Harmonic Analysis and Group Representations (C.I.M.E. Summer Schools, 82)

A. Figà Talamanca ( E d.)

Harmonic Analysis and Group Representation Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Cortona (Arezzo), Italy, June 24 - July 9, 1980

C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy [email protected]

ISBN 978-3-642-11115-0 e-ISBN: 978-3-642-11117-4 DOI:10.1007/978-3-642-11117-4 Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2010 Reprint of the 1st ed. C.I.M.E., Ed. Liguori, Napoli & Birkhäuser 1982 With kind permission of C.I.M.E.

Printed on acid-free paper

Springer.com

C O N T E N T S

L. AUSLANDER and R. TOLIMIERI Nilpotent groups and Abelian varieties

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

pag.

5

M. COWLING Unitary and uniformly bounded representations of some simple Lie groups

"

49

M. DUFLO Construction de representations unitaires d'un groupe de Lie

"

129

R. HOWE On a notion of rank for unitary representations of the classical groups

"

223

A. KORANYI Some applications of Gelfand pairs in classical analysis

...

"

333

V. VARADARAJAN Eigenfunction expansions of semisimple Lie groups

..........

"

349

J. ZIMMER Ergodic theory, group representations and riqidity

.........

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

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

423

CENTRO INTEmAZIONALE

MXTEFlATICO ESTIVO

(c.I.M.E.

NILPOTENT GROUPS AND ABELIAN V A R I E T I E S

L.

AUSLAMDER

AND R.

TOLIMIERI

Lectures on NILPOTENT GROUPS AND ABELIAN VARIETIES by L. Auslander and R. Tolimieri Introduction A. A. Albert, in an immense burst of creative energy succeeded in solving the "Riemann matrix problem." Although this is one of the great mathematical achievements of our century, there are few systematic accounts of Albert's work. Perhaps, C. L. Siegel's account [6] comes the closest to providing us with a view of this marvelous achievement. Albert's and Siegel's treatment are difficult because their arguments are based on matrix calculations. Because a coordinate system has been chosen, there is a hidden identification of a vector space with its dual and matrices play the role of both linear transformations and bilinear forms.

In these notes, we will present a way of using nilpotent groups to formulate the ideas of Abeiian varieties and present part of the existence theorems contained in Albert's work. A full treatment of the existence part of Albert's work will appear in [ 4 ] . Our approach rests o n nilpotent algebraic groups. This enables us to present a matrix-free treatment of the Riemann matrix problem. We hope this approach will reawaken admiration for, and interest in, Albert's achievement.

TABLE OF CONTENTS

I.

Associative Algebras and Nilpotent Algebraic Gr0up.s.

2.

The Jacobi Variety of a Riemann Surface and Abelian Varieties.

3.

Morphisms of Abelian Varieties and the Structure of Riemann Matrices.

4.

Riemann Matrices whose Multiplier Algebras are Totally Real Fields.

5.

The Involution Problem for Division Algebras of the First Kind (Part 1).

6.

The Involution Problem for Division Algebras of the First Kind (Part 11).

7.

Existence of Riemann Matrices for Division Algebras of the First Kind.

I..

Associative Algebras and Nilpotent Algebraic Croups

In these notes the word field will denote either the reals, IR, the coniplcx, algebraic number field, k, containing the rationals, Q and we let

[&,a = h
O

group N(A(v)).

i

A (V) is the radical of It is clear that

3

/\(v)..

i

= Z i>2

Hence, we may form the k-nilpotent algebraic (V) is an ideal in A ( v ) . Hence we may form

Since S 2 ( V ) i s v e r y important in the rest of this paper, we will present another more explicit description or "presentation" of g 2 ( v ) . As a set

and the group law of composition is given by

whcrc v26V and W ~ E V Afor V a = 1,2 b , ( V ) is a 2-step k-nilpotent algebraic group with center (O,w), WCVAV,and it is called thc

T s2-step k-nilpotent

If G

group over V. The reason for the name, "free," is the following:

a 2-step k-nilpotent algebraic group and

:i

is k-lrnear, then there exists a homomorphism

such that the kernel of F is a k-algebraic group and the following diagram is commutative:

The nilpotent algebraic groups N ( A ( v ) ) and g 2 ( v ) exhibit a property that will have enormous implications in our later work. We observe that the representations of N ( A ( v ) ) or g 2 ( V ) arising from the associative algebra structure can be defined by linear equations whose coefficients are in

This will enable us to consider N ( A ( v ) ) and g 2 ( V ) as Qnilpotent

algebraic groups. We will now discuss how this can be done. Let G be a 4-nilpotent algebraic group and assume that a set of equations defining G can be chosen to have coefficients in K c k. We will then say that G is defined over K. Now let V be an m-dimensional k-vector space. If [k:K] = h, then we may consider V as an mh dimensional K vector space that we will denote by V(K). Clearly, k linear transformation of V gives rise to a K linear transformation of V(K). Thus we have an isomorphism

We will call r(K) the isomorphism of reducing the field from k to K . It is easily seen that if G is a k-algebraic group defined over K, then r(K)(G) will be a K-algebraic group. We will call r(K)(G) the K-algebraic group obtained by reducing the field of G. Again, let G be a k-algebraic group defined over K.

K c k. Consider G(K) c G

consisting of those points in GL(m,k), all of whose coefficients are in K. Then G(K) will be a K-algebraic group in GL(m,K). If all the k points of G(K) = G, we will call G(K) a K-form of the k-algebraic group

G.

It should be remarked, that

G

may have non-isomorphic

K-forms. An example may help the reader understand all this better.

Let k

be a totally real

algebraic number field over Q and let [ k : q = h. Consider the k-algebraic subgroup G of GL(2.k) defined by

A set of defining equations for G are given by xl = x 2 = ~ 1 and x 2 ~= 0 where

Clearly, G may also be considered as the k-points of the Q algebraic group

We will now give an explicit map for r ( 0 . Let r denote the regular representation of k over UB Then

where we view the right hand matrix in GL(2h,B. Thus r(Q(G) = G(Q) c GL(2h,Q) @algebraic group. GL(2h.W. Since

Let G(@,

denote the group of &points of G(@.

k is totally real, there exists A

and xi:k---xi(k),

E

is a

Then G ( O R c

GL(h,lR) such that

i = I , ...,h, is an isomorphism of & into B Indeed, x I ,....x,

are distinct

isomorphisms. Now

where D is as above. Now, N(A(v)) and S 2 ( V ) are easily seen to be defined over Q and so both may be considered

- by reducing the field - as

V(Q

V lifts to a morphism

--,

Q algebraic groups. Hence, we have the identity map

There are certain homomorphisms of g 2 ( V ) that will play an essential role in our theory. We will now establish a language with which t o carry out this discussion. We begin by listing some standard notation that we will follow.

If

V

and

W are k-vector spaces, wc use

Hom(V,W) to denote the k-vector space of k-linear maps and V* = Hom(V,rC). the dual vector space. For T

E

Hom(V,W), we have T* E HO~(W'.V*) and we will identify

v*'

with

v. Let BiI(V) denote the vector space of bilinear forms on V x V . For B c Bil(V), define L(B) a H O ~ ( V . V ' ) by (L(B)(u))(v) = B(u,v). u,v L(B)*

E

E

V. Since L(B) c n o m ( v , v * j , we have

H O ~ ( V . V * ) . Clearly, B is alternating if and only if L(B)* = -L(B), and B is

symmetric if and only if L(B)* = L(B).

