO n U1 n U; n U; and C := CoC1, we obtain that for x , y E X with y E x * U , we have x E y * U and thus ( Y O ( X ) W ( y ) / ( a o ( y ) a l(.) I C as claimed. Example 4.1. In 2 3 , Zeuner presented quite general, but technical conditions on a function A E C([O,co[)n C'(]O,co[) with A ( x ) > 0 for x 2 0 which ensures that there exists a unique commutative hypergroup ([0,oo[,*)

261

whose semicharacters are precisely the eigenfunctionsof the Sturm-Liouville operator

LAf := -f”

- (A’/A)f’

with initial conditions f(0) = 1 and f‘(0) = 0; see also Section 3.5 of This hypergroup is called the Sturm-Liouville hypergroup associated with A. Moreover, to the knowledge of the authors, all known hypergroup structures on [O,m[ appear in this way (up to isomorphism); see also for details. We here mention that Zeuner’s approach in particular includes all Chebli-”kimeche hypergroups and thus all double coset hypergroups associated with noncompact symmetric spaces of rank one. We claim that all positive semicharacters on a Sturm-Liouville hypergroup on [0,m[ with A satisfying Zeuner’s conditions are exponential. To prove this, recall from Section 3.5 in that Zeuner’s conditions imply that 1 p := - lim A’(x)/A(x)2 0 2 x--fw exists, and that the positive semicharacters are precisely the unique solutions cpix of

with X 2 0. Moreover, the renormalization ([O,m[,o) of ([0, co[,*) w.r.t. pix is again a Sturm-Liouville hypergroup associated with the renormalized function Ax := ‘p% A where Ax again satisfies Zeuner’s conditions; see Section 3.5.51 of Applying (4.1) to A as well as to Ax, we see that lim,,,&(x)/~pix(x) exists. As supp(b,*b,) c [ ) x - y l , x + y ] for x , y 2 0, it follows from the mean-value theorem that cpix is exponential.

’.

Example 4.2. Let V be a finite-dimensional Euclidean vector space, K C O ( V ) a compact subgroup, and V K the associated orbit hypergroup as in Example 2.1(2). Then, for each p E V , the positive semicharacter ajPwith CX~,(K.Z) = ] K e - ( ~ + x ) d k( x E V) is exponential. In fact, we may take U := {K.x : x E V, 11x112 6 1) c V K as a neighborhood of the identity. For orbits K.x, K.y E V K with K.x E U * K.y we then have representatives X , y E V with llx - y112 6 1 which implies that e - - ( P l k . x ) 5 e-(PrL.u)ellPllz for k E K and thus ai,(K.x) 5 aiP(K.y)ellPllzas claimed. Example 4.3. Let G be a (not necessarily complex) noncompact, connected semisimple Lie group with finite center and K a maximal compact subgroup. Let G = NAK and g = n + a t be the corresponding Iwasawa decompositions. For g E G let A(g) E a be the unique element with

+

262

g E N e z p ( A ( g ) ) K . Let C+ be the set of positive roots (for the order m,a the half sum of positive roots corresponding t o n), and p = CaEC+ with m, as multiplicity of a. Then, by a formula of Harish-Chandra (see Theorem IV.4.3 of 4), the spherical functions on G, i.e., the multiplicative functions on G I I K , are given by

where X runs through w, the complexification of a. Clearly, the cpx for X E i . a are positive multiplicative functions. These functions are also exponential. To prove this, we conclude from Lemma IV.4.4 of that

Hence, for each comact neighborhood U of e there is a constant C that cpx(g-'h) 5 Ccpx(h)for all g E U ,h E G and X E i . a.

> 0 such

Example 4.4. Let R be a (reduced, not necessaryly crystallographic) root system in R" with the standard inner product (. , .), i.e. R c Rn \ (0) is finite with R n lwru = {&a}and ua(R)= R for all a E R, where u, is the reflection in the hyperplane perpendicular to a. Assume also without loss of generality for our considerations that (a,a ) = 2 for all cr E R. Let W be the finite reflection group generated by the u, and let k : R + [0, CG[ be a fixed multiplicity function on R, i.e. a function which is constant on the orbits under the action of W . The (so-called rational) Dunkl operators attached t o G and k are defined by

Here at denotes the derivative in direction 5 and R+ is some fixed positive subsystem of R. The definition is independent of the special choice of R+, due t o the G-invariance of k. As first shown in ', the Tc(k), 5 E Rn generate a commutative algebra of differential-reflection operators. This is the foundation for rich analytic structures related with them. In particular, there exists a counterpart of the exponential function, the Dunkl kernel, and an analogue of the Euclidean Fourier transform with respect to this kernel. The Dunkl kernel Ek is holomorphic on a?' x a?' and symmetric in its arguments. Similar to spherical functions on symmetric spaces, the function Ek ( . ,y) with fixed y E cc" may be characterized as unique analytic solution of the joint eigenvalue problem

T . ( k ) f= ( 5 1 Y ) f

for all

5 E c,f(0) = 1;

(4.3)

263

c.f. ll. Apart from the trivial case k = 0 with Ek(z,y) = e(zJ),EI, is explicitly known in a few cases only like n = 1; see l7 for a survey. The G-invariant counterpart of Ek is the generalized Bessel function

which is G-invariant in 2,y and naturally considered on the closed positive Weyl chamber C associated with R+. For n = 1, J k is a usual Bessel function. Moreover, in the cristallographic case and for certains half-integer multiplicities, the Jk are the multiplicative functions of certain Euclidean orbit hypergroups as in Example 2.1. Here, and for n = 1, the Jk(2,y) (y E P) therefore form the multiplicative functions of some commutative hypergroup on C. It is conjectured that there exist such commutative hypergroups on C for all root systems and multiplicities k 2 0. Only part of this conjecture has been verified up to now in 16. Now fix a root system R and k 1 0 such that the Jk(.,y) (y E P) are the multiplicative functions of a commutative hypergroup (C, *). To find positive semicharacters, we employ the following psoitive integral representation for Ek (and thus Jk): For given R,k 2 0, and s E Rn there exists a unique pribability measure pa on Rn such that

~ k ( z , y= ) / e + > v ) dpx(z)

for y E C .

(4.4)

Moreover, supppz c { z E Rn : llzll2 5 11x112). Thus, for each y E Rn, Jk(., y) is a positive semicharacter on (C, *). We claim that these semicharacters are exponential. To show this, let U := { z E C : llzll2 5 1) and ~ , Z ZE C with 2 1 E U * x 2 , We conclude from Theorem 4.1 of l6 that then x1 E C n n w E W { zE Rn : 1 - 20.221 5 1) holds. As llz - 'wII llz - 20.211 for all 2,z E C and w E W by Ch. 3 of 3, we even have 1\21- 2211 1. In the same way as in Example 4.3 we now obtain from Eq. (4.4) that Jk(z1,y) 5 ellvllJk(z2,y) which proves that Jk(.,y) is exponential for each y E Rn.

<

<

References 1. W.Bloom, H.Heyer, Harmonic Analysis of Probability Measures on Hypergroups. De Gruyter-Verlag, Berlin, 1994. 2. C.F. Dunk], Differential-difference operators associated to reflection groups, Trans. Amer. Math. SOC. 311 (1989), 167-183. 3. L.C. Grove, C.T.Benson, Finite Reflection Groups. Springer-Verlag, 2nd ed., 1985.

264 4. S. Helgason, Groups and Geometric Analysis. American Mathematical Society, 2000. 5. A. Hulanicki, A class of convolution semi-groups of measures o n a Lie group. In: Probab. Theory on Vector Spaces 11, Proceedings. Lecture Notes in Mathematics Vol. 828, Springer 1980, pp. 82-101. 6. R.I. Jewett, Spaces with a n abstract convolution of measures. Adv. Math. 18 (1975), 1-101. 7. A. Klyachko, Stable bundles, representation theory, and H e m i t i a n operators, Selecta Math. (new Series) 4 (1998), 419445. 8. A. Klyachko, Random walks o n symmetric spaces and inequalities f o r matrix spectra, Linear Algebra Appl. 319 (ZOOO), 37-59. 9. A.W. Knapp, Lie Groups beyond an Introduction. Birkhauser, Boston, 1996. 10. A. Knutson, T. Tao, T h e honeycomb model of GLn(@)tensor product I. Proof of the saturation conjecture, J. Amer. Math. SOC.12 (1999), 10551090. 11. E.M. Opdam, Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Compos. Math. 85 (1993), 333-373. 12. C. Rentzsch, A Ldvy-Khintchine type representation of convolution semigroups o n commutative hypergroups. Probab. Math. Stat. 18 (1998), 185198. 13. C. Rentzsch, M. Voit, Ldvy processes o n commutative hypergroups. Contemp. Math. 261 (ZOOO), 83-105. 14. L.C.G. Rogers, D.Williams, Difisions, Markov Processes, and Martingales, Vol. II. Wiley: Chichester - New York, 1987. 15. M. Rosler, Positivity of Dunkl’s intertwining operator. Duke Math. J. 98 (1999), 445-463. 16. M. Rosler, A positive radial product formula f o r the Dunkl kernel. Trans. Amer. Math. SOC.355 (2003), 2413-2438. 17. M. Rosler, Dunkl operators: Theory and Applications. Lecture Notes for the SIAM Euro Summer School: Orthogonal Polynomials and Special F’unctions. (Leuven, 2002). In: Springer Lecture Notes in Mathematics, Vol. 1817, 2003, pp. 93 - 136. 18. M. Rosler, M. Voit, SU(d)-biinvariant random walks o n SL(d,C) and their Euclidean counterparts, preprint 2003. 19. E. Siebert, Continuous convolution semigroups integrating a submultiplicntive function, Man. Math. 37 (1982), 383-391. 20. M. Voit, Positive characters on commutative hypergroups and some applications, Math. Z. 198 (1988), 405-421. 21. M. Voit, A Girsanov-type formula for Ldvy processes o n commutative hypergroups. In: H. Heyer et al. (eds.), Infinite Dimensional Harmonic Analysis, Proc. Conf., Grabner 2000, pp. 346-359. 22. H.Zeuner, T h e central limit theorem f o r Chdbli-lkimeche hypergroups. J. Theor. Probab. 2 (1989), 51-63. 23. H. Zeuner, M o m e n t functions and laws of large numbers o n hypergroups. Math. Z . 211 (1992), 369-407.

Infinite Dimensional Harmonic Analysis I11 (pp. 265-276) Eds. H. Heyer et ol. 0 2005 World Scientific Publishing Co.

AN INFINITE DIMENSIONAL LAPLACIAN ACTING ON MULTIPLE WIENER INTEGRALS BY SOME LEVY PROCESSES KIMIAKI SAITO Department of Mathematics Meijo University Nagoya 468-8508, Japan E-mail: ksaito @ccmfs.meijo-u.ac.jp

In this paper we shall discuss the Ldvy Laplacian as an operator acting on multiple Wiener integrals by some LBvy processes. This space includes regular functionals in terms of Gaussian white noise and it is large enough to discuss the stochastic process. From Cauchy processes an infinite dimensional stochastic process is constructed, of which the generator is the Ldvy Laplacian.

Introduction An infinite dimensional Laplacian was introduced by P. LCvy in his famous book [17]. Since then this exotic Laplacian has been studied by many authors from various aspects see [1-6,18,20,23]and references cited therein. In this paper, generalizing the methods developed in the former works [16,19,24,28,29], we construct a new domain of the LCvy Laplacian acting on some class of multiple Wiener integrals by some LBvy processes and associated infinite dimensional stochastic processes. This paper is organized as follows. In Section 1 we summarize basic elements of white noise theory based on a stochastic process given as a difference of two independent LCvy processes. In Section 2, following the recent works Kuo-Obata-Saitb [16], Obata-Saitb [24],Saitb [30] and SaitbTsoi [31], we formulate the LBvy Laplacian acting on a Hilbert space consisting of some LCvy white noise functionals and give an equi-continuous semigroup of .class (CO) generated by the Laplacian. This situation is further generalized in Section 3 by means of a direct integral of Hilbert spaces. The space is enough to discuss the stochastic process generated by the LCvy Laplacian. It also includes regular functionals (in the Gaussian sense) as a harmonic functions in terms of the LCvy Laplacian. In Section 4, based on infinitely many Cauchy processes, we give an infinite dimensional stochastic process generated by the LCvy Laplacian.

266

1

Functionals of Gaussian noise and Poisson noise

Let E = <S(R) be the Schwartz space of rapidly decreasing R-valued functions on R. There exists an orthonormal basis {ev}v>o of I<2(R) contained in E such that Aev = 1(v + l)ev,

v = 0,1, 2 , . . . ,

A=

For p 6 R define a norm | • \p by \f\p = \Apf\L2(R) tor f € E and let Ep be the completion of E with respect to the norm | • \p. Then Ep becomes a real separable Hilbert space with the norm | • \p and the dual space E' is identified with E-p by extending the inner product (-, •) of I<2(R) to a bilinear form on -p x Ep. It is known that E = proj lim Ep, p-KX>

E* = indlim E-p. P-»°°

The canonical bilinear form on E* x E is also denoted by {•, •}. We denote the complexifications of L2(R), E and Ep by L^R), EC and EC,P, respectively. Let {LpX(t)}t>0 and {L^x(t)}t>o be independent Levy processes of which the characteristic functions are given by E[eizLi-*w] = ethW , t>Q, j = 1,2, h(z) = imz - y z2 + (eixz - 1), where m 6 R, 0 and A e R. Set A ff) x(t) = L\^(t] - L2^x(t) for all t > 0. Then we have = exp {-ta2z2 + t(eaz + e~ixz - 2)} , t > 0. Set (7(0 = exp {^(fctfiOO) + h(-^(u)))du} , f = (^1,6) £ E x E. Then by the Bochner-Minlos Theorem, there exists a probability measure fifft\ on E* x E* such that

where (z,£) = (xi,£i) + (xi,£2), x = (zi,z 2 ) £E*xE*,£= (6,6) £ExE. Let (L2)ff>x = L2(E* x E*,^^) be the Hilbert space of C-valued squareintegrable functions on E* x E* with L2-norm || • H^ with respect to /ZO-,A-

267

The Wiener-It6 decomposition theorem says that:

c 00

(L2)u,x =

(1.1)

@&,

n=O

where Hn is the space of multiple Wiener integrals of order n E N and C. According to (1.1) each cp E (L2),,xis represented as

c

HO=

00

cp =

In(fn),

fn

f2L”cR)%

n=O

where L&(R)&)” denotes the n-fold symmetric tensor power of L&(R)(in the sense of a Hilbert space). An element of (L2),,x is called a white noise functional. We denote by ((-,-)),,A the inner product on (L2),,x. Then, for cp and $ E (L2),,0 we have

n=O

n=O

n=O

where the canonical bilinear form on J ~ & ( R x) @ L&(R)@n ~ is denoted also by (-7

9).

For cp and $ E (L2)o,xwe also have 00

00

is defined by

The U-transform of

Theorem 1.1 [26] (see also [9,14,22]) Let F be a complex-valued function defined o n E x E . Then F is a U-transform of some white noise functional in (L2),,x if and only if there exists a complex-valued function G defined o n E c x E c such that 1) for any ( and q in E c x E c , the function G(z6 + q ) is a n entire function O f Z E C ,

2) there exist nonnegative constants K and a such that IG(J)I I: Kexp

+

3) F(<)= G(ia2& ia2&

[alrl;] , K E E c

x Ec,

+ X(eixcl - e-ixc2)) for all < = ( ( I , & )

E E x E.

268

2

The LQvy Laplacian acting on multiple Wiener integrals

Consider F = U'p with 'p E (L2),,x. By Theorem 1.1, for any <,T,I E E x E the functions z H F ( < zq) admits the Taylor series expansions:

+

where F ( n ) ( [ ): E x x E + C is a continuous n-linear functional. Fixing a finite interval T of R,we take an orthonormal basis {(n},"=o c E x E for L2(T)which is equally dense and uniformly bounded (see e.g. [14,15]). Let DL denote the set of all 'p E (L2),,x such that the limit

c

N-1

L L ( U V ) ( t )=

(U'p)%)(Cn, G),

n=O

exists for any E E x E and A L ( U ' ~is) in U[(L2),,x].The LCvy Laplacian AL is defined by

ALP = U - ~ K L U V ,

'p

E DL.

Given u 2 0,X E R, n E N and f E LE(R)6n, we consider 'p E (L2),,x of the form: V=/,n

The U-transform U p of

UP(<)=

J

f(u1,. 'p

--

(2.1)

r.Iln)dA,,x(ul)...dA,,x(un).

is given by n

f(u1,. '. ,un)

T"

JJ %x(s)(uj)dul-

*

.dun,

s E E x E.

j=1

+

where E , , ~ ( ( ) ( u j )= iu2<1(uj) iu2&(uj)+ X(eixcl(uj)- e - i X c a ( u j ) ) . For any u 2 0, X E R and n E N let E , , A , denote ~ the space of cp which admits an and suppf C Tn.Set expression as in (2.1), where f belongs to L&(R)6n Eo,x,O = C for any u 2 0,X E R. Then E , , A ,is ~ a closed linear subspace of (L2),,x.Using a similar method as in [31], we get the following

Theorem 2.1 [31] (see also [16,29]) For each u 2 0, n E N and X E R the L6vy Laplacian AL becomes a scalar operator o n E,,o,,, U E o , ~such , ~ that AL'p = 0,

'p

E Eu,o,n;

(2.2)

269

Proposition 2.2 [31] Let X E R be fixed. Consider two white noise functions of the form: m

m

n=O

n=O

If cp = 1c, in (L2)0,x,then cpn = lc,n for all n E N U (0). Taking (2.2) and (2.3)into account, we put

~ admits an For N E N and X E R let D Y be the space of cp E ( L 2 ) o ,which expression

n=l

such that

c 00

IIIPl112N,o,x =

amllcpnll:,x <

(2.4)

n=l

By the Schwartz inequality we see that D$X is a subspace of (L2)0,x and becomes a Hilbert space equipped with the new norm 111 IIIN, o , x defined in (2.4). Moreover, in view of the inclusion relations:

-

( L 2 ) o , x3 DY*x3 . . - 3 D$X 3 D$:l we define

DZX= projlimDi OX = N-tw

3

... ,

n DY. 00

N=l

Note that for any X E R we have 00

(JEo,x,n c DZXc (L2)o,x.

n=l

Then A L becomes a continuous linear operator defined on D$l: satisfying

I~~AL(PIIIN,o,A I ~ ~ ~ ~ ~ ~ ~ N +'pl ,EODZx, ,X~ N E N O Summing up, we have the following

into D$X (2.5)

270

Theorem 2.3 [16,31] The operator AL is a self-adjoint operator densely defined in D Y f o r each N E N and X E R. It follows from (2.5)that AL is a continuous linear operator on DSx. In view of the action of (2.3), for each t 2 0 and X E R we consider an operator G; on DZx defined by

n=l

n=l

We also define G: on (L2)u,o.

(L2),,0

as an identity operator I by Icp = cp, cp E

Theorem 2.4 [16,30] Let X E R. T h e n the family of operators {G?; t 2 0) o n Dzx i s a n equi-continuous semigroup of class (CO) of which the infinitesimal generator i s A,. 3

The LCvy Laplacian acting on WNF-valued functions

Let &(A)

be a finite Bore1 measure on R satisfying

JRdYo<m. X4

Fix N E N. Let 9&be the space of (equivalent classes of) measurable vector functions cp = (cpx) with cpx = Cr=lp: E D>x for all X E R \ {0}, and cpo E ( L 2 ) , , 0 , such that

Then 9 ; becomes a Hilbert space with the norm given in (3.1). Let 9; be the space of (equivalent classes of) measurable vector functions cp = (cpx) with cpx = CrZlcp: E (L2)0,xfor all X E R\{O}, and cpo E (L2),,0, such that

Then 9: also becomes a Hilbert space with the norm given in (3.2).