The set of alternating forms will be denoted by

Alt(V), Sym(V) will denote the set of symmetric forms. and Bil(V) = Alt(V) @ Sym(V). If

L(B) is nonsingular. we say that B is non-singular and the space of non-singular bilinear forms will be denoted by Bilx(V). Analogously, we will use the notation Altx(V) = Alt(V) f l Bilx(V) and SymX(V) = Sym(V) fl Bilx(V). Let S 2 ( v ) denote the free 2-step k-nilpotent group over V. The dual space to V

/\ V

is V* A v*, and we have the commutative diagram

where A

E

Alt(V) and I(A)

Now, for A

E

E

V* A

v*;this enables us t o identify Alt(V) and V'

A

v*.

Alt(V), we may define a group structure N(A) on the set V x & whose law

of multiplication is

where vI,v2aV and kl,k2ck. Then N(A) has (o,k), k a k in its center and N(A) modulo its center is Abelian. Hence. N(A) is a 2-step k-nilpotent algebraic group. Define the surjection

by P(v,k) = (v,l(A)w). (v,w)

6

S 2 ( v ) . If i : V

--

V is the identity mapping, the following

diagram is commutative:

We will call such morphisms of g 2 ( V ) polarizations and denote the ser of polarimtiom by

P(V). Clearly, we may identify P(V) with Alt(V) as above. If A

E

Altx(V), all N(A) are

isomorphic and we will call N(A) a &-Heisenberg group. The corresponding polarizations will be denoted by PX(V). If dim V = 2m, we will sometimes use Nzm+,(k) to denote N(A) and call N2m+l (k) the 2 m + l &-Heisenberg group. Fixing an isomorphism of the center 3 of NZm+, (k) with k

. as when we present NZm+, (4) as N(A), will be called an orientation of

* ~ m + , (1' . The presentation N(A) of the Heisenberg N2m+l (4) has the additional property of determining an isomorphism which we will denote by A:V by P:V

-+

v*. This follows from the fact that

--D

V* or, if P corresponds to A,

A is non-degenerate.

2.

The Jacobi Variety of a Riemann Surface and Abelian Varieties

In this lecture we will need two special examples of a general phenomena; accordingly, we will begin with the general case and then specialize to the examples of interest to us. Let M be a compact manifold and let H*(M,BI) and H*(M,P

be the cohomology rings

of M with real and integer coefficients, respectively. If radical of H'(M,W and H*(M,W = R @ group N(H'(M,W).

a.

= Z H1(M,R), then is the r>O Hence, we may form the nilpotent algebraic

which we will henceforth denote by N(M).

Now the Lie algebra of

N(M) is the Lie algebra associated with k% by

Since, for x

E

H~(M,BL)and y

E

HJ(M,D, we have

xy = (

It follows that, if x

E

- i ) " y ~E

e

.

H*~(M,BL),then

C H"(M,H)cB is in the center of as a C H~'+'(M,IR). Then [x.y] E C H"(M,R) . Thus &? is a

and so XJ

H;+J(M,R)

Lie algebra.

Further, if

2-step nilpotent Lie algebra

and so N(M) is a 2-step nilpotent Lie group. By the standard theory of cohomology rings, there is a natural injection

such that ~(H'(M,P)

a.

is a lattice in the vector space H*(M,W . Let B ( p = i ( H ' ( ~ , w n

Then we may argue as before and obtain that

is a subgroup of N(M).

It is then easily verified that T(M)\N(M) is a compact manifold,

called a nilmanifold. Hence. we have functorally assigned to every compact manifold M, the compact nilmanifold T(M)\N(M).

By [ I ] , there exists a unique Qnilpotent algebraic group

N d M ) such that T(M) c Mw(M) c N(M). We will call N d M ) the topological rational form of N(M). (It may happen that N(M) has other rational forms not isomorphic to N d M ) ) . The groups N(M) and P(M) constructed above have an additional structure that we will

now discuss. As a set N(M) = X x Y

. where

X = { 1 + n I ne

C H1(M,BP),

i odd)

Y = (1+n

C H'(M,E),

i even, i > O )

Inr

where X and Y are vector spaces. If (x,y)

E

X x Y , then the multiplication in N(M) is

given by

where B : X x X

--

Y is skew symmetric. Such a presentation of a 2-step k-algebraic

group will be called a grading. Notice that the presentation of S 2 ( V ) as V x VAV in Section I was a graded presentation and N(A) was a graded presentation of the Heisenberg group. The main purpose for introducing the graded structure of 2-step k-nilpotent algebraic groups is the following: If N = X x Y is a graded 2-step k-nilpotent algebraic group and if a : V

--

X is a morphism, then cr has a

unique extension to a morphism

~*:s~(v)--N

that preserves gradings. This is because the composite mapping vx

axa

v--D

xxx---

B

Y

is a n alternating bilinear mapping on V x V and s o we have a unique linear mapping P(B):VhV --. Y that completes the commutative diagram

-A

vxv

It follows that if

A

is any morphism of

V

VAV

then

A

determines a unique graded

morphism of s Z ( V ) . For the rest of this paper, we will restrict ourselves to graded nilpotent algebraic groups and all morphisrns will be grading preserving morphism. ,qruded, hilt it wrll he whur ussures the uniqueness of Jucir.s.sron.

Henceforth, we will drop the word

variotrs morphisms that occur i n the

Let M be a complex manifold. Then a s in [9], the complex structure o n M determines an automorphism J ( M ) of H*(M,U. Further, if

is a complex analytic mapping. then

It is clear that the complex structure determines a n automorphism of N(M), which we will also denote by J(M). It is important to note that J(M) may nor induce a n automorphism of T(M) o r even of N d M ) . T o illustrate this, let us see how all this works for the m complex dimensional torus. Let W be an m dimensional complex vector space and let L be a discrete subgroup of W such that W / L is compact. We will begin by discussing another way of looking a t W. Clearly, W is also a real vector space W ( U of real dimension 2m. Let e , ,...,em be a basis of W. Then el,iel,...,emaie,,,i = v

- 1, is a basis of

W ( m . For w

E

W, the mapping J:W

--,

iW

defines a n automorphism of W ( m which in terms of the above basis is given by

= m Jo where J o =

J = 0

(-

1

o)

Jo

Notice that J has the property that J' =

0

- I, where

I is the identity mapping.

Let A be a real linear transformation of W ( m . When does A induce a complex linear transformation of W? We will now verify that the answer is when

By a straightforward computation, one verifies that

if and only if a = d, b = -c.

But since the regular representation r of C over BL is given

we have that our assertion is true for m = 1. Relative t o the basis e l , ...,em, let C = (Cap) 6 Hom(W,W)

a,fl

= 1 ,...,m

.

Then relative to the basis el.iel, ...,em,iem of W ( U . C is given by C = (r(Ca,j)). It then follows easily that J C = CJ. Now assume that J A = AJ and write

A

as an m x m matrix whose entries are 2 x 2

matrices

By a direct computation, we have that JA = AJ implies that

J o Aap = Aag J , Hencc each A,,p = r(COlc). Cap

6

all

a,fl

.

C a n d we have that A gives a complex linear transforma-

tion of W. Now let V be a 2m dimensional real vector space and let J be an B-linear transformation such that J* = -I.