271

Proposition 3.1 T h e m a p

i s a continuous linear m a p and a bijection f r o m DG into 9:. In view of the natural inclusion: D$,+l c 23% for N E N, which is obvious from construction, we define

The Levy Laplacian A, is defined on the space D& by ~ L c = p (AL‘P’),

cp = (9’)

E 9:.

Then AL is a continuous linear operator from D& into itself. Similarly, for t 2 0 we define

Gtcp = (G,x‘PX),

cp = (‘PA>E

Then by Theorem 2.4 we have the following:

Theorem 3.2 T h e family (Gt;t 2 0) is a n equi-continuous semigroup of class (Co)o n D& whose generator i s given by AL. 4

An infinite dimensional stochastic process associated with the LBvy Laplacian

For p E R let EF be the linear space of all functions X c) (A E Ep x Ep, X E R, which are strongly measurable. An element of EF is denoted by = ( ~ x ) x ~ R . Equipped with the metric given by

<

the space EF becomes a complete metric space. Similarly, let CR denote the linear space of all measurable function X !+ q,E C equipped with the metric defined by

272

Then CR is also a complete metric space. In view of dp 5 dq for p 1 q, we introduce the projective limit space ER = proj limp+m EF. The U-transform can be extended to a continuous linear operator on 9: by

up(<)= (U(pX(b))XER,

< = (6X)XER E ER,

for any p = ( ( p X ) x E ~E D&. The space U[DO,] is endowed with the topology induced from D$ by the U-transform. Then the U-transform becomes a homeomorphism from a&onto U[D&]. The transform U p of p E DL is a continuous operator from ER into CR. We denote the operator by the same notation Up. Let

et be an operator defined on U[D$] by 6t = UGtU-',

t 2 0.

Then by Theorem 3.3, {Gt; t 2 0) is an equi-continuous semigroup of class ((20)generated by the operator A;L. Let { X i } ,j = 1,2,3,4, be independent Cauchy processes with t running over [0, oo),of which the characteristic functions are given by

E[eiZx:]= e-'l2l, Take a smooth function r]T E E with

y;" =

{

z E R, r]T

j = 1,2,3,4.

= 1/ITI on T. Set

(xitqT, - x i t q T ) (x!,tr]T,-X?,tr]T),

if 2 0, otherwise.

Define an infinite dimensional stochastic process { Y t ;t 2 0) starting at ( = (<X)XER ER by

Yt = (b + KX)XER,

t 1 0.

Then this is an ER-valued stochastic process and we have the following

Theorem 4.1 If F is the U-transform of an element in D ,; G t W ) = E [ F ( Y t ) l Y o=

(1,

t 2 0.

we have (4.1)

273

PROOF.We first consider the case when F E U[Iz&]is given by

F(t) = (FX(<X))XER,

F o E U[(L2)u,0],

with f E L&(R)Qn.Then we have

Next let and for any

Then foralmost all is expressed in the following form:

Since Fo E U[(L2),,o]and F," E U[DzA],there exist cpo E (L2)u,oand c p i E DzXsuch that Fo = U[cpO]and F," = U[cpk]for v-almost all X and each n. By the Schwarz inequality, we see that

274

-

where ‘ptx = C(&,)-lei(’itx) for v-almost all X E R and each N E N. Therefore by the continuity of G;, X E R, we get that

Thus we obtain the assertion.

I

Acknowledgments This work was written based on a talk at the 3rd Japanese-German Symposium on “Infinite Dimensional Harminic Analysis” held from September 15th to September 20th 2003. This work was partially supported by JSPS grant 17540136. The author is grateful for the support. References

1. L. Accardi and V. Bogachev: The Ornstein-Uhlenbeck process associated with the L i v y Laplacian and its Dirichlet form, Frob. Math. Stat. 17 (1997), 95-114. 2. L. Accardi, P. Gibilisco and I. V. Volovich: Yang-Mills gauge fields as harmonic functions for the L i v y Laplacian, Russ. J. Math. Phys. 2 (1994), 235-250. 3. L. Accardi and 0. G. Smolyanov: Trace formulae for Levy-Gaussian measures and their application, in “Mathematical Approach to Fluctuations Vol. I1 (T. Hida, Ed.),” pp. 3 1 4 7 , World Scientific, 1995. 4. D. M. Chung, U. C. Ji and K. Sait8: Cauchy problems associated with the L i v y Laplacian in white noise analysis, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 2 (1999), 131-153.

275

5. M. N. Feller: Infinite-dimensional elliptic equations and operators of Ldvy type, Russ. Math. Surveys 41 (1986), 119-170. 6. K. Hasegawa: Ldvy’s functional analysis in terms of an infinite dimensional Brownian motion I, Osaka J. Math. 19 (1982), 405-428. 7. T. Hida: “Analysis of Brownian Functionals,” Carleton Math. Lect. Notes, No. 13, Carleton University, Ottawa, 1975. 8. T. Hida: A role of the L i v y Laplacian in the causal calculus of generalized white noise functionals, in “Stochastic Processes,” pp. 131-139, SpringerVerlag, 1993. 9. T. Hida, H.-H. Kuo, J. Potthoff and L. Streit: “White Noise: An Infinite Dimensional Calculus,” Kluwer Academic, 1993. 10. T. Hida and K. SaitB: White noise analysis and the Ldvy Laplacian, in “Stochastic Processes in Physics and Engineering (S. Albeverio et al. Eds.),” pp. 177-184, 1988. 11. E. Hille and R. S. Phillips: “Functional Analysis and Semi-Groups,” AMS Colloq. Publ. Vol. 31, Amer. Math. SOC,1957. 12. I. Kubo and S. Taken& Calculus o n Gaussian white noise I-IV, Proc. Japan Acad. 5 6 A (1980) 376-380; 5 6 A (1980) 411416; 5 7 A (1981) 433436; 5 8 A (1982) 186-189. 13. H.-H. Kuo: O n Laplacian operators of generalized Brownian functionals, Lect. Notes in Math. Vol. 1203, pp. 119-128, Springer-Verlag, 1986. 14. H.-H. Kuo: “White Noise Distribution Theory,” CRC Press, 1996. 15. H.-H. Kuo, N. Obata and K. Sait6: Ldvy Laplacian of generalized functions o n a nuclear space, J. Funct. Anal. 94 (1990), 74-92. 16. H.-H. Kuo, N. Obata and K. SaitB: Diagonalization of the Ldvy Laplacian and related stable processes, Infin. Dimen. Anal. Quantum Probab. Rel. TOP.5 (2002), 317-331. 17. P. LBvy: “Legons d’Analyse Fonctionnelle,” Gauthier-Villars, Paris, 1922. 18. R. LBandre and I. A. Volovich: The stochastic Ldvy Laplacian and YangMills equation o n manifolds, Infin. Dimen. Anal. Quantum Probab. Rel. TOP.4 (2001) 161-172. 19. K. Nishi and K. Sait6: An infinite dimensional stochastic process and the Lkuy Laplacian acting o n WND-valued functions, t o appear in “Quantum Inofrmation and Complexity” World Scientific, 2004. 20. K. Nishi, K. SaitB and A. H. Tsoi: A stochastic eqression of a semigroup generated by the Ldvy Laplacian, in “Quantum Information I11 (T. Hida and K. SaitB, Eds.),” pp. 105-117, World Scientific, 2000. 21. N. Obata: A characterization of the Ldvy Laplacian in terms of infinite dimensional rotation groups, Nagoya Math. J. 118 (1990), 111-132. 22. N. Obata: “White Noise Calculus and Fock Space,” Lect. Notes in Math.

276

Vol. 1577, Springer-Verlag, 1994. 23. N. Obata: Quadratic quantum white noises and L i v y Laplacian, Nonlinear Analysis 47 (2001), 2437-2448. 24. N. Obata and K. Sait6: Cauchy processes and the Le‘vy Laplacian, Quantum Probability and White Noise Analysis 16 (2002), 360-373. 25. E. M. Polishchuk: “Continual Means and Boundary Value Problems in Function Spaces,” Birkhauser, Basel/Boston/Berlin, 1988. 26. J. Potthoff and L. Streit: A characterization of Hida distributions, J. F’unct. Anal. 101 (1991), 212-229. 27. K. Sait6: It6’s formula and Ldvy’s Laplacian I, Nagoya Math. J. 108 (1987), 67-76; 11, ibid. 123 (1991), 153-169. 28. K. Sait6: A (Co)-group generated by the Ldvy Laplacian 11, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 1 (1998) 425-437. 29. K. Sait6: A stochastic process generated by the L i v y Laplacian, Acta Appl. Math. 63 (2000), 363-373. 30. K. Sait6: T h e Ldvy Laplacian and stable processes, Chaos, Solitons and Fractals 12 (2001), 2865-2872. 31. K. Sait6 and A. H. Tsoi: The L i v y Laplacian as a self-adjoint operator, in “Quantum Information (T. Hida and K. Sait6, Eds.),” pp. 159-171, World Scientific, 1999. 32. K. Sait6 and A. H. Tsoi: The Le‘vy Laplacian acting o n Poisson noise functionals, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 2 (1999), 503-510. 33. K. Sait6 and A. H. Tsoi: Stochastic processes generated by functions of the Ldvy Laplacian, in “Quantum Information I1 (T. Hida and K. Sait6, Eds.),” pp. 183-194, World Scientific, 2000. 34. K. Yosida: “hnctional Analysis (3rd Edition) ,” Springer-Verlag, 1971.

Infinite Dimensional Harmonic Analysis I11 (pp. 277-287) Eds. H. Heyer et al. @ 2005 World Scientific Publishing Co.

LEVY PROCESSES ON DEFORMATIONS OF HOPF ALGEBRAS

MICHAEL SCHURMANN* Instatut fir Mathematik und Infomatik Universitat Greifswald Friedrich-Ludwig-Jahn-Strape15a 17469 Greifswald, Germany Ernail: schurman0uni-greifswald.de

Using the theory of additively deformed *-algebras developed by J. Wirth, we introduce the notion of LBvy processes on such objects and prove that, as in the non-deformed case, these LBvy processses can be realized as the solution of quantum stochastic differential equations in the sense of Hudson and Parthasarathy. Deformations of polynomial algebras, group algebras and of non-commutative coefficient algebras and their Levy processes are considered as examples.

A *-algebra means an associative unital complex algebra A with an involution a I+ a*, i.e. an antilinear self-inverse mapping satisfying (ab)*= b*a*. By @ we denote the tensor product of vector spaces. Notice that the multiplication of an algebra A is given by a linear mapping p : d B A + A. A coalgebra is a triplet (C, A, 6) consisting of a complex vector space C and linear mappings A : C + C @ C, 6 : C + C satisfying the coassociativity and counit conditions respectively. For *-algebras A1 and A2 the tensor product dl @ A2 is considered as a *-algebra with (a1 @ a2)* = at @ a; and (a1 @u2)(bl @ b 2 ) = albl @ u 2 b 2 . A *-bialgebra ( D , * , p , 1,A,6) is a *algebra (f?,*, p, 1) and a coalgebra (I?,A, 6) such that A and 6 are *-algebra homomorphisms; see [4]and also [2]. A quantum probability space is a pair (A,9)consisting of a *-algebra A and a state 9 on A (i.e. a positive normalised linear functional on A). A quantum random variable on the *-algebra 23 over the quantum probability space (A,9)is a *-algebra homomorphism j : f? + A; 6. [4]. *work supported in part by the european community's human potential programme under contract hprn-ct-2002-00279 qp-applications

278

We summarize some of the results of [5]. An additive deformation of a - of mappings *-bialgebra (B, *, p, 1 , A, 6) is a family (pt)t>o pt : B €3 B

+B

such that 0

(a,* , p t , 1 ) becomes a *- algebra Po=P

0 0

(p8~pt)o(A~A)=Aop,+tforalls,t~0 limt,o+ 6 o pt = 6 63 6 pointwise.

T €3 id) o (A €3 A) with 7 the flip. Notice that the Here A 6 A means (id 18 third condition repeats the bialgebra property of t3 if we put s = t = 0. We call a n additive deformation a Hopfdeformation if there exists a family (St)t>o - of linear mappings on t3 sucht that

pt o (St 63 id) o A = pt o (id 63 St) o A = e o 6

where e : C + B,e(X) = X1. Such a family ( S t )-t > ~if, it exists, is uniquely determined.

It is shown in [5] that the limit

exists pointwise for all elements in

B @ B,and that

Pt = P * =P*(tL)

L*p=p*L

L(b El c) = L(c* €3 b*) L ( l €3 1 ) = 0 and

G(a)L(b€3 c)

- L(ab €3 c) + L ( a €3 bc) - L(a €3 b)6(c) = 0

(5)

for all a, b, c E B. The latter equation means that L is a 2-cocycle in the Hochschild cohomology given by the B-bimodule structure a.X.b = S(a)XS(b)

on C. Moreover, for a given linear mapping L : 13 63 B + C satisfying (2), (3), (4) and ( 5 ) we have that (1) defines an additive deformation of the involutive bialgebra B. In our context a hermitian cocycle will always mean a bilinear form on B satisfying (3) and ( 5 ) . A 2-cocycle will be called

279

commuting if also ( 2 ) holds, and normalized if (4) holds. Of course, dl 2-cocycles are commuting if

B is co-commutative.

. . ,d, of the non-commutative We say that the involutive sub-algebras d1,. probability space (d, @)are tensor-independent (cf. [4] and also [l]) if 0

[ak,al]= 0

0

@(a1

. . .a,)

dk and ar E 4, k#I = @ ( a l ) . .@(a,) for ak E dk,k = 1,...,n.

for

ak E

The random variables j k : Bk + d, k = 1,...,n, are called tensorindependent if jl (&), . . . ,jn(Bn) are tensor-independent.

A family ( j s t ) o ~ s of ~ t mappings jst : B + d is called a Lkvy process on the deformed involutive bialgebra B (over the quantum probability space (.A,@)) if 0

0

is a random variable on (D, 1) j t l t a , . . . ,jt,t,+l are tensor-independent for all tl jrs*jst = j r t for all 0 i r 5 s 5 t 9 0 jst = @ 0 j O , t e s for all 0 5 s 5 t limt-+o+ 9 0 jo,t = b pointwise j8t

If we have a Hopf deformation of Levy process on B if j8t

5 ... 5 t,+l

B,then a family (jt),>, - on B is called a

= ( j , 0 S8)* j t

forms a Levy process in the above sense.

For a linear mapping L : B 8 B + C!, B an involutive bialgebra, a linear functional ?,!J on B is called L-conditionally positive if W*b)

+ L(b* 8 b) >_ 0

for all b E B(O) = kerns.

Proposition 1 Let B be an involutive bialgebra. Let L : B 63 13 + C be linear and such that L satisfies (5) (i.e. L is a 2-cocycle). Then for an Lconditionally positive linear functional ?,!J on B

+

(a) (a,b ) = $(u*b) L(a* 8 b) defines a positive sesquilinear form on B(0)

(b) p(a)r)(b)= r)(ab) defines a representation of the involutive algebra B ( O ) on D

280

where D = B(O)/(, denotes the pre-Hilbert space obtained from B(O) by dividing by the null space of ( , ), and where r] : B(O) + D denotes the canonical mapping.

Proof: To prove (b), let b E B(O) with ( b , b ) = 0. Then for a E B(O),using (51,

+ L(b*a* €3 ab) = $(b*(a*ab))+ L(b* €3 a*ab)

(ab,ab) = $(b*a*ab) = (b,a*ab)

wnich shows (ab,ab) = 0 by Cauch-Schwartz inequality.

0

Now extend r] and p to B by putting r ] ( l ) = 0 and p(1) = 1. Denote by H the completion of D and denote by r ( H ) the Bose Fock space over the Hilbert space L2(&.) @ H. For u E D ,T E L(D), the annihilation, preservation and creation processes At (u), &(r) and A! (u) respectively are defined on the dense sub-space

of r ( H ) , where the big union is over all finitedimensional linear sub-spaces of D, by r ( E ) we denote the Bose Fock spaces over La(&) B E considered as a sub-Hi1bel.t-space of r ( H ) , and N is the number operator on r ( H ) ;cf. [4]. The quantum stochastic differential equations (QSDE)

with

WJ) = At(r](b*>) + At(P(b) - W)l)+ A;(r](w have, in the sense of [4], a unique solution on L(&(D)). Theorem 1 Let 23 be an additively deformed involutive bialgebra with pt = p*exp, (tL), and let $ be an L-conditionally positive, hermitian normalized linear functional on 23. Then, in the vaccuum state 9 on A = L(&(D)), the solutions jst of the quantum stochastic differential equations (6, 7) form a LBvy process on B. Moreover,

281

for all b E f?.

Proof The increment property and the independence and stationarity of increments follow from the general theory of [4]. We prove that jst is a *-algebra homomorphism with respect to the *-algebra structure on (B,*) given by the mutliplication pt-s i.e. that jst(ab) = j l r t ( a ( l ) ) j 8 t ( ~ ( l ) ) e ! t - s ) L ( a@( 2b(2)) )

(8)

The differential of the right hand side equals

1) e!t-s)L

d ( j s t (a(i)) j s t

+jsta(l)jStb(l)L(a(2)

(a(2)QD b(2)

@ b(2))dt-s)L(a(3)@ b(3))dt

(b(1))+ djStb(1)) j s t (b(1))

= (jStb(1)

(t-s)L

+djst(a(l))djst(b(l)))e*

b(2) @

fjSt(a(l))jSt(b(l))L(a(2)

QD

b(2))

b(2))ekt-s)L(a(3)@ b(3)Idt

= jSt(~(l))jSt(~(l))

(dAt (“(2) 4 2 ) > *1 + dAt (Pb(2)4 2 ) 1 - 6b(2)42)11) +dAZ(r)(a(2)b(2)))+ ?G3(2)b(2))dt) (t-s)L

e*

b ( 3 ) QD

43))

(t-s)L

= jst(a(l))jst(b(l))%

(a(2) @ b(2))d1t(a(3)b(3))

where we used quantum Ito’s formula and that L is commuting. Now equation (8) follows from the fact that both sides satisfy the same QSDE.

0 The following result of [5]establishes a 1-1-correspondencebetween (equivalence classes) of Levy processes on an additively deformed *-bialgebra f? with generator L and Lconditionally positive hermitian linear functionals on 8. Theorem 2 (J. Wirth [5]; Schoenberg correspondence) For a linear functional 1c, on B the following are equivalent:

(i) J?+! is L-conditionally positive, hermitian and normalized (ii) exp,(t$) is a state on (B, *, pt, 1) for all t 2 0 0 In order to control the possible additive deformations of a given bialgebra we need a description of the class of commuting 2-cocycles. The following is sometimes helpful.

282

Proposition 2 Let B and C be bialgebras. Let 19 : B homomorphism such that

+ C be a bialgebra

(19 @ id) o A = (19 Bid) o T o A.

(9)

Then for each 2-cocycle 1 on C the mapping

L = 1 0 (19 €3 19) is a commuting 2-cocycle on B.

Proof Since 19 is a bialgebra homomorphism L is a 2-cocycle on 13 if 1 is a 2-cocycle on C. It remains to show that L is commuting. However,

( L€3 p ) 0 (A6A) = (1 @ p ) 0 (id @ id @ p ) 0 (19 @ 1 9 B i d @ i d ) 0 T 0 (A €3 A) = ( I 8 p ) 0 (id @ id @ p ) 0 (tY@tY@idBiid)0 7 0 (..A) ~ = ( p a L) 0 (A6A) where we used (9).

( T O A )

0

It is easy to see that under the hypothesis of the preceding proposition 19(C) must be cocommutative. Condition (9) is related t o the center of a semi-group in the case of bialgebras of funtions. For if G is a semi-group, H a sub-semi-group of G, and f?, C denote the algebras of ‘regular’ functions on G, H respectively (see [3]), then with 19 the restriction t o H condition (9) says that H lies in the center of G. Examples 1. Universal enveloping algebras. Let 4 be a complex *-Lie algebra, i.e. a complex Lie algebra equipped with an antilinear self-inverse mapping z + z* such that [z*,y*] = -[z,y]*. The universal enveloping *-Hopf algebra U(6) of 9 is the tensor-*-algebra over the involutive vector space L d i v i d e d b y t h e r e l a t i o n s z @ g - y @ z = [z,y] w i t h A z = z @ l + l @ z and 6z = 0. Since U(G) is cocommutative each hermitian normalized 2cocycle on U(6) gives rise t o a deformation of U(4). For example, if 9 is the abelian *-Lie algebra structure on a *-vector space 6 then U(6) equals the symmetric tensor-*-algebra S(4) over 4 and a hermitian normalized 2-cocycle on S(6) is given by

c n

L(f, 9) =

i= 1

ddf)

m,f,9 E S ( 6 )

(10)

283

where di, i = 1,.. .n are derivations. Here a derivation on S(S) is a h e a r functional d : S(G) + C satisfying d u g ) = G(f)d(g)

+d ( f ) W -

In general, L will not be a coboundary, but it is positive. Suppose now that 0 is finite-dimensional which means that S(G) equals a polynomial algebra C[zl,. . .,zd]with hermitian elements 21,.. . , z d . Denote by 8i(f), a = 1,. . . , d , the ith partial derivative of the polynomial f evaluated at 0. Then (10) becomes d

W,g)=

C Qijai(fPj(g)

i,j=l

with Q a positive complex d x &matrix. Thus we can form the L6vy process with generator 0 on the deformation of C : [ q , . . ,zd]as the solution of the QSDE of Theorem 1 which in the special case of primitive elements simply becomes

xt(4- At(%) + A;(77i) where X i s = - h ( Z j ) and qi = ~(zi) E H for H = C7Q the Hilbert space obtained from the positive form Q. We have

[xii),x,(j)l=min(s, t

) ( -~Q ~~ ~~ )

which are the canonical commutation relations. For more examples of additive deformations of universal enveloping algebras of Lie algebras see [5]. 2. *-Semigroup algebras. Let G be a *-semigroup, i.e. a semigroup (with unit 1) which has also an involution g I+ g*, that is (g*)* = g and (gh)*= h*g*. The *-semigroup algebra CG of G is the *-algebra obtained

from G by passing t o formal linear combinations of elements of G. If we Put Ag=g@g sg = 1

this extends t o a cocommutative *-bialgebra structure on (CG. In the case of a group G we put g* = 9-l. Then (CG becomes a Hopf *-algebra with antipode S(g) = 9-l. Each L-deformation is a Hopf deformation with

284

For G = Z all 2-cocycles are coboundaries and for a hermitian normalized 2-cocycle L on Z there is a function X : Z + C such that

+

Z(k,m) = X(k m) - X(k) - X(m), k,m E Z X(-m) = X(m) X(0) = 0

The deformations are given by pt(k, m) = (k + m)et(X(h+rrs)--X(k)-X(m)).

For G = Z2 the second cohomology group is not trivial. The hermitian normalized 2-cocycle

i

L ((k,m ),(n,r))= -(mn 2

is not a coboundary. The deformation of

- kr)

(11)

UZ2given by this L is

pt((k,m),(n,r))= (k +n,m+r)eit("n-kr)

which yields the Heisenberg group. The linear functional

k2 + m2

+ ( k , n ) = --

4

is L-conditionally positive and we can construct the LBvy process with generator on the deformation of UZ2as the solution of the QSDE

+

dj8t(b) = jst dj,,(b) = G(b)id

(b)

where 1 It(1,O) = -dA;

fi i

It(0,l) = -dA;

a

1 - -dAt d2

i + -dA

fi

t -4

t

z

We put Ut = jot(1,O) and V, = jot(0,l). Then

u,ut

= utu,

V,& = V,v, V,V, = ei(min(8it))) and we obtain the discrete version of the canonical commutation relations in its Weyl form. The continuous version of this is created by starting with

285

R2 instead of Z2 and by taking the same hermitian normalized 2-cocycle given by (11).

3. Free matrix *-bialgebras. Consider the free *-algebra C(xk1,xi1) = c(xkl,xi1;k , l = 1, generated by 8 indeterminates *-bialgebra if we put

xkl,

d E

N.

*

9

d>

This space is turned into a

d

We set

In the same manner the free commutative *-algebra c [ x k l , x z l ] can be turned into a commutative *-bialgebra. Theorem 3 The commuting hermitian 2-cocycles L on (C(Zk1, xgl) are precisely given by

L ( M €3 N ) = 1 ( 6 ( M )€3 6 ( N ) )

(12)

for monomials M , N where 1 is a hermitian 2-cocycle on C ( x , x * ) and where d ( M ) for a monomial M = x Z l 1 . . .x&, means the monomial d(M)X" .. .X'" of q x , x * ) .

Pmof: The mapping x k l I+ bklx extends to a *-bialgebra homomorphism .9 from c ( Z k l , xZl) to C ( x ,x * ) . Moreover,

c d

(6 €3 id) 0 A X k l =

d ( x k n ) 8 Xnl

n=l

= x €3 x k l d

n=l

= (6 €3 id) 0 7 o A X k l

which shows that 6 satisfies (9). By Proposition 2 equation (12) defines a commuting hermitian 2-cocycle. - Now let L be a commuting hermitian

286

,...,ni

nl

=

C ,...

nl

L(x:l,l

x€i * * *

nili,x$181*-*x$j8j)xZnl

xEi

61

6.

kinixr,ml * * * x G m j

,ni

which forces L(X21,1

.-.X&i,X281

=dklll ..*xr:8j) 6.

* * . ~ k i l i 6 r i 8- *1 * ' & j 8 j

l(XC1

. . . xE',x61 . ..x6j)

for some function 1 of pairs of monomials in C(x,x*). Since L is a hermitian 2-cocycle this must also be true for the extension of I to a linear functional x*)EI C(x,x*). 0 on C(x, Non-commutative analogue of generated by the elements

4.

The *-ideal I in cC(xki,x;,)

ud.

d

n=l d

n=l We denote the *-bialgebra C ( X ~ ~ , X by ~ ~K(d). ) / I If we is a *-bi-ideal. start form the commutative free matrix *-bialgebra, division by the corresponding ideal will give the coefficient algebra of the group u d of unitary d x &matrices. Thus K(d)can be regarded as the non-commuative analogue of this coefficient algebra or of u d itself. For a monomial M = x x l l .. .zZln in K(d) we put

c n

[MI = S ( M ) ( n- 2

Q)

i

Theorem 4 The commuting hermitian 2-cocycles L on K(d) are exactly given by

L(M

N ) = ~(IMI, INI)

(15)

for monomials M , N where 1 is a hermitian 2-cocycle on Z.

Proof: The mapping 13 of Theorem 3 maps the relations (13) and (14)in K ( d ) to the relations xx* = 1 = x*x in K(1) = a%. Thus 19 gives rise

287

t o a mapping n : K ( d ) + CZ and n(M) = [MI. By Proposition 2 the mappings L of the form (15) are commuting hermitian 2-cocycles on K(d). - Conversely, if L is a commuting hermitian 2-cocycle on K ( d ) it can be lifted t o a commuting hermitian 2-cocycle i on (C(Z/EI,zil;k,I = 1,. . .,d). By Theorem 3 i is of the form i = l"o (I9 €319)with hermitian 2-cocycle ion C(z, 2;). Since respects the relations for K(d) it follows that I" respects the relations for CZ. 0

It follows that the deformed multiplications are given by p t (8 ~N ) = ~ ~ e ~ ( ~ ( I ~ I + I ~ I ) - ~ ( I ~ I ) - X ( J N I ) ) which implies that the solutions

ut = ( j t ( z k l ) k , l = l , ...,d of the QSDE (6, 7), in general, are not unitary. References

[l]A. Ben Ghorbal, M. Schiirmann: Non-commutative notions of stochastic independence. Math. Proc. Camb. Phil. SOC.133 (2002), 531-561 [2]U. Franz, R. Schott, M. Schiirmann: LCvy processes and Brownian motion on braided spaces. Preprint Greifswald 2003 [3]U. Franz, M. Schiirmann: LCvy processes on quantum hypergroups. Transactions of a JapaneseGerman Symposium, Kyoto 1999 (eds. H. Heyer, T. Hirai, N. Obata). Grabner, Altendorf 2000 [4]M. Schiirmann: White noise on bialgebras. Lect. Notes in Math. vol 1544. Springer, Berlin Heidelberg New York 1993 [5]J. Wirth: Formule de LBvy-Khintchine et dgformation d'algkbre. Thesis Paris 2002

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Infinite Dimensional Harmonic Analysis I11 (pp. 289-311) Eds. H. Heyer et al. @ 2005 World Scientific Publishing Co.

UNITARY REPRESENTATIONS OF THE GROUP OF DIFFEOMORPHISMS VIA RESTRICTED PRODUCT MEASURES WITH INFINITE MASS*

HIROAKI SHIMOMURA ( JOINT WORK WITH TAKESHI HIRAI) Department of Mathematics, Facuty of Education Kochi University Kochi 780-8520 Japan E-mail: shimomu0cc.kochi-u.ac.jp

Let M be a connected, non compact but a-compact, smooth manifold with d := dim(M) > 2, Diffo(M) be the group of all smooth diffeomorphisms on M with compact support, and p be a smooth locally Euclidean measure with infinite mass. Take a restricted product measure V E of the countably infinite copies of p assigning a family E = { E n } n of disjoint Bore1 sets in M which satisfies; 0 < p ( E n ) < +m W

11 - p ( E n ) l < +XI as in the context of Moore

and

[lo].

VE

is quasi-invariant

n=l

under the diagonal actions of Diffo(M). Our main aim in this paper concerns the irreducibility and equivalence of some unitary representations of Diffo(M), proposed by Hirai [4] which are considered to be irreducible components of the natural representations (Diffo(M), LEE (MOO)). The results obtained here are an extension of, and variation on, the work described by [20] on finite direct product space and on configuration space.

1. Introduction

Let M , Diffo(M), p and v~ be as defined above. Many unitary representations of Diffo(M) or its subgroups have been studied and constructed by various authors (cf.[1,2,4,5,6,7,8,11,20]). In particular an extension of the natural representation (Diffo(M), L:(M)) to the one on finite product measure spaces was first described in [20].Let us recall it briefly. For any n E N take a product measure v, of the copies p on M", *This research was partially supported by Grand-in-Aid for Scientific Research (No.14540167), the Ministry of Education, Culture, Sports, Science and Technology, Japan.

290

and take an irreducible unitary representation ( p , W ) of the permutation group 6, of { 1. . . ,n } . Then, the restriction of natural representation (Diffo(M), Lz, ( M " ) ) to a subspace gives an irreducible representation of Diffo(M), where H,,,, consists of all the functions expressed in the following form:

f(m) = p-l(a)f(z)

for

'U

E

B,,

under the notation X U := ($,(I), . ,~ ~ ( ~ 1 ) . In addition, in the same literature an infinite-dimensional version of this result is described via measures on the configuration space. Besides the corresponding group there to 6, is 6,, the infinite permutation group on N, but 6, works more implicitly than 6,. On the one hand, another infinite-dimensional version of the above result is considered by Hirai [4],which is based on the theory of infinite tensor product, with reference vectors attributable to von Neumann (cf.[12]). He constructed an irreducible unitary representation by means of E = {E,}, which are mentioned above, and which he calls p-unitals; by means of the corresponding product measures of PIE,; and finally, by glueing the L2spaces generated by the product measures together into an acting group 6, of all finite permutations on N. His reslults are important and interesting. However it seems that some corrections or more exact arguments to his work are necessary and that direct and clear discussions will be given through restricted product measures in place of the ones above. Therefore, in this paper, we concern ourselves with the unitary representations of Diffo ( M ) on the following representation spaces: Given p, E = {E,}, and an irreducible unitary representation (II,H ) of B,, we consider a Hilbert space %(C), C := ( p , E , l l ) , of all Borel measurable functions f on M" such that

f ( z a )= I I ( u ) - ' ~ ( z ) for and

Ilfll??,:=

'U

E

B,,

LE

Ilf(z)II$VE(dZ) < +O0,

where DE is a Borel set that satisfies: DEU n DE = 8, if u # id, and the complementary set of U,,EG~ DEU is YE-null under the notation DEU := {XUIX E DE}. As a suppliment of our discussions, we recall the following fact: Consider the natural representation T ( g ) of Diffo(M) on L z E ( M w )

291

acting from the left together with the unitary operators R ( u ) , u E 6,; R(a)f(x) := f(m)acting from the right, which is a unitary representation of 6,, as is easily seen. Then the system { T ( g ) , R ( a ) forms } a dual pair (cf [ 5 ] ) . Hence we claim that our representation discussed in this paper is the irreducible component of the natural representation ( T ,L;E ( M " ) ) along the dual pair. However this compelling conjecture has never been proved. In any case we will justify irreducibility and equivalence in section 3 and 4, respectively, as each requires a substantial argument. In particular, we will need a useful lemma (listed as Lemma 4.1), that has many applications to an ordinal check of the irreducibility of natural representations of the normal type. In addition, to complete the proof, we need another lemma (listed as Lemma 4.3) that states the possibility of incompressive transportation of a mass from one part to another part of an open, connected set. However, by the techinical reasons, we have to impose a condition (mcc) on our manifold M , though it is fairly mild (perhaps we will be able to show our results without (mcc), but it seems that they will require a lot of arguments, and in addition, we do not know whether (mcc) generally holds or not for non compact, connected manifolds). Finally, as a rule we will omit the easy proofs of the propositions made in this paper, but t o those which are found hard we will prove them completely. 2. Basic Discussions 2.1. Restricted product measure with infinite mass. We begin by introducing the notion of restricted product measure. Suppose that we are Bn,Pn) and a set E n E Bn given, for each n, a a-finite measure space (Xn, with 0 < Pn(En) < +GO. Denoting the restriction of pn to En by PnlEn, we form a product measure:

~ lim Since fin is increasing for n, we have a a-finite measure f i := n+m the product measurable space ( X " , Bm)of (Xn, B,,).

-

fin

on

Proposition 2.1. Take a sequence {Fn}nwith the same poroperty as that of {En}n.T h e n f i is ~ equivalent t o f i ~ (& f i ~ ) ,if and only

292

where 8 denotes the usual symmetric dif-

4 +m,

follows from the above conclusion, with

ference. Moreover,

In what follows, all the measure spaces (Xn,!Bn,pn) are identical: ( M ,% ( M ) ,p ) , where M is non compact but u-compact, manifold of class C", B ( M ) is the Borel field and p is a smooth, locally Euclidean measure whose total mass is infinite.

Definition 2.1 (Unital sequence) A sequence E = { E n } n of Borel sets in M is said to be p-unital, if it satisfies the following two conditions: (1) ' n , O < p ( E n ) < +m, w

C 11- p(En)I < +m.

(2)

n=l

In addition, i f the En are mutually disjoint, we call it a disjoint p-unital sequence.

n 00

Let us agree that the product

En, which is called a unital product

n=l

set, is denoted by the same letter E , so that no confusion occurs.

Definition 2.2 (Cofinality) Two p-unital sequences, E = { E n } n and 03

F =

{Fn}n

are said t o be cofinal if they satisfy

C p ( E n 8Fn) < +w, n=l

and this is symbolicly denoted by E F. In addition, i f En = Fn f o r suficiently large n, they are said t o be strongly cofinal, and this expression is denoted by E sz F. N

Take a p-unital sequence E = {En}n and form the restricted product measure CE. Because the infinite product of { p ( E n ) } , absolutely converges, 00

p ( E n ) C ~ has a meaning as a measure in M w ,and it depends

YE := n=l

only on the equivalence class of the sequences cofinal to E. It is easy but important to observe that UE(M&) = 0, where we

n 00

put

M E := U;=,(Mn

r(u); x tion.

-+

x

&) and that the action u E 6, on M"

:

n+ 1 X(T

leaves ME invariant. We now introduce one more defini-

293

Definition 2.3 (o-ring) Let E be a disjoint p-unital sequence. (1) The cr-ring generated by all unital product sets F with F N E ( with F M E ) is denoted by M ( p ,E ) ( M " ( p ,E ) ), respectively. (2) Similarly, the cr-ring generated b y all unital product sets F that come from the disjoint p-unital sequences with F N E is denoted by M d ( p ,E ) .

Lemma 2.1. Assume E = {En}nbe a disjoint p-unital sequence. (1) ME E M ( P ,E ) . (2) M ( p , E ) = M " ( p ,E ) mod VE. (3) M ( p ,E ) = M d ( p ,E ) mod VE. Proposition 2.2. Given a disjoint p-unital sequence E = {En}n and a B,-invariant set B E M d ( p , E ) , that is Ba = B for all a E 6,, there exists a set 13 E M d ( p ,E ) such that

B=

Do

: disjoint

union.

UEB,

Corollary 2.1. Given a disjoint p-unital sequence E , there exists a set DE E M d ( p ,E ) such that 'cr#id,

D E U ~ D E = $and

DEU

ME=

modv~.

UE6-

We end this subsection with the equivalence-singular dichotomy between the measures V E . Proposition 2.3.

W e have YE

= VF

or

VE

IVF

for any p-unital sequences E = {En},, and F = {Fn}n. 2.2 Action of Diff,(M) from the left and of 6, from the right. Let g E Diffo(M) and cr E 6 , act on M m in the following manner: g(Z1,.,xn. * .

,

(21,... z n

*.

) + (9x1,... , g z n , ... 1,

*)a + ( x u ( ~ ) .,* . 7 xu(n), *).

Clearly the actions of g and of o are mutually commutative, and they lead to transformations of V E , which will be denoted by gvE and U U E ,respectively.

294 Theorem 2.1. Given a disjoint p-unital sequence E = { E n } n , (1) U E is 6,-invariant, and

is Diff 0 ( M )-quasi-invariant. More precisely, the Radon-Nikodim derivative is expressed in the following form;

(2)

UE

where the infinite product converges in the L:,-sense

n

on each set

00

Bx

Ek ( B € '13(Mn0), p ( B ) < +ca, and no is arbitrary).

no+l

(Of course the above convergence is equivalent to Lz, -convergence for the square roots of the corresponding functions.) By the above theorem, we have two unitary representations, T ( g ) , g E Diffo(M) and R(a),a E emon L;,(Mw), such that

The representations T and R are mutually commutative, and moreover they form a dual pair (cf. [ S] ) . 2.3 Representation space R(C). Using the same notation for p and E as before and taking an irreducible unitary representation ( I I , H ) of B,, where H is the separable representation Hilbert space, we put C := ( p , E,II). Next, take a Borel measurable H-valued function f on M a which possesses the following property:

f ( x a ) = II(O>-' f (x) for t/a E em. Put

llf112 :=

/

Ilf(x)ll%'E(dx>*

(2.1)

(2.2)

DE

Then, the space R(C) consisting of those functions f that fulfill llfll < 00 forms a Hilbert space by the above norm. It is useful to note that, for each f E R(C), the integral domain D E may be replaced by an arbitrary Borel set D that possesses the following two properties:

295

(2) f = O

(

on

c

Dn)'

mod YE.

0€6-

It follows that the following action T (denoted by the same letter, but no confusion ocurrs) of Diffo(M) on X ( C ) is well-defined and ( T , X ( C ) )is a unitary representation;

T ( g ) :f ( x )+

Jdg""cx) f (9-lx). dYE

The continuity will be addressed in the later part of this section. Now take any disjoint p-unital sequence F = {Fn}nthat is cofinal to

QF : L:,(

E. We introduce a map

u 00

F n , H ) -+ R(C) such that

n=l

n

co

00

We recall that vEl Q F ( f ) = 0 on (

c

Fn = ( p x x p x ...)I Fn and agree that n=l n=l Fo)'. In addition, it follows easily from the remark -

1

.

U€6,

above that

is an isometry.

Lemma 2.2. The space spanned by QF(LE,(F,H)), F runs through the family C ( E ) of the disjoint p-unital sequences that are cofinal to E , is dense in X(E). Lemma 2.3. Assume that d := dim(M) ing condition.