From the pair

(V,J), we will construct a complex m-dimensional

vector space W such that W ( U = V and the automorphism J:w --. iw is the mapping J. Let e l # 0. e l

6

V and let f l = J ( e l ) . Let L ( e l , f l ) denote the linear subspace of V

spanned by c l and f l . Then L(el,f,) is J invariant and since J~ = I, there exists V2 such that JV, = V, and

ldcntifying L ( c l . f l ) with C as a real vector space by

wc can solvc our problem by induction.

Henceforth (V,J) will be called a complex vector space and J will be called rhe complex structure. Let us now consider the complex torus V/L, where (V,J) is our complex vector space. It is well known that the 1-forms dxl,dyl,...,dx,,,dy, V = xlel

+ y l f l + ... + xmem + yJm

are a basis of

H'(v/L,IR),

and Je, = f, and Jf, = -el, i = 1. ...,m. If V

where =

H1(V/L,lR), then

Viewing V/L as a Lie group, we may identify the tangent space to V / L at the identity with V and V* may be identified with the dual space t o V. Hence, J induces J* on V* such that (J*)' =

-

I. Thus a complex structure on a torus V/L

is equivalent t o an automor-

phism J* of S 2 ( V * ) such that modulo the center of S 2 ( V * ) , (J*)* = Now let S be a compact Riemann surface of genus m 2-sphere with

m

imply that N(S)

handles.)

> 0.

(Topologically, S is a

The classical facts about the cohomology ring H*(s,R) easily

is isomorphic to N2m+l (B). The orientation of

orientation of N 2 m + l(I@.

- I.

Let J(S)

S then determines an

be the automorphism of N2m+l(B) induced by the

complex structure on S, then J(S) acts trivially on 3, the center of N(S), and if J I denotes the action of J(S) on N(S)/,y, then ( J ~ ) '=

- I.

Finally

where [a.b] = aba-'b-I. Definition: Let N2,+,(IR) be an oriented B-Heisenberg group and let J be an automorphism of N2m+1(lR)satisfying all the above conditions.

We will call J

a positive definite

CR

structure. We are almost ready to define the concept of a Jacobi variety, but it will be convenient to make a slight detour in order to first define the concept of a dual torus. Let V be an n-dimensional BP-vector space and let L be a discrete subgroup of V such that V/L is compact or a torus. Let V' be the dual vector space t o V and let L* c V'

be the subset of V* such that P'

E

L* if and only if P*(L) c Z. where Z denotes the

integers. One verifies that L' is a discrete subgroup of V* such that v'/L' we will denote by (v/L)*.

is a torus which

We call L* the dual lattice to L and v'/L* = (v/L)* the dual

torus to V/L. Since (L')' = L, we have ((v/L)*)* = V/L.

Notice that L c V determines a unique rational vector space V(Q

and L' c v ' ( Q ~ c

v'.

described H*(v/L,B)

Clearly. V*(Q

as A(v').

such that

can be identified with ( ~ ( g ) * . We have already

Let 92 =

normal subgroup of N ( A ( v ' ) ) determined by

Z H'(v/L, I)and consider the ideal g2.Then one verifies 152

G ( f l Z ) , the that

is the dual torus to V/L. Now form N(S)/r(S)3, where 3 is the center of N(S). Then N ( S ) / r ( S ) g is a toms with a complex structure J* determined by the positive definite C R structure o n N(S) and N(S)/T(S)y determine a unique rational form for N(S)/d.. torus t o N ( S ) / r ( S ) p The complex torus (V/L.J)

Let V/L be the complex dual

is called the Jacobi variety of S.

The Jacobi variety of S is related to S by two important mappings. The first is at the cohomology level; the second is at the manifold level. Consider N(S).

Since this is a graded nilpotent group, there exists an A

such that N(S) = N(A), where V*

5

--

dual torus of N ( S ) / r ( S ) j , be the Jacobi variety of S. Now A ( v * ) = H*(v/L,B) have a natural homomorphism N(V/L) ideal

~ l t ~ ( ~ * )

N ( S ) / 3 and 3 is the center of N(S). Let V/L, the s 2 ( v 9 ) with kernel

and so we

~(9) where . 9

is the

Z H'(V/r,I).Since

123

There is a unique surjection

P : s2(vg)

--

N(S) (of course,

P

is the polarization

determined by A ) such that

is a commutative diagram.

Let J(V/L)

be the automorphism of g 2 ( v 1 ) induced by

J.

Then J(V/L) is the same automorphism of g 2 ( v 9 ) as that induced by the complex structure on V/L. The mapping P is the first mapping at the cohomology level that we sought.

It is natural to ask if there exists a complex analytic mapping f : S -+

equals the composition N(v/L)---s2(v0)

P -+

V/L such that

N(S). The answer is yes and the mappingof

is called the Jacobi imbedding. Working this out in detail would take us too far afield from our main object so we will have to refer the reader t o any of the many standard texts (for instance, 171) for the proof of this result. No;w a complex torus that is a Jacobi variety has the remarkable property of having sufficiently many meromorphic functions to separate points. Definition:

A complex torus (V/L,J) is called an Abelian variety if it has sufficiently many

meromorphic functions to separate points. Remarks:

Not every complex torus is on Abelian wriety. Not every Abelian voriety is a Jocobi

variety. We will now formulate necessary and sufficient conditions for a complex torus t o be an Abelian variety. Let (V/L,J)

be a complex structure and let g 2 ( v 0 ) = ~ ( A ( v * ) / H'(v/L.BI)) z and i23

let J* be the automorphism of g 2 ( v 0 ) induced by J. Recall that L c V(Q) c V and let V*(Q be the dual rational vector space t o V(Q) with ~ * ( ca V.

Then g 2 ( v 0 ( a ) c

S2(~*). Definition:

We call P

E

P O I ~ ( V *rational ) if the kernel P in # 2 c ~ ' )

is the closure of a

subspace of s 2 ( v * ( @ ) . We can now state the fundamental theorem of Abelian varieties. Again we will have to leave a proof to outside sources such as 171. Nilpotent proofs can be found in Theorem:

[a] and [3].

A necessary and sufficient condition for (V/L,J) to be an Abelian variety is that

there exists a rational polarization p : g 2 ( v D )--, N2,+,(1R), where N2,+,(B)

is oriented,

such that 1)

The kernel of P is J* invariant.

2)

The automorphism that J* induces on N2,+,(B4) is a positive definite CR structure.

Definition:

If (V/L.J)

is an Abelian variety. J is called a Riemann matrix. If P satisfies

the above theorem, (J.P) will be called a Riemann pair. It should be remarked that for fixed J there may be many rational polarizations such that (J.P) is a Riemann pair for P

E

(PI

{PI. Also for each fixed P there may be many

complex structures ( J ) such that (J,P) is a Riemann pair for J

E

{J)

.

3. Morphisms of Abelin Varieties and the Structure of Riemann Matrices

V2/L2 be a

Let ( V l / L I J l ) and (V2/L2J2) be Abelian varieties and let f:Vl/LI--complex analytic mapping. Then f * : ~v ~(/ T ~ ) - - - N ( v ~ / L ~ ) and

We will call f or, by abuse of language, f*, a morphism of the Abelian varieties. Let End(@ be the ring of morphisms of an Abelian variety A = (V/LJ). Let

and call &(J)

the rational multiplier algebra of J. Notice that d ( J ) depends on the rational

structure of

V

and on

J , but not on the lattice

L.

representation of a rational associative algebra. Since &(J)

Clearly. &(J)

Q &(J)

is actually a

is never trivial.