> 2 and let M satisfy the follow-

(CC) There exist a sequence {U,}, of relatively compact, open sets, U, t M such that E' is connected for every n.

Given a disjoint unital sequence E = {En}n7 there exists G = (Gn}nE C(E) that posesses the following properties:

(F)'

(1) V n , G, is a relatively compact, open, connected set and p ( C \ G n ) = 0 , (2)

'n,

is connected,

= 0, if n # m ,and (3) G,nG, (4) given a compact set K , there exists N K E N such that K

n G,

=

0

296

for all n 2 N K . 03

Moreover given any E

> 0, we can take G such that

p(En8 Gn)

< E.

n=l

Remark 2.1. Let B be an ope set in Rd (d 2 2) which is surrounded by an outer large sphere and by an inner small sphere. The manifold B does not satisfy (cc). Let Co(E)denote the family of G = {Gn}nE C(E)possessing the properties from (1) to (4), which are listed in the preceding lemma.

Lemma 2.4. Under the same assumptions in L e m m a 2.3, the space spanned by { & ~ ( L ~ , ( FH,) ) } F ~ c o ( E is) dense in X ( C ) . Now we discuss the continuity of (T,'H(C)).Let us recall that Diffo(M) K } , where K runs is a union of Diff(K) := {f E Diffo(M)I suppf through the compact subsets in M. Diff(K) is equipped with the group toplogy 7 K derived from uniform convergence of the maps together with their all derivatives. We introduce the inductive limit topology r M of { r K } K to Diffo(M), though it is not a group toplogy (cf. [19]).

Theorem 2.2. Assume that d > 2. The unitary representation ( T , X ( C ) ) of Diffo(M) is continuous with respect t o r M . We end this section with concerning the following condition that will be imposed on M from now on, which is clearly weaker than (cc). (mcc) There exists a closed set S in M such that p(S) = 0, M \ S is connected and satisfies the condition (cc). (mcc) is fairly mild, and usual non compact, connected manifolds (of course the manifold B described in Remark 2.1) satisfy this condition. The next theorem is easily seen, but basic for our discussions.

Theorem 2.3. Assume that d > 2, and let M satisfy (mcc). Put M' := M\S, and denote X ( C ) for M and M' by X M ( C )and X M I(C), respectively. T h e n the unitary representations ( T ( g ) ,'Htln/r(C)) and ( T ( g ) ,X M (C')), ~ where g E Diffo(M'), C := ( p , E , l l ) and C' := (pIM',E n M',II), are unitary equivalent by a n intertwinig operator A,

n 03

A :

f(2)

E ' H M ( C )-+

k=l

X M ~ ( Z ~ ) ~E( XX M ) ~(C').

297

3. Irreducibility The main goal of this section is to prove the following theorem.

Theorem 3.1. Let E be a disjoint p-unital sequence, (II,H ) a n irreducible unitary representation of em.Form a triplet C = (Elp,II), as before. If dim(M) 2 3 and M satisfies the condition (mcc), the unitary representation ( T ,X ( C ) ) is irreducible. The proof consists of the succesive steps as below. First we need the following lemma.

Lemma 3.1. Let G = {Gn}n be a disjoint p-unital sequence, and further assume that G, is connected and open for each n. Then, the natural * representation T of the restricted product group Diffo(Gn) on L:G (G),

n n

defined below, is irreducible.

Next we go to the second stage that describes the successive steps. As before, we can assume that E = {En}npossesses the properties in Lemma 2.3. Put Em,n

.-

(q

(which is a connected open set in M),

k=n+l

En := UoEB,

n

((Em,nIn

Ek)g,

n+l (which is a symmetric set in M") and, pn :f E x ( C ) + XB,

*

f E x(C),

(which is an increasing projection tending to id). In addition, take any non empty disjoint connected open sets GI, . . . G, in Em,n,and form a unital sequence

G E := {GI, * .

*

,Gn, En+' ,

* * *

,Ek,. } * *

and a map Q g E . Now let A be an intertwinig operator of ( T , X ( C ) ) . It is not hard to see the following results.

298

(1) Image(P,AQzE) C Image(QzE) (because of the irreducibility described in the above lemma). (2) P,A&EE = &gE((id)qG,, 63 3 A n , ~ where ), A,,G is a bounded operator on H (due t o the same reason and to the irreducible assumption on

(ITH I ) . (3) A,,G does not depend on a particular choice of ( GI,. . . ,G,), so we simply write A, := A,,G (due to the connectedness of E,,,). (4)

' 0

E 6,, II(n)A, = A,II(c).

(5) vn, A, = An+l(=: A,), some calculations). (6) 3c E C , A,

(Both (4) and (5) are easily shown by

= c id (merely because of the irreducibility of (II,H ) ) .

(7) PnAQgE = cQzE (due to (2)). (8) P,AP, = cP, (because {Image(Q",))G, G := (GI,... ,G,) generates the space P,(X(C)), whenever G runs through all possible pairs of sets in (Em,,),). This is a rough sketch of the proof of Theorem 3.1. 4. Equivalence

This section is devoted to a study of the mutual equivalence of ( T , X ( C ) ) and lets E run chiefly through p-unital sets. Theorem 4.1. Assume that dim(M) >_ 3 and let M satisfy the condition (mcc). Given C1 = ( p , E,IIl) and C2 = ( p , F , I I z ) , ( T , X ( C 1 ) )and ( T , X ( & ) ) are unitary equivalent, if and only if (1) there exists a permutatation a on N (maybe an infinite permutation) such that E Fa-' and is equivalent to "II2 defined by (2)

-

"rII,(a):= rIz(a-laa)

for

v0

E6 .,

First we prove sufficiency. A space transformation defined via a; x + xu is a measure-preserving map between vFa-l and v ~ thus ;

Proof.

299

it induces a unitary operator from L:Fo-l(Mw) to L:,(Mw); f(x) + f ( x a - l ) . Furthermore, it also induces an intertwining operator A from (T,?l(Ei)) to ( T , X ( & ) ) , where Ci := ( P , F ~ - ~ , ~=I (IP~, )E , ~ I I ~ ) . By the assumption there exists an intertwinig unitary operator U from (aI12,H2) to (HI, H I ) , so ((id )p E €3 U ) o A-l gives a desired operator 0 To prove necessity, as before, we may assume that M possesses (cc). We need many lemmas. Lemma 4.1. Let M satisfy (cc), and suppose that ( T , X ( C 1 ) ) and (T,?t(&))are unitary equivalent and let A be a n intertwining unitary operator; A : X ( C 1 ) --+ X ( C 2 ) . Given a Bore1 set B E B ( M ) , we introduce a projection PB o n X(&) (i = 1,2) by

(PBfI(2):= l - K l X B ( 4 m . Then, we have

APE = PEA. Proof. The proof of this lemma is somewhat long and complicated owing to the parts concerning infinite-dimensional arguments. Thus it would be better to observe its finite dimensional version and to show how to prove the main parts. So suppose that we have as given a relatively compact, open set D. Given ~ 1 take , a compact subset K of D such that p(D\K) < 71 and cover K with a finite collection of relatively compact, open sets {Wt}r=l that are diffeomorphic to disks in Rdim M:

K

c urz1Wt

D.

Without loss of generality, we may assume that the image of p(Wt by the coordinate map q5t : Wt --+ Rdim is the restriction of the Lebesgue measure to q5t(Wt)and that p(wt\Wt) = 0. Put

v1 :=W1,

V,:=Wt\(W1U...UWt-1)

(t=2,... ,T).

Then V, (t = 1 , e . e , T ) are mutually disjoint, open sets,

p(UT=lWt\ UT=, V,) = 0 , and hence p(K\ UT=, I$) = 0. Given

r]2

> 0, take an open set Ut such that Ut C V, and p(V,\Ut) < 59 (t = 1 , . . . ,T ) .

300

Moreover, for a given 0 such that

< a < 1, and for each t , take

(j:>q)-l(x) = ax on

j,"Iv

E Diffo(4t(Wt))

+t(Vt).

Finally, put

n T

g:lq := 4;'

o j:lv o q5t,

and ga?l?:=

g:'q.

t=l

Then, we have ga>vE Diffo(D) and

Hence, letting first, a -+ +O and second,

771, r/2

-++0, we get

Thus formally we have

(%771,772

--++O).

Hence, it follows from a trivial equality

< T(ga'v)A$,A4 >=< T(ga>"+, 4 > that we get

< P D ~ A ~ , >=< A + P D C ~>, ~ as the limit when a, r/1,7~tend to 0, and PO,A = APD.. Now we can replace D" with a general Bore1 set B in Rd by standard arguments in measure theory. The proof of Lemma 4.1 will be carried out in a similar way, but by using infinite-dimensional techniques.

Lemma 4.2. Under the same assumptions and the same notation as in Theorem 4.1 and the assumption (cc) o n M , the following holds: f o r suficiently large n E N, there exists a(n) E N such that lim p ( E n 8Fc(n))= 0.

n+m

301

Sketch of the Proof. As before, we may assume that E = {En}nand F = {Fn}npossess the properties expressed in Lemma 2.3. Take vh E H1 with ilhllHl = 1 and put 03

d x ) := AQ",'(II

XE,

h)(x).

@I

n=l

Applying the above lemma to B := Ei for each k , we have PE$

= 0.

Next, approximating g with a sum of QF-image of tame functions such that M

QFM z i ,

* * *

,x i )

I-J

. h'),

X F , (2,)

1+1

where p is a square summble function and h' E Hz, we find that

n 00

'E

> 0, ' N ,

EN

s.t., ' k 2 N,,

p ( E i n F,) < 6 .

n=l

It follows that

3 4 N E N,

P(Ei n F o ( k ) )

< CKU,

with a universal constant Ku. By proceeding in a similar fashon, but changing E to F , we can show that p(Ek n F:ck,) is so small, if Ic is so large. 0

For the proof of Theorem 4.1 we need more analysis on M . The following lemma is quite useful for our discussions, and it shows the possibility of incompressive transportation of a mass from one part to another part with slim tubes in a connected open sets in M . Lemma 4.3. Assume that d := dim(M) 2 2. Let F be a connected, open subset of M and Ui (i = 1,2) be open subsets of F such that U1 n Uz = 0 and p(U1) < p(U2) < +oo. Then, given E > 0, we have a p-preserving diffeomorphism ge E Diffo(F) and a Bore2 subset B, c U1 such that

P(Ul\&)

< E and

!A(&)

c u2.

The Proof of this lemma is based on the local considerations that are guaranteed by the next lemma.

Lemma 4.4. There exists a Lebesgue measure preserving daffeomorphism with a compact support that realizes local displacement in Rd (d 2 2).

302

Proof.

Put n := d - 2 and take compact intervals

[Q',P'l c [a,PI,

[^/',6'1 c [Y,61, and

We take a vector field

defined by

wi E 0 (i = l , . - . n ) , where fi, f2 and gi (i = 1,. - . n )are C" functions on R1with a compact support such that

fl(z)= 1 on [a',p'], = O

on [a,/3]',

It is clear that

n

v =(i,~ 0 , ,. . . ,o) on T := [a',p'] x [TI,6'1 x n [ a ; , & l . i= 1

Therefore, exp(tv)(zo) = zo provided that zo morphisms.

+ t(1,0,0,..- ,0)

+ t ( l , O , 0,.. . ,0) E T.

for all

20

E T,

This is one of the desired diffeo0

Lemma 4.5. Under the same assumptions and the same notations in Lemma 4.2, we have

where the summation is taken over the indices except for a finite number of the n 's.

303

Proof. We continue to assume that E = {En}n and F = {Fn}n possess the properties in Lemma 2.3. For each n E N put 00:

a n := maxp(Ek fl Fc(n)), k#n

and GZ :=

Diffo,,(FO(k)) k=n

(: restricted direct product), where Diffo+(Fk) is a group of all p-preserving diffeomorphisms in Diffo(Fk). It is easy t o see that T(g) ---+ id, g E Gg strongly in 3t(&), as n --+m. It follows from the assumption that for a fixed unit vector h E Hi

Take sufficiently large n and m > n. As below, we choose E{ and Fj, and then arrange them in the following manner. First, put Ei := En+l,F; := Fc(n+l), and Ei is one of the Ek's that attains the maximum value of p(Ek n Fc(n)), k # n. Let Fi be the corresponding set to E i . Namely, if Ei = E k for some k , we put Fi := F,,(k). Going these procedures, we have the following sets of class E and F:

E: , F; ,E;, F211.. * ,EAl,FA1,EL, where EL(= Ep) is the first appearing set that posseses the properties: 5 n or p > m, or the property that Ep coincides with a set Et already appeared in this line. Next, we begin the second line with Ef(= E k ) that does not appear in the first line and that has the minimum index k , n < k 5 m. In a similar manner we end the second line with EL(= E p ) , where Ep is the first appearing set in the second line that possesses the properties: p 5 n or p > m, or the property that Ep coincides with a set E{ ( j = 1,2) or EL already appeared in these lines. Continuing these processes, we have the following sets of class E and F:

p

E:, F;,. . . 7 EA1,FA1,EL

.....................

304

E: ,F,",. . * ,EL, ,FL,,EL, where ELl is the last set choosen from {En+1,.. ,E m }by this procedure. Now we apply Lemma 4.3 to the sets F j ( = F ) E: n F j ( = U2)and F{ n E!+l(= U1) (i 1 = 2 , . . . , n j , or oo), and get a diffeomorphism 91 E GE with an arbitrary small E { ( = E ) (If p(F{ n E!+l) = 0 , we take the identity map as 91). Pu t

+

EL).

a{ := p ( ~nj E!+~)(i < nj), afj := p ( ~ ni ~

By the proof of Lemma 4.3 (but, here it is omitted), we see that 91 leaves invariant the other parts, namely F j n (E! u E!+l)c,except possibly small subsets with p-measure. From now on till the end of this proof, we will carry the proof through somewhat rough arguments, because the exact estimations are highly complicated and they disturb to see the essential parts. step 1. First we take the sets Fii-lwith odd indices in each line and work only these gii-l. In addition, if for the final set FAi in each line, ni is an odd number, we omit g&. It is evident that a map g composed of these gii-l is in GE. Let us estimate the following value: pg

:=

1fi

reem

n 03

XEk ( z k )

k=l

X g - l E , ( & ) (zk)vE(dz)-

k=l

It follows from the choice of gii-l that we may neglect the terms in pg except for the terms corresponding to a product of transpositions I- = ( p ,q), where

n 03

Ep = Eii-l and E, = Eii. Thus, p,/ than a product of the following terms:

Since

1

pg > -

n

2 k=l

p(Ek) is approximately smaller

k=l

p ( E k ) for large n,m, it follows from an elementary in-

equality: 1- t

5 - logt for all t E

(0,1],

305

that we have

This demonstrates that

for large n and m. step 2. In a similar fashon, taking only g& correponoding to even numbers, we have

c

a;i

< 2 log2.

id

+

step 3. Consider E L that is expressed as Eq (n 1 5 3q 5 m) and that does not coincide with any Ehk ( k = 1 , . . . ,I ) . Suppose that

Ej = EG = . . . = Ejk 00. We work only gel, . . . ,g k k and set gp,i := g g l these gp,i and form a map g .

n

0.’ ’ o

g k k . Let us compose

0

This time, the term pg/

P(Ek)is approximately smaller than a prod-

k=l

uct of the following terms:

It follows from the same arguments as before that for large n and m we have

which leads t o

cak,,.<

2log2 for large n and m.

P>S

+

step 4. Consider E L that is expressed as Eq ( n 1 5 j q 5 m ) , and that coincides with some E t k ( k = 1,.. . , I ) , and form a class & of all such EL. Let E L E E . Then E L = EAj for some j < Ic. Again, if E L E E , we have E L = EAi for some i < j. Continuing these processes, we finally

306

reach t o a set EEp such that EI$ 4 E . Thus, we are led to a tree diagram: The first(top) stage consists of the above EKp's. The second stage consists of all EAj such that E L = EEp,and then we connect Egp and ELj by a segment, and so on. Taking only gLj corresponding to the set ELj (e Fij) in the even stage, and composing a map g of these g A j , we see that the summation over the terms p(Fij nEE,) is smaller than 2 log 2. A similar result is obtained from the odd stage. step 5. Finally, consider E L which is expressed as Ep (p 5 n or p > m). Let HI, . . . ,H , be a family of the maximal (disjoint) sets of the above ones: H a. -- E2.i = . . . = EGi.' 7ai.i We take g::i,4,. . . ,gnYsi,; (i = 1 , . . . , q), and compose a map g of these maps. It follows from the same discussions as before that

a,k

for large n and m. Since n and m are arbitrary, provided that they are so large, we have < +co. 0

Can n

Lemma 4.6. Under the same assumptions and the same notations in Lemma 4.2, we have

where the summation is taken over the indices except f o r a finite number of the n 's. Proof. We continue to assume that E = {E,}, and F = {F,}, possess the properties in Lemma 2.3. As before, put a, := maxkf, p(Ek r l Fu(,l). 1 Take any positive number 6 such that S < min(-,inf p(Ej)), and take an 4 3 0

integer JO such that

aj j=Jo

< -. 2

By the assumption on E and F , there exist J1, J2,J3 E N such that JO < J1 < 5 2 < J3 which possess the following properties:

307

Take integers p and q such that J 3 5 p < q, and take all of Fi which intersects at least one of Ep +l,. . . ,Ep. Evidently we have i 2 J z . Further, we add all of other El that intersects at least one of these Fi ( I 2 J1). We may assume that Fi = F r ( ~(i<= ) l , . . ., r ) , and hence Ki 2 J1. Now let Es1, *

* *

,ESm

(s1

< . . . < sm)

be a list of all Ei which intersect to F r ( ~n , )Ek, with positive p-measures. If necessary, cutting off small parts of E,, (k = 1,.. . ,m ) we may assume that there exists a unit u (depending on K1) such that P(Es, n F r ( K l ) ) / U =: Nk E N. We divide each ESkfl F,,(K,) into Nk pieces with p-measure u and then number them as E k , l , . ,E k , N k . Now applying Lemma 4.3 to these pieces, we have maps

n

M *

gl E GJ, :=

1 = 1;-. ,rn ax (N l,-.. ,Nm) such that

Diffo,p(Fj),

j = Jo

gl(Ek,l) C Ek',z approximately holds (except a subset of E k , l with an arbitrary small p-measure), where E k , l satisfies 1 5 Nk, and k' is to be the smallest integer such that k' > k and 1 5 Np . If such E k t .1 does not exist we take EK,n F r ( ~ in , ) place of E k l , 1 in the above inclusion reration. On the other parts gE acts as the identity map. Pu t g1 := n g l for F r ( ~ (, If we have no such E,; with the above properties for F r ( ~n, )Ek, , we put g1 := id). For each 1 = 1,. . . ,max(N1,. . ,Nm), let Ml be the largest integer of M such that 1 5 NM. We remove these EM,,[and gl(EMl,l) (C E K n ~F r ( ~ from E K and ~ E,, , . ,E,,,, . Going on the same procedures for F r ( ~ ;(i) = 1 , 2 , . . . ,T ) , we have the maps g l , . . . ,gp E Gj,, and a new unital sequence {Ek}n (Ek C En). Put +

g := g1 0 . . . og', Clearly we have

/3j

and

= 0 for all j

/3j

:= p (Ej\Ei)

for each j .