Let (J,P) be a Riemann pair and let A be the alternating form corresponding t o P. Then A determines an isomorphism A:V(Q that A-'M* A

E

&(J)

--

v*(Q.

For M r &(J),

one verifies [6]

and hence

is an involution of &(J)..

This involution, called the Rosati involution , is also positive; i.e.,

the trace M M *is positive if M # 0. Let us now state a lemma due t o Poincare' that will enable us t o completely structure

4J). Poincare' Lemma: Let --

(V/L,J)

be an Abelian variety and let Vl(Q) c V(Q) be such that

V I = V1(Q Z) O m c V is J invariant. Then there exists V 2 ( a

c

V(Q) such that

1)

V(Q=V~(QQV~(O

2)

V2 = V 2 ( 0 @

3)

If L2 is a lattice in V2(Q), then (V2/L2, J I V2) is an Abelian variety.

Remark:

The existence of

operator or idempotent.

R c V is J invariant.

V I ( @ is equivalent t o &(J)

containing a proper projection

We may find

V 2 ( Q as follows.

subgroup of N2,+,(lQ

Let (J,P) be a Riemann pair and let G be the

generated by P(VI). Let

' % ( G ) / j . where 3 is the center of N2,+l(R),

O(G)denote the centralizer of G. Then

will be V2. A proof of these assertions can

he found in (33. Chapter 111. Remark:

Clearly, the ~ o i n c a r e 'Lemma implies that &(J)

is completely reducible and so is

semi-simple. We say that J is airreducible if &(J)

has no non-trivial projections. Clearly, the

~ o i n c a r e 'Lemma implies that

where J(Vi) = Vi and if Ji = J ( Vi, then Ji is Qirreducible, all i. We say that Ji and Jj are Qequivalent if there exists D e &(J)

and DJ, = J,D

.

such that

We may group together all the @equivalent 3,'s and change the indexing to

write

We call m, the multiplicity of Ji. It follows that

Now, it is easily seen that if Ji is irreducible, then &(Ji) is a representation p of a division algebra 8, . Further, &(miJi)

is the mixmi matrix algebra over P(Oi). Thus t o determine

all Riemann matrices J, Albert had t o first solve the following algebraic problems. I)

Determine the set A of all rational division algebra with a positive involution.

2)

For 0 r A determine the set of all positive involutions.

Since the solution to these algebraic problems have many good expositions [I], we will just pull the algebraic results out of the hat as we need them. language that has become customary in this subject. Let

0. and let

o

9

We adopt the following

A and, let k be the center of

he a positive involution of 0 . Then a(&) = k and a 1 k is a positive involu-

tion. 9 is said to be of the first kind if a ( k ) = k , all k

E

k and if this fails. 8 is said to

be of the second kind.

In these notes, we will not discuss the problem of ((Birreducible Riemann matrices, Wt discuss the fdiowing simpler problem. Let 8 E A and let p be a right Qrepresentation of

Main Problem: --

matrices J such that &(J)

8. Find all Riemann

= @).

We will now outline our approach to this problem: By the general representation theory, we know that every right representation p of 8 that could be a candidate for an irreducible J can be considered a s pr. where r is the right regular representation of P over Q and p and form

S2(v), noting

E

p. Assume

p acts on the @vector space V

that p induces a representation of 9 as morphisms of g Z ( V )

that we will also denote by p. We next determine all P

E

Polx(V) such that if A is the

alternating form corresponding to P then A - ' P ' ( ~ ) A = p(a(d))

d e 8 , a r {a)

where ( a ) is the set of positive involutions of 9. We let d ( 8 , p , o ) denote the set of such polarizations.

In other words, we first find the polarizations that can be candidates for a

Rosati involution. For each P

2)

E

d ( 8 , p,o), we produce a Riemann matrix J(P) such that

(J(P),P) is a Riemann pair.

Finally, from J(P) we determine all Riemann matrices (JIp such that if J

2)

(J,P) is a Riemann pair.

E

(JIp:

4.

Riemann Matrices Whose Multiplier Algebras are Totally Real Fields

The simplest examples of division algebras with positive involution are the totally real fields with the identity mapping as positive involution. Indeed, the identity mapping is the only positive involutions for totally real fields. Recall that & is totally real, [ & : a = h, if and only if k has h distinct isomorphisms into & or, if and only if the regular representation r of

4

over

is diagonalizable over B

Assume for the rest of this section that k is totally real,

k over Q

representation of

representations p of the form qr, q r Let V(Q

[&:a = h, and

r is the regular

Up to rational equivalence, we may restrict ourselves to

2; i.e.,

to multiples of the regular representation.

be the @vector space for the representation p. Then dim V ( a = hq. But V ( a

can also be considered as a k-vector space by defining

As a k-vector space we will denote V ( 8 by V(k). Of course, the k-dimension of V(k) is q. We will now solve the problem of determining all polarizations of g2(V(UB) that induce the positive involution on k. For this argument, it will be convenient to adopt the following notation: For A r Alt(V), let r ( A ) r Pol(V) be the polarization of s Z ( V ) corresponding to A. Let A c Altx(V(k)) and let r = r ( A ) r PolX(V(k)). Let t : k mapping and set B = t E

V(k) and let a

E

0

-+

Q be the trace

A. Then B E Altx(V(0)) and r' = r(B) c P o l X ( V ( a ) . Let x,y

4. then

and

Thus B-'~*(O)B= ~ ( 0 ) . Let d ( k . p , o ) , where o is the identity mapping, denote all polarization r(B) such that B-'p'(a)B

= p(o(a)) = p(a)

.

Then the image of Polx(V(k)) under t in PolX(V(Q) is contained in d(k.p.0).

We will prove that t(Polx(V(k)) = d ( k , p . o ) .

Suppose n

'E

Polx(V(Q), n' = n '(B), B

E

AltX(V(@), and that n'

E

d ( k , p . a ) or

~ - ' p * ( a ) B = p(a) . The equation is equivalent to B(ax,y) = B(x,ay), a &,:a-+B(ari,vj)

E

&, x,y c V(k). Then the mappings aak

are Qlinear mappings of & to Q , where v l ,....vq define a basis of V(&). Since the trace form is non-singular. there exists fij = & such that

Since B is alternating,

tij=

-f ... Let x = JI

4

XI a i

vi and y

4

P

f: b, v, where ai.b, > 0 .

remains t o verify that it is positive or, equivalently, trace r(dgd) totally positive. By hypothesis, Now trace r(d*d) = trace (M-'A-'(M

.-I

.

r (S)M')A(Mr(d)))

= trace (A-I(M*-~~*(~)M*)A(M~(~)M-'))

Letting C* = M*-'r*(S)M9 we see that trace r(dgd) = trace (A-'C'AC) But the latter is totally positive by hypothesis and we have proven our assertion.

It

denotes

totally positive. By hypothesis.

race(^-'r*(G)Ar(G)) > > 0 . Now trace r(dDd) = trace (M

-1

A

-1

(M

*-I

r (G)M9)A(Mr(d)))

= trace (A-I(M'-~~'(~)M*)A(M~(~)M-~))

Letting C* = M * - ' r ' ( 6 ) ~ ' we see that trace r(dUd) = trace (A-'C'AC) But thc latter is totally positive by hypothesis and we have proven our assertion.

6. The Involution Problem for Division Algebras of the First K i d (Conclusion

Let p:K

-+

End(V(Q) be a right representation of K as a Qalgebra.

Since & is

central in 5, we may view V ( a as a &-vector space which we will denote by V and induces a representation p:K

p

End(V). Identifying the simple right K-module over &

-+

with K itself, we have

Clearly, dim K = 4, dim V = 4p. and dim V(Q = 4ph. By our usual convention. we have a representation p l ( m on H O ~ ( V , V ' ) . We will view V* as

Ox%*.

Let Brs denote the space of elements of H O ~ ( V . V * )which satisfy B,($)

c


0. Hence in K + ( R i ) , --exists, and its square is -I.

our problem in that special case.

JNO

xi

And so we have solved

NOWlet ~ ~ d ( 6 r . r . where ~ ) . ~ ( 8= ) 6()-'a(6) 6,. and ~ ( 6 =~ -6,. ) Let

wherc ~ ( 6 )= 6 ~ ' a ( 6 ) 6 , and 0(6()) = -8,.

Then, let

S = TP(6,) r(6,) and A = Tr(6,). Consider A-'S = P(6,,). Clearly, P(6(,) commutes with r(62), 6

E

pi. Again, ~ ( 6 may ~ )not ~

be -1. but we may complete the argument exactly as in the K+ case, above. We will now produce a canonical form theorem that will reduce the general problem to the two special cases we have just verified.

Consider the matrix

where IK is a division algebra with involution

Since (c")'

= C and (cD)'

= D'c,'

7.

Define

we have if A" = -A the (c" AC)* =

will now prove the following result. Given A such that A' = -A.

-c'

AC. We

Then there exists C

such that

where

To prove this, begin by noting that if a,b.c,cK are such that ~ ( a = ) -a#O and ~ ( c = ) -c, then

By repeated applications of this fact we may assume there exists C I such that

where 6,' = -6, and

Now there exists C2 in the entries P ( 1 ) and ((0) such that

and thcrc clearly exists C3 such that

If B ' ~o r B', are not both the zero matrix. we can rearrange the matrix such that the upper left hand corner looks like

If 6 1

+0

then if a =

-3we have 61

=

(

p(0) -

1

P(l a

-

p ( ~ ( 6 ~ ) ) P(0)

P(6,) a

p(0)) P(0)

I f ?(a)-a # 0. we may return t o the above argument t o reduce B2 by 1 row and column and

s o hc ahlc t o apply an induction to finish the argument. If ?(a)-a = 0 we have reduced t o lhc casc

and wc have easily a matrix F such Lhat

A repeated use of the above argument proves our canonical form theorem. Clearly if

Then there exists El,..

B such that if

is an automorphism of N(B) for E = K+ or K. Since C'AC

= B. J ~ - ( C * ~ ~ Cis- 'a

Riemann matrix such that (J I.A) is a Riemann pair in G. This proves the existence of a Riemann matrix. REFERENCES [ I]

A. A Albert, Structure of Algebras, American Math W e t y , 1939.

[ 21

L. Auslander, An exposition of the structure of s o h d o l d s . Part I. Algebraic Theory. Bull. A.M.S. 79 (1973) 227-261.

[ .3]

L. Auslander, Lectures on Nil-Theta Functions, C.B.M.S. Regional Conference Series, American Math. Soc. No. 34.

[ 41

L. Auslander and R. Tolimieri, A matrix-free treatment of the problem of Rlemam matrices, to appear.

[ 51

L. Auslander and R Tolimieri, Abelian harmonic analysis, Theta

[ 61

C. L. Siegel, Lectures on Riemann Matrices. Tata IndMe, Bombay (1963).

[ 71

H. P. F. Swimerton-Dyer, Analytic Theory of Abelian Varieties, Lodon M& Soc (1974).

[ 81

R Tolimieri, Hekenberg manifolds and Theta functions, Tram. A.M.S., 239 (1978) 293-319.

[ 91

A. Weil, Introduction a I'etude des ~ r i e t e s ,Kahleriemes, Paris (1958).

[lo]

A. Weil, Sur certxines groupes d'operateurs 143-211.

unitaires. Acts Math. 111 (1964)

CEN TRO INTERN AZIONALE MATEMATICO ESTIVU

(c.I.M.E.)

UNITARY AND UNIFORMLY OF SOME

BOUNDED REPRESENTATIONS

SIMPLE L I E GROUPS

MICHAEL COWLING

We denote by F the real or complex numbers ( R or C) or the quaternions (Q).

We consider R as a subfield of C, and C as a subfield of Q.

If z E F,

then we may write z=s+ti+ui+vk, with s, t, u, and v in R.

Note that zS = zz = lzl

The conjugate

.

2

the formulae:

is now described thus:

-z = s - t i - u i - v k .

The real and imaginary parts of z in F are given by

-

ZRe(z)=z+z,

2Im(z)=z-z

-

.

This is not the usual imaginary part in the complex case. n+l We shall consider the vector space F ; in the quaternionic case, the scalars act on the left. We choose a basis (eO, el, , en) for Fn+1 over F, n+1 , given by the formula: and consider the sesquilinear form q on F n 2, 6 E F ~ + ~ 2 , C = z C.) c0 j J n+1 We shall be interested in the group O(q) of all linear transformations of F

...

( 1

-

5-

.

which preserve this form. One of our principal aims here is to describe sane aspects of the harmonic analysis of O(q).

Along the way, we shall meet other

groups, on which we shalldescribe some aspects of harmonic analysis, some of which will be directly pertinent to our study of O(q) and some of which will be complementary.

These groups are compact, abelian, and nilpotent. Harmonic

analysis on compact groups has been under intensive investigation during the last half-century, and on abelian groups for much longer, but harmonic analysis on nilpotent groups is less well known. We shall therefore dedicate the first three sections to this. The main thrust of our development of harmonic analysis on nilpotent Lie groups owes much to E.M.

Stein. In particular, much of what we present is al-

ready contained in work of R.A. Kunze and E.M. A.W. Knapp and E.M.

Stein

( [K~s]),

Stein ( [KS~] , [KS~], [KS~]) , of

and of G.B. Folland and E.M.

Stein ( [FOS]).

We have simplified and even improved the methods of the above authors, but who sower is at home with the above works will find no real surprises. The last five sections are dedicated to some semisimple groups of rank one. In the fourth section, we discuss some general properties of these groups, and indicate how they are related to certain geometric objects, which appear in various contexts and guises.

This section may be considered as a study of some

aspects of the geometry of semisimple Lie groups, together with a few easy consequences of a measure-theoretic nature. In the fifth section, we consider certain unitary representations of these groups and their analytic continuations.

In fact, we construct analytic fami-

lies of representations n which, if the complex parameter 5 lies in the tube UrS (strip) T

-

act isometrically in certain ~'-s~aces, where l/p = 5/2r + 112. We also describe the intertwining operators A(w, p , 5) which express the equivalence between n and n us 5 u,-c* 1 The sixth section contains some results on the L - and LP-harmonic analyoften abbreviated to G. Here the questions treated are about the Fourier transforms of LP-functions and of certain convolution algesis of the group O(q),

bras on G. In the penultimate section, we study the question of uniformly bounded representations, and touch on the problem of complementary series.