< 51, and T

T

M

308

In addition, on the estimation of the term

/n M

DL := rEem

n 00

X E ; (zk)

k=l

Xg-'E:(,) (zk)vE(d%)

k=l

we can neglect the terms corresponding t o all nation from E = {En}n. By elementary calculations, we have

k=l

T

k=l

# id in virtue of the elimi-

k=l

1 where we assume that p ( E j ) > - for all j 2 Jo, and K1 is a universal 2 constant such that 1 -1ogt 5 K ~ ( I t ) for - < ' t 5 1. 2 Let €g

:= IIT(g)QE(XE @ h) - QE(xE @ h)ll,

for an arbitrary but fixed unit vector h E HI. Of course S whenever JO --+ 00, and set

U

cg

+~ 0,

Then, M

k=l

Y

and hence we have

It leads to

Since E is so small, if necessary, by taking large JO and small 6, we can assume € 1 < -. 2 2 P(Ek)(l - 2K16)

nE1

~

309

It follows from some calculations that

On the other hand, we have n(gEk n Eck) < n(gEk\gE'k) + »(gE'k n Ekc) = »(Ek\E'k) + »(gE'k n E'k). Thus, n£

Since

we get

El) <

» + .infj, ^(Ej) .1 . - dy ,} - 9y 2/iid)

=:?? In particular,

k=p+l v

For p + 1 < fc < q we have Ek = Ekn Fa(Kl} + --- + Ekn Fa(Kr) + Ek n Fc, oo

and A = Ki0 for 1 < 3z0 < r (Recall that F = V Fk). In addition, for any k=l

i 7^ i0, we can assume that


and

^(ff£;fc n Eg) > ]£ /z(^ n F ff(lfj) ) = MEfc n ^ (jt) ) - ^(£fc n FC). i^io

Therefore, we have

k=p+l

k=p+l

310

c 0

Since

p(Ek

n F " ) < +oo and p , q is arbitrary, provided that

J3

5 p < q,

k=l

we conclude that n

Finally, we have n

n

< +oo.

n

n

0

Proof of Theorem 4.1 continued. The rests of the proof are that g is extended to a permutation on N and to show the equivalence relation of the irreducible unitary representations (II,H ) . However, since the space is lacking for a full explanation and these proofs are rather easier than those what we have seen, we will omit them. We end this issue with the following remark, which is seen by the above discussions. Our results obtained on the irreducibility and the equivalence also valid for any disjoint unital sequences whose cofinal equivalence classes including other sequences possessing the properties described in Lemma 2.3 in any non compact, connected manifold M with dim(M) 2 2, even if M does not satisfy the condition (mcc). References 1. G.A.Goldin, Non relativistic current algebra as unitary representations of

groups, J.Math.Phys. 12 (1971) 462-488. and D.H.Sharp, Non relativisitic current 2. G.A.Goldin,J.Grodrick,R.T.Powers, algebra in the N / V limit, J.Math.Phys., 15 (1974), 217-228. 3. T.Hirai, Construction of irreducible unitary representations of the infinite symmstric group Boo,J.Math.Kyoto Univ., 31 (1991), 495-541. 4. T.Hirai, Irreducible unitary representations of the group of diffeomorphisms of a non-compact manifold, ibid., 33 (1993), 827-864. 5. T.Hirai and HShimomura, Relations between unitary representations of diffeomorphism groups and those of the infinite symmetric group or of related permutation groups, ibid., 37 (1997), 261-316. 6. R.S.Ismagilov, Unitary representations of the group of diffeomorphisms of a circle, Funct.Anal.Appl., 5 (1971), 45-53 (= Funct.Anal., 5 (1971), 209-216 (English Translation)).

311 7. RSIsmagilov, On unitary representations of the group of diffeomorphisms of a compact manifold, Math.USSR Izvestija, 6 (1972) 181-209. 8. R.S.Ismagilov, Unitary representations of the group of diffeomorphisms of the space Rn, n 2 2, Funct.Anal.Appl., 9 (1975), 71-72 (= Funct.Anal.,S (1975), 154-155 (English Translation)). 9. R.S.Ismagilov, Representations of infinite-dimensional groups, Transl. Math. Mono, 152, Amer.Math.Soc., Providence, R I (1996) 10. C.C.Moore, Invariant measures on product spaces, 5th Berkeley sympo. Univ.of California Press, 2 part 2 (1967) 447-459. 11. Yu.A.Neretin, The complementary series of representations of the group of diffeomorphisms of the circle, Russ.Math.Surv., 37 (1982), 229-230. 12. J.V.Neumann, On infinite direct products, Compositio Math., 6 (1938) 1-77. 13. H.Omori, Infinite-dimensional Lie groups, Transl. Math. Mono, 158, Amer.Math.Soc., Providence, RI (1997). 14. E. Shavgulidze, Mesures quasi-invariantes sur les groupes de diff6omorphismes des variht6s riemaniennes, C.R.Acad.Sci., 321 (1995) 229-232. 15. H.Shimomura, Poisson measures on the configuration space and unitary representations of the group of diffeomorphisms, J.Math.Kyoto Univ., 34 (1994) 599-614. 16. H.Shimomura, Ergodic decomposition of probability measures on the configuration space, ibid., 35 (1995) 611-630. 17. H.Shimomura, 1-cocycles on the group of diffeomorphisms, ibid., 38 (1998) 695-725. 18. H.Shimomura, Quasi-invariant measures on the group of diffeomorphisms and smooth vectors of unitary representations, J.Funct.Anal., 187 (2001) 406-441. 19. N.Tatsuuma,H.Shimomura and T.Hirai, On group topologies and unitary representations of inductive limits of topological groups and the case of the group of diffeomorphisms, J.Math.Kyoto Univ., 38 (1998) 551-578. 20. A.M.Vershik,I.M.Gel’fandand M.I.Graev, Representations of the group of diffeomorphism, Usp.Mat.Nauk., 30 (1975), 3-50 (= Russ. MathSurv., 30 (1975), 3-50). 21. Y.Yamasaki, Measures on infinite dimensional spaces, World Scientific (1985).

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Infinite Dimensional Harmonic Analysis I11 (pp. 313-324) Eds. H. Heyer et al. @ 2005 World Scientific Publishing Co.

AN APPLICATION OF THE METHOD OF MOMENTS IN RANDOM MATRIX THEORY

MICHAEL STOLZ Ruhr- Universitat Bochum, Fakultat fur Mathernatik N A 4/30 D-44780Bochum Germany E-mail: [email protected] The Method of Moments says that under certain circumstances it is possible to prove the weak convergence of probability measures by establishing the convergence of their moments. The present paper reviews the fundamentals of this method insofar as they are relevant for the proof of multivariate Central Limit Theorems. As an application, Central Limit Theorems for the joint distributions of the traces of different powers of random matrices from the compact classical groups are proven.

What is sometimes called ”Method of Moments” (cf. Billingsley’ , Diaconis3) is the systematic use of the fact that under certain circumstances it is possible to prove the weak convergence of probability measures by establishing the convergence of their moments. Although this method is largely folklore, it is surprisingly difficult to find a systematic presentation of it which also covers the multidimensional case. Therefore it seems worthwhile t o provide a rigorous introduction to the method with applications to multivariate Central Limit Theorems in mind. This is the content of Section 1. Those who are interested in the moment problem per se may wish t o compare the deep and far-reaching results obtained by de Jeu6. In Section 2 the method is applied to derive Central Limit Theorems for random vectors which are constructed from random elements of the compact classical groups as follows. For n E N let K , be one of On, Sp, or U,, i.e., the compact group of orthogonal, symplectic or unitary n x n-matrices, respectively. (In the symplectic case we have to assume that n = 2m be even.) Let r n be a random variable with values in K n , and suppose that the distribution of r, is given by Haar mesure. In the first two cases, for

314

any

T

E

N,we consider the R'-valued random vector

(Tw),l-w2),.. . ,Wr")), whereas in the last case we consider the C2"-valued random vector

(n(r), . . . ,n(r"),W,. . . ,q r q ) . It will be shown that these random vectors tend in distribution to a normal limit as n + M (resp. m --+m in the symplectic case). 1. Moments and Weak Convergence

In this section we will introduce moments of probability measures on the euclidean spaces and see in which way they are related to weak convergence. Let M1(R', D') (T E N) denote the set of all probability measures on the measurable space (R", a"). Let a = (al, . . . ,a,) E NL,and set

and

n

T

and

exist and are finite. aa(p) is called the a-moment of p, Pa(p) the absolute a-moment of p.

If X = (XI,.. . ,X,): (0,-4, P ) + (R',B") is a random vector with distribution Px E M:(RT,Dr), then we set

a a ( X ):= cy,(Px).

(5)

315

Note that

The absolute moments of a random vector are defined in an analogous manner. cp :

Ng + R is said to be a moment function if there exists p E such that for all a E N; : a a ( p ) = cp(a). In this case we say

M*(R',B')

that cp is represented by p. A moment function cp is said t o be determinate if cp is represented by one, and only one, p E M*(R', B'). We will then also say that p is determinate. There seems no informative condition available which is both necessary and sufficient for a moment function to be determinate, not even in the case T = 1. On the other hand, there exists a useful link, discovered by L. C. Petersen, between the one-dimensional and multidimensional problems. Let ri (i = 1,.. . , T ) denote the projection R' + R : z = (X I,.. . ,x,) ++ zi. Theorem 1.1. p E M*(R', B') is determinate if for all i = 1,.. . , r the image measure 7ri(p) E M*(R,B) is determinate.

Proof. Petersenlo, Thm. 3 The cited paper Petersen" contains an example which shows that the converse of the theorem is false. We now consider the case T = 1, fix p E M*(R,B), and write a k = a k ( p ) and P k = P k ( / l ) . Then a certain analyticity condition on the Fourier transform fi of p guarantees that p is determinate. This criterion, an explicit statement of which is contained in Theorems 1.2 and 1.3 below, appears under several guises in a number of monographs and textbooks. Let us remark that Lukacs8, p. 198, gives an example of a probability measure which does not meet the criteria below, but is nevertheless determinate in view of Carleman's theorem (see Shohat and Tamarkin13, Thm. 1.11). Let X be an algebraic indeterminate. The formal power series

316

will be called the moment generating function of p.

Theorem 1.2. For p E M*(R,B), the following conditions are equivalent:

(4

(ii) There exists 6 > 0 such that M, converges on the open disc B6(O), (iii) There exists 6 > 0 such that Mp is a holomorphic continuation of Res]-6,6[(fi) to Ba(0). (iv) There exist 6 > 0 and a holomorphic function f o n the strip { z E C : IIm zI < S} such that Resw(f) = fi. Theorem 1.3. If p E M* (R, 23) satisfies one of the conditions in Theorem 1.21then it is determinate. Corollary 1.4. For all m E R, u

2 0, N(m,a2)is determinate.

Corollary 1.5. For all r E R, m E R', C E R'" definite, N(m, C ) is determinate.

symmetric and positive

Proof. For any linear form X : R' + R the image measure X(N(m,C)) is a one dimensional normal distribution. Now Corollary 1.4 and Theorem 1.1imply what was claimed. 0 Now we wish to show that weak convergence to a determinate distribution can be established using moment arguments. This will follow from Prohorov's theorem. Since it is not very easy to find an explicit treatment of the multidimensional case in the literature, we have chosen to include full proofs.

Definition 1.6. A sequence ( p n ) n in ~ ~M1(R',Br) is said to be uniformly tight, if for every 6 > 0 there exists a compact set K , C R' such that for all n E N: pn(K,) > 1 - 6. Theorem 1.7. If ( p n ) n E is ~ a uniformly tight sequence in M1(R',23'), then it has a weakly convergent subsequence.

Proof. P a r t h a ~ a r a t h y Thm. ~, I1 6.7

0

Lemma 1.8. Let ( p n ) n E be ~ a sequence in M*(R',f?') which converges weakly to p E M1(RT,23'). Suppose that for all a E N; there exists 0 <_

317

K , < 00 such that supnENIa,(pn)l 5 K,. Then p E M*(R',B'), and for all a E Nh we have the convergence a a ( p n )--+ a,(p) as n -+ m. Proof. By Skorohod's theorem (see Kallenberg7, Thm. 4.30, for a version in sufficient generality), we can choose a probability space (Qd,P ) and random variables X,,X : (fl,d,P) -+ (R',f?') with Px, = p, ( n E W), Px = p such that X , + X almost surely as n -+ 00. For all a E Nb we set

n T

fa : R'

+ R,

,

(Zl,. . . z,)

H

z4'.

j=1

Then for all a E Ng we have that a,(pn) = E(f a o X n ) and f a o x , a.s. . Now for all p > 0

- 1 - - E ( f ( 2 a 1,...,2 0 , )

P

1 1 O X , ) = - a ( 2 a 1 ,...,2 a , ) ( ~ n )5 P

P

-+

K ( 2 a 1 ,...,2a,)

faoX

< m,

Hence (fa o X n ) n E is ~ uniformly integrable. A standard theorem on uniformly integrable sequences implies that ( f a o x ) is integrable, and therefore E(If a l o X ) = ,&(p) < 00, i.e., a,(p) exists. In addition, this theorem yields the convergence aQ(pTI) = E ( f c 2

n-icc --+

x71)

x , = aa(p).

E(fa

Theorem 1.9. Let ( P ~ ) ~ E N p, be in M*(R',B'), determinate. If for all a E Nb n+cc

aQ(pn)

0

and suppose that p is

aa(p)>

then ( p n ) n Econverges ~ weakly to p . Proof. By assumption, the sequences (q2,o,o, ...,O)(Pn))nEN,

(Q(0,2,0,

...,o)(Pn))nEN,

are convergent, hence bounded. Let 0 5 R Then for all n E N, p > 0, we have that

...

7

( y o , o , o , ...,2

)bn))nm

< cc be a bound for their sum.

318

This implies that (P,),~N is uniformly tight, because for

[

fl]

E

> 0 we can

take K , = I1 . 11 5 in Definition 1.6. In view of Theorem 1.7 it remains only to show that every weakly convergent subsequence of ( p , ) , € ~ converges t o p. Suppose that the subsequence (p,,) converges weakly to v E M1 ( E X T , B T ) ,as k tends to infinity. Since for all a E NE (a,(p,,))k€~ is bounded, we obtain from Lemma 1.8 that v E M*(RT,B T )and that cya(pn,) + aa(v).On the other hand we know that a a ( p n k )+ aa(p).This implies cy,(v) = aa(p). As a E NE was arbitrary, determinacy implies p = v. 0 2. Application to Random Matrices Fix n E N,and let K , be one of the compact matrix groups U,, O,, Sp, (n even in the last case). A random element of K,, hence a random matrix, is a random variable r, with values in K,. In what follows we only consider the case that the distribution of I?, is the normalized Haar measure WK, on K,. If I?, is a random matrix in this sense, then its eigenvalues, trace, and related functionals are complex valued random variables, and it makes sense t o study their behavior as n -+ co. Depending on the functional considered, different methods of investigation apply. See, e.g., D’Aristotile, Diaconis and Newmanl for a problem which can be treated using elementary Fourier analysis. We will concentrate on a case which is particularly well suited for applying the Method of Moments. To be specific, we consider random vectors of the form (Tr(T),Tr(r2),. * * (T

E

, n-(rr)>

(7)

N) in the cases K , = 0, and K , = Sp,, and of the form (Tr(r),. . . , Tr(rT),m,. . . , Tr(r7))

(8)

in the case K , = U,. For these vectors, the method yields multivariate Central Limit Theorems. Of course, one has to compute the joint moments of the random vectors (7) and (8) in the first place. This is in fact possible and can be done using elementary combinatorics. The ultimate reason for the friendly character of this situation is the fact that the joint moments of (7) and (8) have natural interpretations in terms of the tensor invariants of the classical groups O(n, C), Sp(n, C) and GL(n, C), respectively (see Stolz14 for details). The moment formulae were first obtained by Diaconis

319

and Shahshahani4, who chose an approach via special functions and used the character theory of Brauer algebras (Raml1>l2)in the orthogonal and symplectic cases. Let us explain these moment formulae. Fix r, q E N , a = (a1 , .. . ,a,) E NE, b = @ I , . . . , b q ) E Ni. Set 7.

k , := C

j U j

(9)

j=1

and define kb analogously. In the orthogonal and symplectic cases we write k := k,. Define f a by setting for all j E N

f a ( 8 :=

1;

if aj = 0, if j a j is odd, aj 2 1, if j is odd and aj is even, aj 2 2,

- I)!! 1+ C:zJ j d ( 3 ) ( 2 d- l)!! if j is even, aj 2 1.

jqaj

0 .

Here for m E N (2m - l)!! := (2m - 1)(2m - 3 ) . . . 3 . 1 .

(10)

If K , = On, we then have the following Theorem 2.1. If k = k , is odd, then

If k = 21 is euen and 2 n 2 k, then this expectation equals r j=1

By elementary computations, one gets from this the following more intuitive result:

Theorem 2.2. Consider a family 21,.. . ,Z, of iid standard normal random variables. If 2n 2 k ,

then

(11)

320

where r)j

:=

1, if j is even, 0 , i f j is odd.

Note that if k is odd, then (12) holds regardless of whether condition (11) is met or not, both sides being equal to zero in this case.

If K , = Sp, (n even), then we have Theorem 2.3. If k = k , is odd, then

If k = 21 is even and n 2 k , then this expectation equals T

j=1

Again elementary computations teach us that this is tantamount to Theorem 2.4. If

n 2 k, then

If K , = U,, we have the following Theorem 2.5.

If k, # kb, then

and if k , = kb and n 2 k,, then

321

To provide an interpretation in terms of the moments of a suitable normal distribution, note that if Z is an R2-valued random vector with distribution N(0, $I2), then it has the same distribution as a product RU of independent random variables R and U , where U is uniformly distributed on the unit circle and R has the same distribution as the euclidean norm llZll2 of 2 , R2 thus being distributed according to what might be called a "squeezed" X2-distribution with 2 degrees of freedom. In any case, this distribution belongs to the family of gamma distributions, and an appropriate specialization of parameters shows that it is in fact an exponential distribution with parameter 1. Partial integration and a trivial induction show that E((R2)k)= Ic! for all Ic E NO. Now regard Z as a complex random variable and call it a standard complex normal random variable. Write U = eiT, where T is distributed according to &.X1~0,2~[.We are now going to compute the joint moments of Z and Let a , b E NO. Then E ( Z " 2 ) = E(R"eiaTRbe-ZbT) = E(R("+b))E(ei("-b)T)by independence. Now

z.

Hence E(Z"zb) = dab E(R2") = dab E((R2)") = dab a! ~ iid standard complex normal random variables. Now let ( Z j ) j E be Then

Comparison with (16) yields

After this preparatory work, we can apply the Method of Moments to yield Central Limit Theorems. Theorem 2.6. For n E N let r, be a random variable with values in 0, whose distribution is the normalized Haar measure won on 0,. Fix r E N.

322

Then, as n

+ 00, yn := (Wrn),wr3,..

. 7

n(r3)

converges in distribution to

+ 17 d i ~ 3 , ... ,

z := (21, where

2 1 ,. . .

, Z, is an aid family Vj

:=

&zT

+

vT),

of standard normal random variables and

1, if j is even, j is odd.

{ 0 , if

This means that the distributions of the Yn converge weakly to 1

0 1 0 1

N

0

2

3

4 0

.. r

VT

Proof. Given a E N;, (11) will be fulfilled if n is la (rge enough. Hence by (12) the a-moment

will eventually be equal to aa(Z), so a fortiori we have the convergence aa(Yn)

n+oo ---f

aa(Z).

a being arbitrary, the claim follows by Theorem 1.9 and Corollary 1.5. 0 Using (14) instead of (12) the same argument yields the analogous result for the symplectic group. Theorem 2.7. For m E N let I'2, be a random variable with values in Sp,, whose distribution is the normalized Haar measure wsP,_ on Sp,,.

Fix r E N. Then, as m

+ 00,

n(r;,), . . . ,n ( G m ) )

Ym := (n(r2,),

converges in distribution to

2 := (Z17fizz

- 1,&z37..* , &ZT - V T ) ,

323

that is, the distributions of these random vectors converge weakly to 0 -1

N

0

-1

What makes the unitary case slightly more difficult is that a straightforward application of Theorem 1.9 and Corollary 1.5 using the identification of C with R2 requires information about the joint moments of the real and imaginary parts of the traces. Therefore we must extract this information from equation (17) in the first place. To this end consider endomorphisms cp and I)of the polynomial ring C[S, TI, where cp is given as the evaluation homomorphism determined by S w S + iT,T w S - iT,and I)as the Tw A routine evaluation homomorphism defined by S i--+ computation verifies at once that cpI)= I)cp = id. So cp and I)are automorphisms of C [ S , T ] . Now fix T E N, and for j = 1,.. . , r let &, 0, be real random variables and <j := <j + iej. Then for any a , b E N; the monomial

y,

9.