Both these

questions arise when one attempts to develop a calculus of analysis and synthesis of representations which would extend the direct integral theory for unitary representations to include the bounded Banach representations of G. The last section is dedicated to the study of an extrapolation principle. We show that, metamathematically, the finite-dimensional representations of G determine its harmonic analysis. We conclude this introduction with a disclaimer.

This is not an attempt

to provide an unbiased version of the representation theory of semisimple Lie groups.

Rather, we have some hopes of offering the "comnutative harmonic

analyst" a vision of some aspects of the "noncommutative theory".

Conse-

quently, the references at tile end are only those works cited in the text.

Horror of horrors, the labours of Harish-Chandra, without which this work would never have been written, have not been mentioned.

It seems worthwhile to add

G. Warner's tomes [War] to the list so that the reader may obtain a less slanted view of the literature and the subject. It is a pleasure to thank Professor Alessandro FigB-Talamanca for his invitation to present this material at C.I.M.E.,

and to thank M. Cristina for

her understanding during the preparation and writing up of it all.

1.

ANALYSIS ON HEISENBERG GROWS. I

We s h a l l work on a g r o u p V, whose e l e m e n t s a r e o f t h e form ( x , y ) , where x E Fn"

The p r o d u c t of ( x ' , y 1 ) and ( x , y ) i n V i s g i v e n by t h e

and y E Im(F).

rule (x',yl)(x,y)

= (XI+ X , y l + y

*

where we c o n s i d e r x a s a row v e c t o r , and x c o r r e s p o n d i n g column v e c t o r .

-

2b(X'X*))

,

i s thus the conjugate of t h e

I n p a r t i c u l a r , we n o t e t h a t (0,O) i s t h e i d e n t i t y

I f F = R, t h e n h ( F ) i s n-1 t r i v i a l , and t h e group V i s j u s t t h e E u c l i d e a n s p a c e R ; otherwise V is a

o f V, and t h a t (-x,-y)

i s t h e i n v e r s e of (x,y) i n V.

two-step n i l p o t e n t group. There a r e two i m p o r t a n t groups of automorphisms of i?, which we s h a l l now d e n o t e by M and A.

Ivl

-

An e l e m e n t of M i s w r i t t e n m, o r m(u,v), where v E F and

1, and u i s a n (n-l)x(n-1)

n-1 in F

.

matrix over F such t h a t

xu

=

x

for a l l x

We make t h e f o l l o w i n g d e f i n i t i o n s : 1 -1 (1.1) R ~ ( X , Y )= ( v xu, v yv) (1.2)

-

2

D ~ ( X , Y =) ( s x , s Y ) m

Then M "is" t h e group of r o t a t i o n s R

, and

s E R+. A "is" t h e group of d i l a t i o n s DS.

The s i g n i f i c a n c e of t h e s e g r o u p s w i l l become c l e a r i n 14.

We a l s o d e f i n e a

"norm" o n V , which i s s u g g e s t e d by t h e geometry of t h e s i t u a t i o n ( s e e Lemma

When F = R , t h i s i s t h e u s u a l E u c l i d e a n norm, and i n a l l c a s e s , i f ( x , y ) E V,

I t i s n e c e s s a r y t o d i s c u s s c e r t a i n s i n g u l a r i n t e g r a l o p e r a t o r s o n V.

s h a l l h e r e develop and e x t e n d work of A.W. A . Kordnyi and S. V6gi [ K O V ] , of G.B.

We

Knapp and E.M. S t e i n [ K ~ s ], of

F o l l a n d and E.M. S t e i n [ F O S ] , and of many

others. Thc p r i n c i p a l o b j e c t of s t u d y w i l l be c e r t a i n f a m i l i e s of homogeneous

distributions.

A f u n c t i o n f on V i s c a l l e d homogeneous of degree d (d E C) i f

s £OD =

d

S

E

s

~

~

+

;

i t i s c l e a r t h a t , i f we ignore t h e p o i n t (0,0), then any such f u n c t i o n may be

w r i t t e n i n t h e form d

f ( x , ~ )= Jl(x,y)N(x,y) where fl i s homogeneous of degree zero.

(x,y) E V,

We s h a l l d e a l w i t h f a m i l i e s of hmo-

where, f o r 5 i n C , 5' K5(x,y) = Q ( x , ~ ) N ( x , Y ) ' - ~

geneous d i s t r i b u t i o n s K

( x , ~ ) .E V ,

and r, c a l l e d t h e hmogeneous dimension of V, i s given by t h e r u l e r = p + 2q, n-1 and h ( F ) a s r e a l v e c t o r spaces. To where p and q a r e t h e dimensions of F avoid some f i n i c k y provisos, we assume t h a t r

>3

i n what follows.

The f i r s t r e s u l t we need about such d i s t r i b u t i o n s i s of a t e c h n i c a l We w r i t e E f o r t h e u n i t sphere

nature, but i s v e r y important f o r our study. ( e l l i p s o i d ) i n V:

Z = {(x,y) E V: N(x,y) = 1 )

.

There i s a unique smooth measure on E, da say, such t h a t f o r any f i n C (V), C r-1 f (D'u) Svdxdy f (x, y) 5 Edo jR+ds s We abuse n o t a t i o n , and consider n i n d i f f e r e n t l y as a f u n c t i o n on Z and as a

-

.

f u n c t i o n on V, homogeneous of degree 0. 1 Suppose t h a t Q is i n L (E).

LEMMA 1.1. f2N5-'

i s l o c a l l y i n t e g r a b l e and d e f i n e s a d i s t r i b u t i o n on C The cW (V)'-valued f u n c t i o n .5 C+ N

by i n t e g r a t i o n . morphic i n C

w

C

It; E C

(V)'-valued

: Re(5)

w

C

(V)

~ i- s ~holo-

and extends meromorphically t o a

f u n c t i o n whose only p o s s i b l e poles a r e simple poles

a t the p o i n t s 0,-1,-2,... Proof.

> 01,

> 0,

Then, i f Re(c)

.

We omit t h e easy c a l c u l a t i o n t h a t , i f Re(5) a0

> 0,

then

N

is locally integrable.

We choose any f i n C

i n a Taylor expansion.

By grouping together a l l the t e r n s of the same hano-

~

(V), and develop f about (0,O)

geneity, we may w r i t e f i n the form

where f

j

i s homogeneous of degree j i f 0

f i e s the c o n d i t i o n t h a t

< j < J,

and the e r r o r term f

J

satis-

-

~

fJ(x,y) m

We take a C 0.

= O(N(x,y)

J

.

(R)-function $ which t a k e s t h e v a l u e 1 i n a neighbourhood of

C l e a r l y , i f Re(5)

>

0, t h e n

IVdxdy Q ( x , ~ ) N ( x . y ) ' - ~ f ( x . y ) = I V d x d y Q ( x , y ) ~ ( x , y ) ~ $(N(x,y)) -~

+ Ivdxdy

~ ( x , y ) N ( x , y ) ~[- 1 ~

-

f (x,y)

+(N(x,y))l f ( x , y )

-

The second i n t e g r a l on t h e r i g h t hand s i d e c o n t i n u e s a n a l y t i c a l l y i n t o t h e whole complex plane.