T

j=1

can be expressed as a linear combination of monomials of the form

j=1

with c , d E

Ni.

Now consider sequences ( @ n ) ) n E ~ ,(B("))nEN of RT-valued random vec:= 5'") iB(") for all n E N and suppose that for all c, d E N; tors. Set <("I

+

Interchanging the limit with the finite linear combination from the last

324

paragraph one sees that for all a, b E

NE

Specializing to the data of the unitary case yields the following central limit theorem:

Theorem 2.8. For n E N let I?, be a random variable with values in U, whose distribution is the normalized Haar measure wu, o n U,. Fix r E N. Then, as n -+co, Y, := ( R e ( T r ( r , ) ) , I m ( ~ ( r , ) ) ,. . . ,Re(Tr(rL)),I m ( n ( r L ) ) )

converges in distribution to

N 0 References 1. A. D'Aristotile, P. Diaconis and C. M. Newman, Technical Report 2002-18, Stanford University, Dept. of Statistics 2. P. Billingsley, Probability and Measure, New York (Wiley) 31995 3. P. Diaconis, Proc. Symp. Appl. Math. 37 125-142 (1987) 4. P. Diaconis and M. Shahshahani, J . Appl. Probab. 31 A 49-62 (1994) 5. W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, New York (Wiley), '1971 6. M. de Jeu, Ann. of Prob. 31, 1205-1227 (2003) 7. 0. Kallenberg, Foundations of Modern Probability, New York (Springer) 22002 8. E. Lukacs, Characteristic Functions, London (Griffin) '1970. 9. K. R. Parthasarathy, Probability Measures on Metric Spaces, New York (Academic Press) 1967 10. L.-C. Petersen, Math. Scand. 51 361-366 (1982) 11. A. Ram, Paczfic J . Math. 169 173-200 (1995) 12. A. Ram, European J. Combin. 18 685-706 (1997) 13. J. A. Shohat and J . D. Tamarkin, The Problem of Moments, New York (AMS) 1943 14. M. Stolz, On the Diaconis-Shahshahani Method in Random Matrix Theory, Ph.D. thesis, University of Tubingen, 2004

Infinite Dimensional Harmonic Analysis I11 (pp. 325-351) Eds. H. Heyer et al. @ 2005 World Scientific Publishing Co.

ISOTROPY REPRESENTATION FOR HARISH-CHANDRA MODULES

HIROSHI YAMASHITA Department of Mathematics, Hokkaido University Sapporo 060-0810, JAPAN E-mail: yamasitaOmath.sci.hokudai.ac.jp We study the isotropy representation attached to an irreducible Harish-Chandra module with irreducible associated variety. It is shown that, under some assump tions, the dual of the isotropy representation in question can be characterized by means of the principal symbol of a differentialoperators of gradient type. By using this, the case of Harish-Chandra module of discrete series is more closely examined.

1. Introduction

Let g be a complex semisimple Lie algebra with a nontrivial involutive automorphism 8 of g. We write g = 0 @ p for the symmetric decomposition of g given by 8, where C and p denote the +1 and -1 eigenspaces for 8, respectively. Let Kc be a connected complex algebraic group with Lie algebra 0. We assume that the natural inclusion C v g gives rise to a group homomorphism from Kc to G;d through the exponential map. Here GEd denotes the adjoint group of g. Then, this homomorphism naturally induces the adjoint representation Ad of Kc on g. We say that a finitely generated g-module X is a (8, Kc)-module, or a Harish-Chandra module, if the action on X of the Lie subalgebra 0 is locally finite and if it liits up to a representation of Kc on X through the exponential map in such a way as ( k . X - k - l ) v = (Ad(k)X)v for X E g, Ic E Kc and v E X. It is a fundamental result of Harish-Chandra that the study of irreducible admissible representations of a real semisimple Lie group essentially reduces to that of irreducible (8, Kc)-modules. Let X be an irreducible (8,Kc)-module. A Kc-stable good filtration of X naturally gives rise to a graded, compatible (S(g), Kc)-module grX annihilated by 0 , where S(g) denotes the symmetric algebra of g. By BorhoBrylinski [l]and Vogan [22], [23], the associated cycle C ( X )of X is defined to be the support V ( X ) of g r X combined with the multiplicity at each

326

irreducible component of V ( X ) .The support V ( X )is called the associated variety of X. It is a Kc-stable affine algebraic cone contained in the set of nilpotent elements in p, and each irreducible component of V ( X )is the closure of a nilpotent Kc-orbit 0 in p. As we have shown in [6] and [27], the variety V ( X )controls some fundamental properties for X. The algebraic cycle C ( X ) describes a sort of asymptotic behavior of X (cf. [21]). Moreover, it is shown by Vogan [22, Theorem 2.131 that the multiplicity of X at an irreducible component of V ( X )can be interpreted as the dimension of a certain finite-dimensional representation ( w g ,W ) of the isotropy subgroup Kc(X) of KC at an X E 0. We call wg an isotropy representation attached to X . In terms of wo, the associated cycle C ( X ) of X is expressed as dim wg

C ( X )=

- [B].

0

Now, we assume that the associated variety V ( X ) of X is the closure of a single nilpotent Kc-orbit 0 in p. This assumption does not exclude important (g, Kc)-modules related to elliptic orbits. In reality, it is wellknown that the Harish-Chandra modules of discrete series (more generally Zuckerman derived functor modules) and also the irreducible admissible highest weight modules of hermitian Lie algebras satisfy this hypothesis. The purpose of this paper is to study the associated cycle C ( X ) and in particular the isotropy representation w g attached to a (g, Kc)-module X with irreducible associated variety, by developing our arguments in [30] for unitary highest weight modules and also those in [29] for discrete series. To do this, we first look at in Section 2 a relationship between the (S(g), Kc)-module g r X and the induced representation r ( W ) = Ind$(x,(wg,W) of Kc equipped with a natural S(g)-action. This amounts to a survey of some aspects of Vogan’s work ([22, Sections 2-41 and [23, Lectures 6 and 71) in a slightly modified and simplified form (but for limited X ’ s ) . A reciprocity law of Frobenius type for such an induced module (Proposition 2.2) plays an important role. In fact, it is effectively used to prove an irreducibility criterion for wg (Theorem 2.1). Also, we include a remarkable result (Theoren 2.2) on the isotropy representations for singular unitary highest weight modules, given in [30], [33] and [24]. In order to identify the isotropy representation w o , it is useful to consider not only gr X but also its Kc-finite dual realized as a space of certain (vector valued) polynomial functions on p. We present this idea in Section 3. A sufficient condition is given in Proposition 3.1 for grX being annihi-

327

lated by the whole prime ideal I of S(g) defining 6. In such a case, the isotropy representation, more precisely, its dual w & , can be described by means of the principal symbol of a differential operator on p of gradient type (see Propositions 3.2 and 3.3). In the last part of this paper, Section 4, we focus our attention on the irreducible Harish-Chandra modules X of discrete series. As is well known, such an X has irreducible associated variety (cf. [28], [29]). The multiplicities in the associated cycles for discrete series have been intensively studied by Chang [2], 131, by means of the localization theory of Harish-Chandra modules. He succeeds to describe C ( X )explicitly for the real rank one case. Taniguchi applies in [19] and [20] the results of Chang in order to specify Whittaker functions associated with discrete series for S U ( n ,l ) , Spin(n,1) and SOo(2n,2). Here in this paper, we exploit another way to identify C(X), by using a realization of X as the kernel of an invariant differential operator of gradient type on the Riemannian symmetric space (cf. [9], [MI; [26], [34]). Based on our results in Section 3 and also on the discussion in [29], we can construct a certain Kc(X)-submodule U,(QC)of the representation (w&,W ' ) contragredient to wo. Moreover some evidences are given for this subrepresentation being large enough in the whole w&. The gained results are summarized as Theorem 4.2 and Corollary 4.1. This article is an updated and enlarged version of the informal reports [31] and [32] appeared in RIMS Ktjkyiiroku.

2. Graded module g r X and induced representation r ( W )

As in Section 1, let X be an irreducible (g, Kc)-module with irreducible where 0 is a nilpotent Kc-orbit in p. For associated variety V ( X ) = later use, this section introduces some elementary aspects of Vogan's theory on the associated cycle and the isotropy representation attached to X . The results in this section may be read off from [22] and [23] with some effort. Nevertheless, we include the proofs for these important results in order to make this paper more accessible.

n,

2.1. Associated cycle and isotropy repwsentation First, we introduce our key notion precisely. Take an irreducible Kcsubmodule (.,Vr) of X , which yields a Kc-stable good filtration of X

328

in the following way:

X Oc X I C

c X, c

, with

x , := V,(g)V, ( n = 0,1,2,. ..).

(2.1)

Here U ( g )denotes the universal enveloping algebra of g, and we write Un(g) (n = 0,1,. ..) for the natural increasing filtration of U ( g ) . This filtration gives rise to a graded (S(g),&)-module M = g r X , annihilated by S(t), as follows: 00

M = grX = @ M n

with M, := X n / X n - 1 ( X - 1 := (0)).

(2.2)

n=O

We note that

M, = S"(g)V, = Sn(p)V. and MO= V,,

(2.3)

where S,(D)is the homogeneous component of the symmetric algebra S(a) of degree n. By definition, the associated variety V ( X ) of X is identified with the afEne algebraic variety of g given by the annihilator ideal Annqa)M in S(g) of M :

V ( X )= (2 E g 1 f(2)= 0 for all f E Anns(a)M) c p,

(2.4)

where S(g) is viewed as the ring of polynomial functions on g by identifying g with its dual space through the Killing form B of g. Throughout this section, we assume that V ( X ) is irreducible. The Hilbert Nullstellensatz tells us that the radical of Anns(,)M coincides with the prime ideal I = I ( V ( X ) )defining the irreducible variety V ( X ) : I = So we see I n M = (0) for some positive integer n, and we write no for the smallest n of this nature. Then, one gets a strictly decreasing filtration of the (S(g), &)-module M as

d w .

M =P M

2I'M 2 2 InoM = (0).

(2.5)

By the multiplicity multI(X) of X at I is meant the length as an S ( g ) p module of the localization MI of M = grX at the prime ideal I . Then, the associated cycle C ( X ) of X turns to be

C ( X )= multI(X) - [8] with V ( X )= 8.

(2.6)

Note that this cycle does not depend on the choice of a good filtration (2.1) of

x.

329

Now, let us explain how the multiplicity multI(X) can be interpreted as the dimension of an isotropy representation. For this, we take an element X in the open Kc-orbit 0 C V(X). Set Kc(X) := {k E KcI Ad(k)X = X}, the isotropy subgroup of Kc at X. We write m(X) for the maximal ideal of S(g) which defines the one point variety {X} in g: m ( x ) :=

C(Y- B(Y,x))s(~)

for x E 0.

(2-7)

y a

For each j = 0, . .. ,710 - 1, we introduce a finite-dimensional representation a o ( j ) of (S(d,&(X)) acting on

W ( j ):= IjM/m(X)IjM,

(2.8)

in the canonical way, and we set no-1

( a o , W := @

( ~ O ( j ) , W ) ) .

(2.9)

j=O

We call wo the isotropy representation attached to the data ( X , V , , U ) , where V, yields the filtration (2.1) of X. The following lemma (cf. [22, Corollary 2.71; see also [31, Remark 2.21) is essential for our succeeding discussion.

Lemma 2.1. Let N be a finitely generated (S(g), Kc)-module such that I N = (0). Then, the length of S(g)I-module NI is equal t o the dimension of the vector space N/m(X)N f o r every X E 0. This lemma tells us that the length of the localized S(g)I-module (IjM/Ij-+'M)1 equals d i m a o ( j ) , by noting that the ideal I annihilates the subquotient IjM/Ij+lM of M. Together with the exactness of localization, we immediately get the following

Proposition 2.1. One has multI(X) = dimmo. Moreover, the equality multI(X) = dim wo(0) = dim M/m(X)M

(2.10)

holds i f and only if the support of the S(g)-module I M i s contained in the boundary 3 0 = B\0. Remark 2.1. The representation ao(0)in (2.10) never vanishes because the annihilator ideal Anns(g)M/IM is equal to I (cf. [30, Lemma 3.41). Moreover, the equality (2.10) holds for a number of unitary (g, Kc)-modules

330

X with unique extreme Kc-types V,. See Example 2.1 and Theorem 4.2 (1). 2.2. Induced module

r(Z)

We consider a finite-dimensional (S(g), Kc(X))-module (a,2) with X E 0, where &(X) acts on 2 holomorphically. Let r(2)denote the space of all left Kc-finite, holomorphic functions f : Kc + 2 satisfying

f

(Yh)= w ( h ) - l f ( y ) (Y E Kc, h E Kc(X)).

Namely, r(2)consists of all &-finite, holomorphic cross sections of the &-homogeneous vector bundle Kc X K ~ ( X )2 on Kc/Kc(X) N 0. Then, r(2)has a structure of (S(g), &)-module by the following actions:

(D* f ) ( Y ) := d A d ( Y ) - l D ) f ( Y ) , (k * f ) ( Y ) := f ( k - l y ) , for D E S(g), k E Kc and f E r(2).We call I'(2) the (S(g),Kc)-module induced from w. We note that, if 2 is annihilated by the maximal ideal m(X), the S(g)-action on r(2)turns to be the multiplication of functions on the orbit 0: (2.11) In this case, the annihilator in S(g) of any nonzero function f E r(2) coincides with the prime ideal I defining 8. Let M be any (S(g),Kc)-module. If p is a homomorphism from M to 2 as (S(g),Kc(X))-modules, we define a function T, : Kc + 2 for each m E M by putting Tm(Y) := P ( Y - ~* m) (Y E Kc).

(2.12)

Then it is standard to verify that Tm lies in r(2)and that the map T : m H T, ( m E M ) gives an (S(g),Kc)-homomorphism from M to F(2). More precisely, one readily obtains the following reciprocity law of Frobenius type.

Proposition 2.2. Under the above notation, the assignment p up a linear isomorphism Homs(g),Kc(x)(M,2 ) = Homs(g),Kc(M,V)).

HT

sets

(2.13)

Here, for R-modules A and B , we denote by Homn(A, B ) the space of Rhomomorphisms from A to B.

33 1

2.3. Homomorphism T = @jf(j) We now return to our setting in Section 2.1, where M = g r X for an irreducible (g,Kc)-module X with V ( X ) = 8. Take an integer j such that 0 5 j 5 no - 1. Let p ( j ) denote the natural quotient map from I j M to W ( j )= I j M / m ( X ) I j M . Correspondingly, we get an (S(g), Kc)homomorphism T ( j ): I j M -+ I'(W(j))by Proposition 2.2. It follows that KerT(j) =

n

m ( Y ) I j M 3 Ij+'M,

(2.14)

YE0

by the definition of T ( j )together with m ( Y ) 3 I (Y E 0 ) .

Proposition 2.3. KerT(j) is the largest (S(g),&)-submodule of I j M among those N having the following two properties: (i) N 3 Ij+'M, and, (ii) the support of N / I j + l M is contained in ao. Proof. First, we show that KerT(j) have two properties (i) and (ii). The inclusion (2.14) assures (i). As for (ii), we consider a short exact sequence of (S(g), Kc)-modules: 0 + KerT(j)/Ij+lM

+ IjM/Ij+lM -+ IjM/KerT(j) + 0.

(2.15)

Each module is annihilated by I. In view of Lemma 2.1, we find that the multiplicity of IjM/KerT(j) at I is equal to the dimension of vector space

( I jM/Ker T ( j ) ) / m ( X(IjM/Ker ) T(j)) 21

+

I j M / ( m ( X ) I j M KerT(j)) = W ( j ) . (2.16)

Here, the last equality follows from KerT(j) C m ( X ) I j M (see (2.14)). This shows that the length of S(g)I-module IjM/Ij+'M and that of IjM/Ker T ( j ) coincide with one another. Hence (Ker T ( j ) / l j + ' M ) ~ vanishes by (2.15). This means that the support of KerT(j)/Ij+lM is contained in do. Second, let N be any (S(g), Kc)-submodule of I j M with two properties (i) and (ii) in question. (2.14) tells us that T ( j ) naturally induces an (S(g), Kc)-module map from I jM /I j+'M to I'(W(j)) which we denote by f'(j).Then, f ' ( j ) ( N / I j + ' M )must vanish by virtue of (2.11) together with the property (ii) for N . This proves N c KerT(j).

As for the injectivity of T ( j ) , one gets the following consequence of Proposition 2.3.

332

Corollary 2.1. The homomorphism T(j) : I j M + r ( W ( j ) )is injective if and only i f Annq,) m = I f o r all m E IjM\{O}. In this case, one has I j +‘ M = {0}, i.e., j = no - 1. Example 2.1. We encounter the situation in the above corollary with j = 0, for example, if X is a unitary highest weight module of a simple hermitian Lie algebra g, and V, in (2.1) is the extreme Kc-type of X.Note that the associated variety of such an X is the closure of a “holomorphic” nilpotent Kc-orbit in p. See [30, Section 3.21 for details. Summing up f’(j)’s on I j M / I j + l M ( j = 0,. . . ,no - l), we obtain an (S(g), Kc)-homomorphism T := @ j f ’ ( j ) :

&q~ :=)@ I ~ M / I ~ + 3 ~ M @ WW))= rw), j

(2.17)

j

where the support of the kernel KerT is contained in 80.

Remark 2.2. By using the “microlocalization technique”, Vogan constructed a new Kc-stable Z-gradation on X such that the corresponding graded module embeds into r ( W ) as a representation of Kc (see [22, Theorem 4.21). Thanks to this result, one always has X c) r ( W ) as Kcmodules. Noting that G(I)N X as Kc-modules, we find that the above T : G(I)+ r ( W )must be an isomorphism if T is surjective.

2.4. ImdUCibility of wo The results in Sections 2.1-2.3 lead us to prove the following natural criterion for the irreducibility of isotropy representation ( wg,W)of K c ( X ) (cf. [23, Proposition 7.61; see also [31, Section 51).

Theorem 2.1. T h e following two conditions o n X are equivalent t o each other. ( a ) ( w g , W ) i s irreducible as a K c ( X ) - m o d u l e . (b) If N i s any (S(g), Kc)-submodule of M = g r X , either the support of N or that of the quotient M / N i s contained in 80. In this case, we have wg = wg(O), or equivalently, the support of I M is contained in 8 0 by Proposition 2.1. Proof. The implication (a) + (b) is an easy consequence of the exactness of localization. In what follows let us prove (b) (a). First, we note that

+

333

the condition (b) together with Remark 2.1 implies that the support of I M is contained in 80. Thus one gets wg = wg(O), or,

W = W ( 0 )= M/m(X)M. Now, suppose by contraries that W is not irreducible. Then, there exists a Kc(X)-stable subspace C of M such that M 2 C 2 m(X)M and that 2 := M / C is irreducible as a Kc(X)-module. The condition C 3 m(X)M assures that C is S(g)-stable. Thus 2 becomes an (S(g), Kc(X))-module annihilated by m(X). Next, we consider two induced (S(g), Kc)-modules r ( W ) and r(Z). The quotient map W = M / m ( X ) M + 2 = M / C gives rise to an (S(g), Kc)-homomorphism, say y, from r ( W ) to r(2) in the canonical way. Set T' := y o T(O),where T ( 0 ) : M + r ( W ) is the (S(g),Kc)homomorphism defined in Section 2.3. Then, as shown in the proof of [23, Proposition 7.91, the image T ' ( M ) of T' is a finitely generated (S(g), Kc)submodule of r(2)whose isotropy representation is isomorphic to 2. This combined with T ' ( M ) N M/Ker T' tells us that the multiplicity of Ker T' at the prime ideal I is equal to dimW - d i m 2 > 0. By the assumption (b), we find that the support of M/KerT' N T ' ( M ) is contained in 80. This necessarily implies Ker T' = M , i.e., 7'' = 0, because the S(g)-module T ' ( M ) (C r ( 2 ) ) admits no embedded associated primes by (2.11). Finally, the resulting equality T' = 0 means that y-l. m

+ m(X)M = T ( o ) , ( ~E) C/m(X)M

for all y E Kc and m E M . This contradicts C

# M.

2.5. Case of unitary highest weight representations

Let X be an irreducible unitary highest weight (8, Kc)-module of a simple hermitian Lie algebra g, with extreme Kc-type V,.