The f i r s t i n t e g r a l may be w r i t t e n thus:

IP X ~ Y

Q ( X . Y ) N ( X , Y ) ~$(N(x,Y)) -~ ~(X,Y)

The r a d i a l i n t e g r a l of t h e f i r s t J terms e x t e n d s meromorphically t o t h e whole complex plane, with p o l e s a t 0, -1, term i s i n t e g r a b l e i f Re(5) ( 5 E C: Re(

-J).

> -J

-2,...

, while

t h e i n t e g r a n d of t h e f i n a l

and so t h e l a s t term extends a n a l y t i c a l l y t o

Since J i s a r b i t r a r y , we have indeed a meromorphic con-

t i n u a t i o n t o t h e whole complex plane.

We remark t h a t the c o n t i n u a t i o n t h u s

o b t a i n e d does n o t depend on t h e choice of t h e f u n c t i o n $I,

by t h e uniqueness of

meromorphic c o n t i n u a t i o n . REMARK 1.2.

Lemma 1.1 can be g e n c r a l i s e d $0 d e a l w i t h f a m i l i e s of

d i s t r i b u t i o n s Q N ' - ~ , where Q i s a meromorphic f u n c t i o n of 5 . 5 5 s-r REMARK 1.3. I t i s c l e a r from t h e proof t h a t t h e d i s t r i b u t i o n QN i s of o r d e r a t most J i n ( 5 E C: Re(6) REMARK 1.4.

>

-J)

.

We observe t h a t , i f 52 i s even, i n t h e sense t h a t

Q ( x , y ) = Q(-x,y) f o r (x,y) i n V , then t h e o n l y p o s s i b l e p o l e s of S ~ N ' - ~ a r e where 5 i s even, while i f Q i s odd, i n t h e sense t h a t Q(x,y) = -a(-x,y), any p o l e s of C ~ N ' - ~ l i e i n (-1,

-3,

-5,.

..) .

then

This h o l d s because, i f t h e i n t e -

grand i s odd, then

IE

d n(o) ~ f .(u) =

J

o

.

We s h a l l now examine t h e d i s t r i b u t i o n s K

f o r c e r t a i n v a l u e s of 5, v i z 5 , ) t h e u s u a l i n n e r product on L2 ( V ) , by

5 = i q , q E R\ CO}. We denote by ( 11 11 t h e u s u a l LP-norm, and by f * g t h e c o n v o l u t i o n of f u n c t i o n s f and g : P

(when t h i s makes s e n s e ) .

1

m

Suppose t h a t R E L (21,

THEOREM 1.5.

and t h a t f , g E C c(V).

Then, i f rl E R\{o),

ki2 ,

I ( ~ ~ ~ * f , g,

and t h e c o n v e r s e of Hijlder's i n e q u a l i t y c o m p l e t e s t h e p r o o f .

REMARK 2.5. volution operators K

The above proof a l s o shows t h a t t h e norms of t h e conremain bounded a s

is12

s a p p r o a c h e s 0.

A v a r i a t i o n of t h e above proof e s t a b l i s h e s t h e f o l l o w i n g p r o p o s i t i o n . PROPOSITION 2.6.

Suppose t h a t

Then, away from t h e p o l e s of K

HA^^^

K

A t t h e p o l e s of K

5 5'

*

5' £ 1 1c ~ ctn,s)

m

E C (E),

II~II,

we have t h e f o r m u l a

and t h a t K

f

5

e

= RN'-~.

cz(v)

.

*

Res(K ) 5

*

f 1 2 CC(R.5)

lf12

where Res(K ) i s the r e s i d u e of K a t 5. 5 5 Proof.

We omit the proof, which involves no new i d e a s .

The s i g n i f i c a n c e of t h e preceding p r o p o s i t i o n i s t h a t i t shows t h a t t h e operators K

a r e no more s i n g u l a r than t h e o p e r a t o r s We can improve 5 both t h i s r e s u l t and P r o p o s i t i o n 2.4 by using a theorem which w i l l be proved later (in

§

8 ) , namely, i f 5 i s n o t a n even i n t e g e r , then NS-' N-5-r = c(5)6

.

(2.1)

2 Suppose t h a t R E L (21, and t h a t MV(O) = 0.

PROPOSITION 2.7.

Then convolution with R N - ~ extends f r m 2 on L (V). Proof.

(V) t o a bounded o p e r a t o r

2 I n o r d e r t o show t h a t R N - ~ convolves L (V) i n t o i t s e l f , i t

s u f f i c e s t o prove t h a t

f o r then t h e e q u a l i t y above (2.1) clusion.

OD

Now l e t ,$ be a C

and Theorem 1.2 would g i v e t h e d e s i r e d con-

(V)-function which i s 1 near (0,O) and has small

support; then

The f i r s t term i s a d i s t r i b u t i o n with small support, the second i s the conm

v o l u t i o n of a compactly supported d i s t r i b u t i o n with a C (V)-function, and the 2 2 t h i r d i s i n L (V) by Theorem 1.2. It follows t h a t t h e r e e x i s t s fl' i n L (V) such t h a t

Ni-r

r R N - ~ = R'N

i-r

,

by a homogeneity argument and the above d i s c u s s i o n .

Now Theorem 1.2 may be

applied t o complete the proof. The a f f i c i o n a d o s of HP-spaces w i l l be pleased t o know 1 t h a t P r o p o s i t i o n 2.7 extends t o those R which l i e i n a s o r t of H (E) ( i n f a c t ,

REMARK 2.8.

l e t JI be a C

m

(R)-function with support i n (1,4) which i s equal t o 1 on (2,3), 1 and l e t @ be @ON: then we r e q u i r e t h a t , $ R N - ~ l i e i n H (V)). To show t h i s , one i -r 1 1 uses t h e f a c t t h a t N convolves H (V) i n t o L (V), proved by R.R. Coifman and

G. Weiss [COW] o n p. 599.

I t i s a l s o p o s s i b l e t o o b t a i n LP- e s t i m a t e s by s u i t -

a b l y modifying Theorem 1.2

-

i n d i c a t i o n s o f how t o do t h i s c a n be found i n t h e

paper of E.M. S t e i n and S. Wainger [SW2] ( s e e p. 1253). 512 * NS-r Convolution w i t h t h e d i s t r i b u t i o n A 2 is a bounded i n v e r t i b l e o p e r a t o r o n L (V) p r o v i d e d t h a t 5 i s n o t a n PROPOSITION 2.9.

even i n t e g e r . Proof.

We have a l r e a d y s e e n t h a t

u n l e s s 5 E {O, -2,

.

-4,...}

The i n v e r t i b i l i t y of t h e o p e r a t o r i s t h e new

f e a t u r e of t h i s proposition. From t h e formula ( 2 . 1 ) , A5/2

*

we deduce t h a t NC-r N-~-r

*

*

A

-512

= c(06

,

and s o we may w r i t e

*

Nc-r)-l

-1 N - ~ - r = ~ ( 6 ) * A-512

I n o r d e r t o e s t i m a t e t h e r i g h t hand s i d e , we c o u l d u s e L e m a 1.7 t o r e d u c e t h e problem t o t h a t of e s t i m a t i n g A-"' P r o p o s i t i o n 2.6,

*

N-'-'

, which

i s r e s o l v e d by a p p l y i n g

o r we c o u l d r e p r o v e P r o p o s i t i o n 2.6 f o r A a c t i n g on t h e r i g h t

r a t h e r t h a n on t h e l e f t . A l t e r n a t i v e l y , we c o u l d o b s e r v e t h a t [ c ( c ) - l N-c-r A - ~ / 2 1 *= z(5)-1 A -w/2 N-~-r

*

where w = 6-,

*

and a p p l y P r o p o s i t i o n 2.6.