Example 2.2. In [30, Section 51, we have described the isotropy representation wg = wo(0)explicitly, when X is the theta l i t of an irreducible representation of the compact groups G' = O(lc),U ( k ) and Sp(lc) with respect to the reductive dual pairs (G, G') = (Sp(n,R), O(lc)),(Sub,q), U(lc))and (SO*(2n),Sp(lc)),respectively. In particular, one finds that the representation a g is irreducible if the dual pair (G, G') is in the stable range with smaller member G'. In this case, X H a& essentially gives the Howe duality correspondence ([lo], [ll],[8,Part 111 etc.).

334

An irreducible highest weight (g, &-)-module X is called singular if the Gelfand-Kirillov dimension dimV(X) is strictly smaller than one half of the dimension of the corresponding hermitian symmetric space. Recently, we have described the isotropy representations by using the projection onto the PRV-component (cf. Proposition 3.3), for all singular unitary highest weight modules which can not be obtained by the Howe duality correspondence. This work is in collaboration with Wachi (see [24] and [33] for details). As a result, we establish the following remarkable theorem.

Theorem 2.2. The isotropy representation is irreducible and explicitly described f o r every irreducible singular unitary highest weight representation of a simple Lie group of hermitian type. The above result for DI and EVII gives a clear understanding of some multiplicity formulae obtained by Kato and Ochiai ([13], [14]). For EVII case, we get two distinguished series of isotropy representations which decompose the quasi-regular representations for the compact symmetric spaces S8 E S 0 ( 9 ) / S O ( S ) and P2(Cay) 7 Fq--52)/Spinn(9),respectively. These decompositions can be related to tensor products of singular unitary representations [4],by using the generalized Whittaker vectors [30].

3. Utility of the dual ( S ( g ) ,&-)-module In this section, we do not assume a priori that the associated variety V ( X ) of X is irreducible. Let M = gr X be, as in (2.2), the graded (S(g), Kc)module attached to an irreducible (8, Kc)-module X through the filtration (2.1). For any nilpotent element X E p, we can define the maximal ideal m(X) of S(g), the Kc(X)-module W ( 0 )= M / m ( X ) M , and (S(g),Kc)homomorphism T(0): M + I'(W(O)), just as in Section 2. Here I'(W(0))= I n d z ( x ,(W(0))is the (S(g), &)-module induced from W(0). The purpose of this section is to make some simple observations concerning the associated cycle of X,in connection with the Kc-finite dual M * of M realized as a space of V;-valued polynomial functions on p. This is done by developing our arguments in [29], [30] in full generality. A sufficient condition is given in Proposition 3.1 for the annihilator Arms(,) M of M being equal to the prime ideal of S(g) defined by the &-orbit 0 := Ad(&)X through X. Furthermore, we characterize M * as the kernel of a differential operator V on p of gradient type (Proposition 3.2). Then the principal symbol of V allows us to describe the Kc(X)-module W(O)*dual to W(0)(Proposition 3.3). The observations made in this section will be

335

used effectively in Section 4, in order to describe the associated cycles for (g, Kc)-modules X of discrete series. 3.1. Kc-finite dual M * and its submodule

*

First, the tensor product S(p)@V, admits a natural structure of (S(g), Kc)module so that t annihilates the whole S(p) €3 V,:

{

D1.(D€3 v) := D1D €3 v (DlE S(p)), D2.(D€3.) := 0 (D2 E S ( t ) ) , k . (D€3 V ) := Ad(k)D €3 kv (k E Kc),

(3.1)

where D €3 v E S(p) @ V, with D E S(p) and v E V,. Since M = S(p)V, with V, = Mo, there exists a unique surjective (S(g), Kc)-homomorphism 7r : S(p)

€3

v, +M

such that ~ ( 1 v) 8 = v for v E V,. We write N for the kernel of 7r. This is a graded (S(g), &)-submodule of S(p) €3 V,. On the other hand, we identify S(p*) €3V,* canonically with the space of polynomial functions on p with values in V,*,where U ' denotes the dual space of a vector space U.S(p*) €3 V,* also becomes an (S(g), Kc)-module on which g acts by directional differentiation through the quotient map g -b g/e

2(

p:

{

(Dl. f)(Y) := (Wl)f)(Y) (DlE S(P)), (D2 * f)(Y) := 0 (D2 E s(e)), (k * f)(Y) := k * f(Ad(k)-'Y) (k € Kc),

(34

for f E S(p*) €3 V,* and Y E p. Here D1 ++ a(&) denotes the algebra isomorphism from S(p) onto the algebra of constant coefficient differential operators on p, defined by d a(Z)f(Y) := zf(Y t Z ) ~ t = o for 2 E p. (3.3)

+

Note that the action of S(g) on S(p*) €3 V: is locally finite. The assignment

*

p3X X' E p* with X*(Y):= B(X,Y)(YE p). gives a Kc-isomorphism from p onto p*. Now it is standard to verify that (Sb) €3 VT) x (S(P*)€3 v3 3 (D€3 v, f )

(D@ % f := ) ((TD*f)(0),v)v;xV,,

* (D€3 v ,f ) E c,

(3.4)

(3.5)

336

gives a nondegenerate (S(g), K&invariant pairing, where denotes the principal automorphism of S(p) such that TY = -Y for Y E p, and ( - , - ) v ; ~ vis, the dual pairing on V !, x V,. Let M* denote the &-finite dual space of M , viewed as an (S(g), &)-module through the contragredient representation. We write N L for the orthogonal of N in S(p*)@V' with respect to (. , .). Then, (3.5) naturally induces a nondegenerate invariant pairing

( * , -)l:MxN'-+cC,

(3.6)

which gives an isomorphism of (S(g), Kc)-modules:

M* N N'-

c S(p*)@ V.:

(3.7)

For an integer n 2 0, we denote by ( N l ) , , the homogeneous component of N' of degree n: ( N l ) , , := N'- n (Sn(p*)@ V):. Let X be a nilpotent element in p. Noting that M = V, + m(X)M, we have a natural Kc(X)-homomorphism V,

+W(0)= M/m(X)M -+

0.

This induces an embedding of W(O)*into V,* as

W(O)*N (V,/(V, n m(X)M))*c)V,*,

(3.8)

by passing to the dual. In this way, we regard W(O)*as a Kc(X)-submodule of v,. For each integer n 2 0, let q,,be the Kc-submodule of Sn(p*) @ V,! generated by the vectors (X*),, €4 w* (w* E W(O)*): q n := ((X")" @ w* I v* E W ( O ) * ) K , .

Note that the V:-valued

(X')"

(3.9)

polynomial function ( X * ) n€4 w* on p is defined by @ w* : p 3

2 c)B ( X , Z)%* E

cc.

We set 00

9 := @ Qn

c S(p*)€4 v.:

n=O

Then, we can show the following Lemma 3.1. (1) 9 is an (S(g),&)-submodde of S(p*)8 V' contained in N'-.

337

*

(2) W e write L* for the orthogonal of in M with respect to the pairing ( . , . )1 on M x N I . Let T(0): M + r(W(0))be the (S(g), Kc)homomorphism defined in Section 2.3. Then, one gets KerT(0) n M,, = L9n M,,

for every integer n 2 0 ,

(3.10)

In particular, I* = @, L!P n M,, is contained in Ker T(0). Proof. (1) It is easy to see that q is (S(g), Kc)-stable by noting that

Ye (((Ad(y)X)*),,8 y v*) = nB(Ad(y)X,Y)((Ad(y)X)*)"-'@ y - v*

*

lies q n - 1 for Y E g, v* E W(O)*and y E Kc. To prove C N I , let &* denote the linear form on S(p) 8 V,, which is the pull back of v* E W(O)* through the quotient map

S(p)8 V, 1 ,M

+W ( 0 )= M/m(X)M.

+

Then $J"* is zero on the subspace m(X) 8 V, N c S(p) 8 V,. If f i = C j @ v j (y3 E p, vj E V,) is a homogeneous element of N of degree n, it follows that

qn

8 v*> = C(-l)nn!B(X,Yj)n(vj,v*)v,xV:

(fi,

j

= (-l)'h!& (6) = 0,

by noting that y - B ( X , Y j ) " E m(X). Hence one gets ((Ad(y)X)*)n8yv* E y N L = N' for all v* E W(O)*,y E Kc and n 2 0. Thus we obtain (1). (2) Let m = C j vj be an element of M,, with y j E p and vj E V,. Just as in the proof of (l),we see that rn E if and only if

-

y.

0 = ( m, ((Ad(Y)X)*)" 8 y

*

v*)1

= x(-l)nn!B(y3,Ad(y)X)n (9-l

- v j ,v*)v,~v:

j

= (-l)nn!(w)m(Y), v*)w(o)xw(o)* ,

for all v* E W(O)*and y E Kc. This means m E KerT(0).

0

3.2. A suflcient condition for I = Anns(@)M

Let 0 = Ad(Kc)X be the nilpotent Kc-orbit through X. We write I for the prime ideal of S(g) defining the Zariski closure of 0,and let (Oil i = 1 , . . . ,r } be a finite set of homogeneous elements of S(p) which generates the ideal I. We set n(i):= deg Di (1 5 i 5 r) and n ( 0 ) := 0.

338

Proposition 3.1. Assume that Q n ( i ) = ( N l ) , ( i ) f o r i = 0,. . . , r . T h e n we have I = Anns(,)M. Therefore, X has irreducible associated variety V ( X )= 8, and the corresponding associated cycle C ( X ) of X turns t o be

C ( X )= dimW(0). [a]. Proof. We see for every v E V, that T(0)Div(!l)=

( 0 2*

T(0)v)(!l)= oi(Ad(y)X)T(O)v(!l)

= 0 (Y E &),

since Di E I and Ad(y)X E 0. It then follows that Div = 0 by Lemma 3.1 = ( N L ) n ( i )which , is equivalent to (2) together with the assumption Qn(i) L!PflMn(i)= (0). Hence, Di annihilates V, and so the whole M = S(p)V,. This shows I c Anns(,)M. Now, W(O)*cannot vanish by the assumption QO= ( N L ) o N- V.: This implies m(X) 3 Anns(,)M. In reality, if m(X) $ Anns(,)M, one gets S(g) = m(X) Anns(,)M by the maximality of m(X). This yields

+

M = S(g)V, = m(X)V, Hence we deduce

I=

n

c m(X)M,

i.e., W ( 0 )= (0).

m(Ad(lc)X) 3 Anns(,)M,

kEKc

because Anns(,lM is stable under Ad(&). Thus we find I = Anns(,)M and in particular V ( X ) = 8. The last assertion follows immediately from Proposition 2.1. 0 Under the assumption in Proposition 3.1, the (S(g), &)-module !P is almost equal to N L N M*, in the sense that the support of the orthogonal l!P C KerT(0) is contained in the boundary 8 0 by Proposition 2.3 and Lemma 3.1.

3.3. Difleerential opemtor of gmdient type We wish to characterize N L N M* as the kernel of a certain differential operator on p of gradient type. For this, we first take an orthonormal basis (Xl,. . . ,X,) of p with respect to the Killing form B l p x p . Then, (X; ,... ,Xz) (cf. (3.4)) gives a basis of p*, dual to (XI ,... ,X,). For a multi-index (Y = ((~1,.. . ,a,)of nonnegative integers ( ~ (1 i 5 i 5 s), we set

Da := XFl.. .x;4,

(D*)* := (X;)al.. * ( X y .

339

Then, the elements D Q (resp. (D*)") with (a1= n form a basis of Sn(p) (resp. Sn(p*))for every integer n 2 1, where la1 := a1 a, denotes the length of a. Note that

+ +

a ( D a ) ( D * ) a= a! 6,,p (Kronecker's a(DQ)(Z*)= " n!(D*)a(Z)

and

(3.11)

--

for all a,/3 of length n, and 2 E p. Here, we set a! = al! . a,!.Note that the above functions 8 ( D Q ) ( D * ) pand a ( D a ) ( 2 * ) "are constant on p. We now introduce a gradient map V" of order n by

(V"f)(Y) :=

c

-+*)a a.

8 a ( D a )f (Y),

(3.12)

la[="

for f E S(p*) @ V,!. It is easy to observe that Vn f is independent of the choice of an orthonormal basis ( X i ) l l i l , . Furthermore, V" gives an (S(g), Kc)-homomorphism

S(p*) 8 v; 3 f

s V"f

E S(p*) 8 (Sn(p*)8 v;),

where S(p*) 8 (Sn(p*)@ V ): is looked upon as the space of polynomial .: functions on p with values in Sn(p*)8 V

Lemma 3.2. It holds that Vnf = 1 8 f for every f E Sn(p*)8VV,,where 1 denotes the identitp element of S(p*). Namely, V" f is the constant function on p with the value f E S" (p*) 8 V,* . Proof. It is enough to prove the lemma for f = (D*)a8 v* with 1/31 = n and 'u* E V .: In view of (3.11), Vnf turns to be 1 V" f (Y)= +*)" 8 (a(DQ)(D*)~)(Y)v* = (D*)P8 v* = f , a.

c

lal=n

which proves the lemma.

0

We note that our submodule N of S(p) 8 V, is finitely generated over

-

S(g), since the ring S(g) is Noetherian and since S(p) 8 V, = S(p) V,.

Hence, there exist a finite number of homogeneous Kc-submodules W, (u = 1 , . . . ,q ) which generate N over S(g):

N = S ( g ) . W l + . . . + S ( g ) . W , with W, c S i U ( p ) @ V ,

cN

(3.13)

for some integers i, 2 0 arranged as il < . .. < i,. For each u = 1 , . . . ,q, let P, denote the Kc-homomorphism from Si-(p*)@ V,! to W; defined by

Pu(h)(w) := ( w , h ) (wE Wu)

(3.14)

340

for h E Siu(p*) €3 V;. Here, (. , - ) is the dual pairing in (3.5). We now set a

and let us introduce an (S(g), Kc)-homomorphism

D : S(p*) €3 v: -+ S(p*) €3w*,

(3.15)

by putting Q

Pu(Viuf(Y)) (Y E p; f

( D f ) ( Y ):=

E S(p*) €3 V:).

(3.16)

u=l

Definition 3.1. We call D ' the differential operator of gradient type associated with (V,",W * ) . The space of solutions of the differential equation Df = 0 is characterized as follows.

Proposition 3.2. One gets N L = Ker'D. Hence, the kernel of the differential operator 2) is isomorphic to the Kc-finite dual M* of M N (S(p) €3 V T ) / N as , (S(g), Kc)-modules. Proof. Let f be a homogeneous element of S(p*) @V; of degree n. We are going to show that f lies in N L if and only if D f = 0. This will prove the proposition because both KerD and N L are graded (S(g), Kc)-submodules of S(p*) €3 v;. First, the condition f E N L is written as (Dwu, f ) = 0

(3.17)

for all D E Sn-iu(p) and w, E W, (u = 1 , . . . ,q), by noting that Sm(p)€3 V, is orthogonal to f E Sn(p*)€3 V; if m # n. Here Sn-iu(p) should be understood as ( 0 ) if n - i, < 0. Since the pairing ( - , is S(g)-invariant, (3.17) is equivalent to a )

P,(a(D)f) = 0 for all D

E

5 F i ~ ( p ) (u = 1 , . . . ,q).

(3.18)

We set 'D,f := Pu((Vif)(.)) E Sn-*,(p*) €3 W,*. Then, in view of Lemma 3.2, the left hand side of (3.18) turns to be

P u ( W > f )= Pu((ViUa(D)f)(Y)) = ( W ) ( D U f ) ) ( Y >(YE PI-

341

N L if and only if a(D)Vuf = o for all D E Sn+(p) (u = 1 , . . . ,q ) , or equivalently, V,f = 0 (u = 1 , . . . ,q). This means V f = 0 as desired.

We thus find that f lies in

0

Let us define a map u from p* x V,+to W *by 9

u ( X * , w * ):= x P u ( ( X * ) i u@v*) for (X*,w*) E p* x V * ,

(3.19)

u=l

which we call the symbol map of V. For any fixed X E p, we get the following characterization of W(O)*. Proposition 3.3. The Kc(X)-submodule W(O)*of V: (cf. (3.8)) is described as

W(O)*= Keru(X*, - ) := (w* E V,* I u(X*,w*) = 0},

(3.20)

where X * E p* corresponds to X E p by (3.4). This proposition can be proved just as in the proof of [30, Lemma 3.101 (see also the proof of Lemma 3.1 (1)). We omit the proof here. 4. Isotropy representation attached t o discrete series

In this section, we assume that g = t @ p is an equi-rank algebra (cf. [17]), i.e., rank g = rank t. By using our results in Section 3, we study the isotropy representations attached to irreducible (g, &)-modules of discrete series. This develops our work in [29]. 4.1. Discrete series

We begin with a quick review on the discrete series representations, and let us fix our notation. As is well known, the complex Lie algebra g has a &stable real form go such that go = to@ po

with to := t n go, po := p n go,

gives a Cartan decomposition of go. Such a real form go is unique up to Kc-conjugacy. Take a maximal abelian subalgebra of to, and we write t for the complexification ofto in t. Since g is an equi-rank algebra, t turns to be a Cartan subalgebra of g. We write A for the root system of (g, t). The subset of compact (resp. noncompact) roots will be denoted by Ac (resp. An).

342

Let G be a connected Lie group with Lie algebra of go such that Kc is the complexification of a maximal compact subgroup K of G. An irreducible unitary representation u of G is called a member of discrete series if the matrix coefficients of u are square-integrable on G. We are concerned with the irreducible (g, Kc)-modules X of discrete series, consisting of K-finite vectors for such u's. For example, we refer to [9], [18], and also [26, I, Section 11 for the parametrization and realization of discrete series representations. Now, let X = X Abe the (g, Kc)-module of discrete series with HarishChandra parameter A E t*. Since the parameter A is regular and real there exists a unique positive system A+ of A for which A is on dominant:

mt,,

A+ := { a E A I ( A , a ) > 0 ).

(4.1)

We denote by (T,V,) the unique lowest Kc-type of X which occurs in X with multiplicity one. Set A$ := A+ n A, (resp. A: := A+ n An).The A$-dominant highest weight X (say) for T is called the Blattner parameter of X.Then, X is expressed as X = A - p, pn with p, := (1/2) . CaEa:a and Pn := (1/2) * &.a;t P.

+

4.2. Results of Hotta-Parthasamthy

In what follows, we always assume that the Blattner parameter X of X is far from the walls (defined by compact roots) in the sense of [26, I, Definition 1.71. Let M = grX = en>oMn be the graded (S(g),&)-module defined through the lowest Kc-type V,. As in Section 3, we have a natural quotient map K : S(p) @ V, + M with N = Kerr. This subsection explains the structure of graded modules M , N , and M * N N L by interpreting the results of Hotta-Parthasarathy in [9]. For this, we first decompose the tensor product p €3 V, as

p @ V, = V,' @ V;

as Kc-modules,

where 'V denotes the sum of irreducible Kc-submodules of p 8 V, with highest weights of the form X f P (P E A:), respectively. The inclusion V,- L) p @ V, naturally induces a quotient map of Kc-modules:

P : p* €3 v: = (p €3 V,)*

+ (v,-)*.

(4.2)

Hereafter, we replace p' by p through the identification p = p* by the Killing form of g restricted to p x p.

343

Let B, be the Bore1 subgroup of Kc with Lie algebra 6, = t@CffEAcg a , where ga is the root subspace of g corresponding to a root a. We set

Then, we have p = p+ @ p- as vector spaces, and p- is stable under the action of B,. If U is a holomorphic representation of B,, the i-th cohomology space Hi(Kc/Bc; U )of Kc/B, with coefficients in the sheaf of holomorphic sections of the vector bundle Kc XB, U has a structure of Kc-module. The following theorem can be read off from the proof of [9, Theorem 11 by taking into account the Blattner multiplicity formula [7] for discrete series. (See also “1; [29].)

Theorem 4.1. (I) One has N = S(p)V,-. (2) The orthogonal N L of N in S(p)@V; coincides with the kernel of the diflerential operator 2) o n p of gradient-type associated with (V;, (VT-)*) :

for f E S(p) @V,*.Here {Xi}1siss is an orthonormal basis of p with respect to the Killing form. (3) For every integer n 2 0, the dual M i of M , is isomorphic to the cohomology space: Hq(Kc/Bc; Sn(p-) €3 (c--x--spc)

with q := dim Kc/B,,

as a Kc-module. Here ( c - ~ - z ~ =denotes the one dimensional B,-module corresponding to -A - 2p, E t*. Note that the claim (2) can be deduced from claim (1) by Proposition 3.2.

Remark 4.1. The maximal globalization of dual (g, Kc)-module X* of discrete series can be realized as the kernel space of an invariant differential operator V of gradient type (on the Riemannian symmetric space for (go, to)). The operator D in the above theorem gives the “polynomialization” of 8.

344

4.3. Description of associated cycle

We are going to apply Theorem 4.1 in order to describe the associated cycles for (g, Kc)-modules of discrete series. For a positive number c, we say that a linear form p on t satisfies the condition (FFW(c)) if (p, a) 3 c for all

a E A:.

(FFW(c))

Theorem 4.1 coupled with the Borel-Weil Bott theorem for the group Kc leads us to the following proposition(cf. [29, Section 6.1]),which is crucial to describe the associated cycle of X. Proposition 4.1. ( I ) Let vl; be a nonzero lowest weight vector of V,* of weight -A. Then, N L = Ker?) contains the Kc-submodule ( S ( p - - ) @ v ; ) ~ ~

generated by S(p-) @ vl;. (2) For any integer n 2 0, there exists a positive constant c, such that

holds if the Blattner parameter X satisfies the condition (FFW(c,)). Now, let 0 be the unique nilpotent Kc-orbit in p which intersects pdensely. Then one sees that 8 = Ad(Kc)p-. As before, we write I for the prime ideal of S(g) defining 8. It follows from the the claim (1) in Proposition 4.1 that Annq,)M C I, i.e., V ( X )3 8. Also, the same claim shows p- @ v; E Ker P,which can be easily verified by noting that -A - /3 (p E A:) cannot be a weight of (V;)*. Take an element X E 0 n p-. By Proposition 3.3, we find that the Kc(X)-module W(O)*= (M/m(X)M)* c V: consists exactly of all the vectors v* E V,* satisfying P(X 8 v*) = 0. Let N K ~ ( Xp-) , be the totality of elements k E Kc such that Ad(k)X E p- (cf. [3]). For any subset R of NK~(X, p-), we denote by Ux(R) the Kc(X)-submodule of V,* generated by R-' * v;: Ux(R) := ( R - l * ~ ; I ) ~ c ( x ) . Then, we readily find from p- 8 vl; E Ker P that

Ux(R)c W(O)*, and so ( Xn @UA(R))K,C 9, C ( N l ) , i , for every n 2 0. Moreover one gets the equality ( X n @ Ux(R) )K@ = (S"(p-)@v?, ) K ~ ,

(4.6)

(4.7)

if Ad(R)X c p- is Zariski dense in p-. This is true when R equals the whole N ~ ~ ( x , p -because ), Ad(NK,(X,p-))X = 0 0 p- is dense in p-.

345

As in Section 3, we take homogeneous generators Di (i = 1 , . .. ,T ) of the ideal I such that degDi = n(i). We set c ( I ) := maxi(c,(i)). By virtue of Proposition 3.1 together with (4.5), (4.6) and (4.7), we come to the following conclusion.

Theorem 4.2. Assume that the Blattner parameter X of discrete series X is far from the walls and that it satisfies the condition (FFW(c)) with c = c(I). (1) One gets I = Anns(,,)M and so V ( X ) = Ad(KC)p- = 8.Moreover, the K c ( X )-module W *contmgredient to the isotropy representation (wo,W ) is described as

w *= W(O)*= {v* E v; I P ( X €3 v * ) = O } ,

(4.8)

where X E 0 n p-, and P : p €3 V,* + (V;)* is the Kc-homomorphism in (4.2). (2) Let R be a subset of N K C ( X ,p-) such that Ad(R)X is Zariski dense in p-. Then, the Kc(X)-submodule Ux(R) = (R-' . V ; ) K ~ ( X )c W *is

exhaustive in the following sense: for every integer n 2 0, one has

(X"@ W * ) K=~(x"C3 % ( R ) ) K ~

(4.9)

i f X satisfies FFW(c,).

Remark 4.2. (1) The assertions I = Ann.q,lM and V ( X )= Ad(Kc)phave been obtained in [29]. But, in that paper, we did not discuss the possibility of applying the results to describe the isotropy representation. (2) One should get a result similar to Theorem 4.2, more generally for the derived functor modules d4(X). (3) Compare Theorem 4.2 (1) with Chang's result [3, Proposition 1.41 established by means of the localization theory of Harish-Chandra modules. 4.4. Submodule

Ux(Qc)

In this subsection, we give a natural choice of R c N K ~ ( X , ~ for - ) which we expect to have the property (4.9). Let ll be the set of simple roots in A+. We write S = 1T n A, for the totality of compact simple roots. Then, there exists a unique element Hs E t such that

346

The adjoint action of H s yields a gradation on the Lie algebra g as g=

@ g(j)

with g ( j ) := ( 2 E gl (adHs)Z = j Z } .

j

Here j runs through the integers such that Ijl 2 ~ ( H s with ) the highest root 6. Note that

e = @ dj), j:even

P = @ g(j) with j:odd

Pk =

@

g(fj).

j>O,odd

Now, we set

q := @ g ( j ) , I:= g(0)

ct

and u := @ g ( j ) .

j 50

j
Then, q = I@ u gives the Levi decomposition of the standard parabolic subalgebra q of g associated with the subset S of rI. We write Q (resp. Q,) for the parabolic subgroup of Gc := GEd (resp. of Kc) with Lie algebra q (resp. q n t). The group Q (resp. Q,) admits the Levi decomposition Q = LU (resp. Qc = L,U,), where L and U (resp. L, and Uc) are the connected subgroups of Q (resp. Q,) with Lie algebras I and u (resp. I and u n t) respectively. Note that Ad(Lc) = L. The parabolic subgroup Q acts on its nilradical u, and so Q , acts on p- = p nu by the adjoint action. Thus, Qc is contained in N K @ ( Xp-) , for all X E p-, and the corresponding Kc(X)-submodule Ux(Qc)of V: turns to be (4.10) Here, (Vfc)*= U(I)vf, denotes the irreducible L,-submodule of V: generated by the lowest weight vector vf,. We can now apply Theorem 4.2 to deduce Corollary 4.1. Under the assumption in Theorem 4.2, the Kc(X)submodule Ux(Qc) of W * is exhaustive in the sense of (4.9), i f p- i s a prehomogeneous vector space under the adjoint action of the group Q,, and if X E 0 n p- lies in the open Q,-orbit in p-. 4.5. Relation to the Richardson orbit

We end this article by looking at the condition for p- in Corollary 4.1, and also some related conditions, in relation to the Richardson Gc-orbit associated with the parabolic subalgebra q. First, let us recall some basic facts on the Richardson orbit (cf. [12, Chapter 51). The Gc-stable subset Gc u (C g) forms an irreducible affine

347

variety of g whose dimension is equal to 2 dimu. Noting that G c - u consists of nilpotent elements only, there exists a unique Gc-orbit d such that

by the finiteness of the number of nilpotent Gc-orbits in g. d is called the Richardson Gc-orbit associated with q. The parabolic subgroup Q acts on u prehomogenenously, and d n u turns to be a single Q-orbit in u. Moreover, the centralizer in g of any element X E d n u is contained in q. Now, we havetwo nilpotent Gc-orbits Gc . 0 and d with the closure relation G c - O C 6. By virtue of a result of Kostant-Rallis [15, Proposition 51, this relation implies that

1 1 dimO= -dimGc.O< -dimd=dimu. 2 2

(4.11)

In particular, we find that the Gelfand-Kirillov dimension dimV(X) = dim 0 of discrete series X cannot exceed dim u. The following proposition tells us when these two orbits turn to be equal. Proposition 4.2. The following three conditions (a), (b) and (c) o n the positive system A+ = (a1 ( h , a )> 0) are equivalent with each other:

(a) Gc . O = d,

(b) dim 0 = dim u,

(c)

d n p- # 0.

In this case, 0 n p- is a single open Q,-orbit in p-, and so one gets the conclusion of Corollary 4.1. Proof. The equivalence (a) H (b) is a direct consequence of (4.11). The condition (a) immediately implies (c), since 0 (C Gc .0 = 6 )contains an element of p-. Conversely, if d n p- # 0, this is a nonempty open subset of p-, since d n p- = (d n u) n p- with d n u open in u. Hence, 6 n pintersects 0. We thus get (c) 3 (a). This proves the equivalence of three conditions in question. Next, we assume the condition (b) (H (a) % (c)), and let X be any element of 0 n p-. We write a,(X) for the centralizer of X in a Lie subalgebra 5 of g. By noting that a,(X) c q, the dimension of the Q,-orbit Ad(Q,)X is calculated as

dimAd(Q,)X = dimq n t - dimasne(X) = (dime-dimunt) -dimJe(X) = d i m 0 - dimu n t = dimu - dimu n t = dimp-,

348

where we used the condition (b) for the forth equality. This shows that the orbit Ad(Q,)X is open in p- for every X E 0 n p-. We thus find that 0 n p- forms a single Q,-orbit, because of the uniqueness of the open Q,-orbit in p-. 0

Remark 4.3. Each of the conditions (a), (b) and (c) in Proposition 4.2is equivalent to Assumption 2.5 in [2]concerning the generically finiteness of the moment map defined on the conormal bundle Ti,(Gc/Q), where 21 is a closed Kc-orbit in Gc/Q through the origin eQ. Suggested by Corollary 4.1 and Proposition 4.2, let us consider the following three conditions on p- which depends on the choice of a positive system A+:

b n p - # 0 (w d i m U = d i m u 0 n p- is a single Q,-orbit,

Gc.U=b),

p- is a prehomogeneous vector space under Ad(Q,).

(C1) (C2) (C3)

Proposition 4.2 says (Cl)=+ (C2),and the implication (C2)=+ (C3)is obvious. As for the conditions (C2)and (C3),we can show the following

Proposition 4.3. One gets (C3)if 0 ng( -1) Ad(Q,)(O n g(-l)) = 0 n p- assures (C2).

# 0. Moreover, the equality

Proof. Let X E 0 n g(-l). Since 0 = Ad(Kc)X contains a nonempty open subset of p-, we find that [t,X]3 p-. We set t+ := @>0g(2j). Then t = t n q @ t+ is a direct sum of vector spaces. Then it follows from the assumption X E g(-1) that [e+,X] C p+ and [t n q,X] C p-. We thus obtain

p- = [e, X] n p- = [en4, XI. Hence Ad(Q,)X is open in p-, and one gets (C3). The above argument shows that any element X E 0 n g(-1) lies in the 0 unique open Q,-orbit in p-. This proves the latter claim, too. Following Gross-Wallach [5],we say that a discrete series (g, &)-module X is small if ~ ( H s 5) 2, or equivalently, g ( j ) = (0) if Ijl 2 3. Here 6 is the highest root of A+ as before. In this case, one has p- = g(-1), and so the above proposition implies

349

Corollary 4.2. The subspace p- corresponding t o a small discrete series admits the property (C2). Remark 4.4. By case-by-case analysis, Chang [3] proved the property (C2)for any discrete series representations of simple Lie groups of R-rank one.

4.6. Condition (Cl)f o r S U ( p , q )

It should be important to study when p- admits the properties (Cl),(C2) and (C3),respectively. Toward this direction, we end this paper by giving an explicit, combinatorial criterion for the condition (Cl)in case of G = S U ( p , q) with n = p q. In this case, we have g = sI(n,C), and let

+

a1

= € 1 - €2,

a2

= €2 - €3,

. . . , an-1

= €,-I

-&,

be the simple roots for (a, t) by the standard notation of Bourbaki. Then p-’s in question are in one-one correspondence to the set of noncompact .. . , where nl, . . . ,nt-1, and positive roots {a,,, a,,+,,, nt := n - (nl . . . nt-1) are positive integers such that

+ +

C

nj

= p or q.

j:odd

Theorem 4.3. p- satisfies the condition (Cl)if and only i f the corresponding partition (nl,n2,.. . ,nt) of n is unimodal, that is, there exists a positive integer k (15 k 5 t ) such that n1

5 - .- 5 n k - 1 5 n k 2 n k + 1 2

2 nt.

If p , q 5 3, every partition (nl,. . . ,nt) is unimodal, and we always get the property (Cl). On the contrary, the partition (2,1,2)for SU(4,l)is not unimodal, and (Cl)fails in this case, where d i m 0 = 7 < 8 = dimu. We will discuss the detail elsewhere. Acknowledgments The author thanks Hiroyuki Ochiai for stimulating discussion.

References 1. W. Borho and J.-L. Brylinski, Differential operators on homogeneous spaces. I11 Characteristic varieties of Harish-Chandra modules and of primitive ideals, Invent. Math., 80 (1986), 1-68.

350 2. J.-T. Chang, Characteristic cycles of holomorphic discrete series, Trans. Amer. Math. SOC.,334 (1992), 213-227. 3. J.-T. Chang, Characteristic cycles of discrete series for R-rank one groups, Trans. Amer. Math. SOC.,341 (1994), 603-622. 4. A. Dvorsky and S. Sahi, Tensor products of singular representations and an extension of the 0-correspondence, Selecta Math., New Series 4 (1998), 11-29. 5. B. Gross and N.R. Wallach, Restriction of small discrete series representations to symmetric subgroups, in: The mathematical legacy of HarishChandra (Baltimore, MD, 1998), 255-272, Proc. Sympos. Pure Math., 68, Amer. Math. SOC.,Providence, RI, 2000. 6. A. Gyoja and H. Yamashita, Associated variety, Kostant-Sekiguchi correspondence, and locally free U(n)-action on Harish-Chandra modules, J. Math. SOC.Japan, 51 (1999), 129-149. 7. H. Hecht and W. Schmid, A proof of Blattner’s conjecture, Invent. Math. 31 (1975), 129-154. 8. T. Hirai and H. Yamashita, Hyougenron nytimon seminar (Introductory Seminar on Representation Theory), in Japanese, Yiiseisha, Tokyo, 2003, pp. i-xi and 1-329, ISBN4-7952-6898-3. 9. R. Hotta and R. Parthasarathy, Multiplicity formulae for discrete series, Invent. Math. 26 (1974), 133-178. 10. R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. SOC., 313 (1989), no. 2, 539-570. 11. R. Howe, Transcending classical invariant theory, J. Amer. Math. SOC.,2 (1989), no. 3, 535-552. 12. J.-E. Humphreys, Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, 43,American Mathematical Society, Providence, RI, 1995. 13. S. Kato, On the Gelfand-Kirillov dimension and Bernstein degree for unitary highest weight modules (in Japanese), Master Thesis, Kyushu University, 2000, 77 pages. 14. S. Kato and H. Ochiai, The degrees of orbits of the multiplicity-free actions, in: “Nilpotent orbits, associated cycles and Whittaker models for highest weight representations”, Astkrisque, 273 (2001), 139-158. 15. B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math., 83 (1971), 753-809. 16. K. Nishiyama, H. Ochiai and K. Taniguchi, Bernstein degree and associated cycles of Harish-Chandra modules - Hermitian symmetric case, in: “Nilpotent orbits, associated cycles and Whittaker models for highest weight representations”, Astkrisque, 273 (2001), 13-80. 17. H. Rubenthaler, A new proof of the Borel-de Siebenthal theorem, the classification of equi-rank groups, and some compact and semi-compact dual pairs, to appear in Hokkaido Math. J., 33,no. 1, 2004. 18. W. Schmid, Some properties of square-integrable representations of semisimple Lie groups, Ann. of Math., 102 (1975), 535-564. 19. K. Taniguchi, Discrete series Whittaker functions of SU(n,1) and Spin(2n, l ) ,

351 J. Math. Sci. Univ. Tokyo, 3 (1996), no. 2, 331-377. 20. K. Taniguchi, On discrete series Whittaker functions SOo(2n, 2) (Japanese), in: ‘LAutomorphicforms on Sp(2; R) and SU(2,2), 11 (Kyoto, 1998)”, RIMS Kakytiroku, 1094 (1999), 11-27. 21. D.A. Vogan, Gelfand-Kirillovdimension for Harish-Chandra modules, Invent. Math. 48 (1978), 75-98. 22. D.A. Vogan, Associated varieties and unipotent representations, in: “Harmonic Analysis on Reductive Groups (W.Barker and P.Sally e&.)”, Progress in Math., Vol. 101, Birkhauser, 1991, 315-388. 23. D. A. Vogan, The method of coadjoint orbits for real reductive groups, an: “Representation Theory of Lie Groups (eds. J. Adams and D. Vogan)”, 177238, IAS/Park City Mathematics Series 8, AMS, 2000. 24. A. Wachi and H. Yamashita, Isotropy representations for singular unitary highest weight modules (tentative), in preparation. 25. H Yamashita, Finite multiplicity theorems for induced representations of semisimple Lie groups 11: Applications to generalized Gelfand-Graev representations, J. Math. Kyoto. Univ., 28 (1988), 383-444. 26. H. Yamashita, Embeddings of discrete series into induced representations of semisimple Lie groups, I: General theory and the case of S U ( 2 , 2 ) , Japan. J. Math.,l6 (1990), 31-95; 11: generalized Whittaker models for S U (2, 2), J . Math. Kyoto Univ., 3 1 (1991), 543-571. 27. H. Yamashita, Criteria for the finiteness of restriction of U(g)-modules to subalgebras and applications to Harish-Chandra modules: a study in relation to the associated varieties, J. Funct. Anal., 1 2 1 (1994), 296-329. 28. H. Yamashita, Associated varieties and Gelfand-Kirillov dimensions for the discrete series of a semisimple Lie group, Proc. Japan Acad., 70A (1994), 50-55. 29. H. Yamashita, Description of the associated varieties for the discrete series representations of a semisimple Lie group: An elementary proof by means of differential operators of gradient type, Comment. Math. Univ. St. Paul., 47 (1998), 35-52. 30. H. Yamashita, Cayley transform and generalized Whittaker models for irreducible highest weight modules, in: “Nilpotent orbits, associated cycles and Whittaker models for highest weight representations”, Asthrisque, 273 (2001), 81-137. 31. H. Yamashita, Associated cycles of Harish-Chandra modules and differential operators of gradient type, RIMS Kijkyiiroku, 1183 (2001), 157-167. 32. H. Yamashita, Isotropy representations attached to the associated cycles of Harish-Chandra modules, RIMS Kakyiiroku, 1238 (2001), 233-247. 33. H. Yamashita, Isotropy representation and projection to the PRVcomponent, RIMS Kakyiiroku, 1294 (2002), 62-71. 34. T. Yoshinaga and H. Yamashita, The embeddings of discrete series into principal series for an exceptional real simple Lie group of type Gal J. Math. Kyoto Univ., 36 (1996), 557-595.