,

I n any case, the proposition i s cer-

t a i n l y proved. REMARK 2.10.

I n S e c t i o n 8, we s h a l l s e e t h a t o u r proof o f t h e equa-

l i t y ( 2 . 1 ) a l s o y i e l d s P r o p o s i t i o n 2.9 d i r e c t l y . f a m i l i e s of o p e r a t o r s S2N5-'

However, t h e r e a r e o t h e r

-

(the intertwining operators

a t l e a s t a t t h e p r e s e n t t i m e , a n analogous i n e q u a l i t y -5-r $IN'-~*$I'N =c1(5)6 -

s e e below) f o r which,

-

c a n be proved o n l y by a method which d o e s n o t imply t h e analogue o f P r o p o s i t i o n 2.9 e x c e p t by t h e argument p r e s e n t e d h e r e ( s e e A.W.

Knapp and E.M.

S t e i n [KnS]

for further details). P r o p o s i t i o n 2.9 i s a two-edged a b o u t t h e k e r n e l s NS-'

sword: on one hand, we may o b t a i n r e s u l t s

by u s i n g t h e group p r o p e r t y of

A , v i z , A'

*

A~ = A ' + ~ ,

and on t h e o t h e r , we may u s e t h e k e r n e l s N ' - ~ kernel)

*-'I2.

a s a p p r o x i m a t i o n s t o ( t h e unknown

The f o l l o w i n g r e s u l t s , o b t a i n e d i n j o i n t work w i t h A.M.

tero, w i l l i l l u s t r a t e t h i s principle. h a r t z [ S t r ] and N.

Man-

Our r e s u l t s e x t e n d work of R.S.

Stric-

Lohou6 [Loh]. The n o n i n c r e a s i n g r e a r -

We s h a l l need t o u s e t h e L o r e n t z s p a c e s L " ~ .

rangement o f a m e a s u r a b l e f u n c t i o n f on V i s t h e n o n i n c r e a s i n g n o n n e g a t i v e right-continuous

equimeasurable f u n c t i o n f

i n c r e a s i n g and r i g h t - c o n t i n u o u s , m((v E V:

lf(v)

I > XI)

If n


0

-

(3.4)

for all k (this expression could not possibly be negative for all k).

This

observation will be of use later in our understanding of the "complementary series".

Now we note that, if F = C, then the condition (3.4) is satisfied if

0 axtends meromorphically t o t h e whole complex plane, with p o s s i b l e p o l e s a t 0, -2,

I t may be worthy of n o t e t h a t , using t h e compact p i c t u r e , we may a l s o e s t a b l i s h t h e meromorphic c o n t i n u a t i o n of t h e o p e r a t o r , b u t i t i s h a r d e r , and c e r t a i n f i n e p o i n t s , such a s t h e conclusion t h a t -1, -3, -5, e t c . a r e n o t p o l e s , a r e n o t a t a l l obvious. The seventh s e c t i o n w i l l be devoted t o , amongst o t h e r t h i n g s , some a p p l i c a t i o n s of t h e e a r l i e r a n a l y s i s on V t o t h e r e p r e s e n t a t i o n theory of G v i a t h e intertwining operators.

The r e s t of t h i s s e c t i o n w i l l be used t o d e s c r i b e t h e

asymptotic behaviour of m a t r i x c o e f f i c i e n t s . Before we d i s c u s s t h e asymptotics, l e t u s n o t e t h a t , i f f E H

and

v15

f' E H

lI,-Ef

then we may form t h e i n n e r product ( f , f ' )

in H

v

, and

we thus ob-

t a i n a f u n c t i o n on G which s a t i s f i e s t h e r e l a t i o n : (f

, f '>(mang)

= ( f (mang) , f ' (mang))

We may t h e r e f o r e form t h e i n n e r products ( f , f') (K) and ( f , f ') ('I, d e f i n e d by t h e formulae ( f , f',

(K)

( f , f')(')

=

J,

dk ( f (k) , f ' (k))

= j v d v ( f ( v ) , f1(v))

,

and, a s i n Lemma 5.1, we f i n d t h a t ( f , f','K'

= CG ( f , f')

(V)

We dcduce t h a t t h e i n n e r products above a r e G-invariant,

naturally dual.

and H

U, 5

and H

lip = 5/2r + 112 and i f l l p '

= -5/2r

+ 112, then

l / p + l / p ' = 1, and the Lebesgue spaces a s s o c i a t e d t o t h e d u a l spaces H

up-z

are

I n particular, it follows t h a t

It i s i n t e r e s t i n g t h a t , i f

H

-

IJs-5

a r e dual i n t h e f u n c t i o n a l a n a l y t i c sense.

lJ,5

and

We conclude t h i s s e c t i o n by d e s c r i b i n g the asymptotic behaviour of the matrix c o e f f i c i e n t s of the r e p r e s e n t a t i o n s n P,5. m

m

and f be i n H and H with 5 p o s i t i v e . 1 2 P9-c U,S + Then t h e r e e x i s t s a p o s i t i v e S-dependent number E such t h a t , i f t E R , LEMMA 5.6.

Proof.

Let f

Omitted .

-

m

m

and f be i n H and H with 5 equal t o 0 . I 2 P, -5 P,C Then t h e r e e x i s t s a p o s i t i v e number E such t h a t , i f t E R , LEMMA 5 . 7 .

Let f

+

Proof.

Omitted.

6. L'-HARMONIC

ANALYSIS 08 SOHE

SIt.PLE GROUPS

In this section, we investigate the role played by the analytic families of representations a

in the harmonic analysis of G. First we consider the P convolution algebra L (G), and later we discuss L -analysis on G.

"lS

There has been some interest in the spectral properties of the algebra

L'(G).

L. Ehrenpreis and F.I. Mautner [EM11 , EM^]

for certain particular groups G.

.

EM^] studied the algebra

They showed that it has two strange proper-

ties: first, it is not symmetric, which means that there exist functions f in

-

L'(G)

such that f = ' f (here is the involution of the convolution algebra -1 ) .for unimodular groups) but vhose spectrum is not real, L (G): f'(g) = f-(g 1

and second, that it contains "non-Tauberian ideals", i.e. proper closed ideals which are annihilated by no irreducible (Banach) representation of the group. 1 M. Duflo, in a letter to H. Leptin, gave a quick proof that the algebra L (G) is not symmetric for any noncompact semisimple Lie group G (see [L~P]).

At

least for the real-rank-one groups (essentially those which we are considering) the result on the existence of non-Tauberian ideals was apparently obtained by R. Krier in an apparently unpublished thesis.

The most recent work on this

argument is presumably that of A. Sitaram [sit], in which partial results for general semisimple Lie groups are obtained.

It may be supposed that the recent

work of Y. Weit [~ei]will stimulate some further development in the study of non-Tauberian ideals. It seems that LP-analysis requires noncommutative techniques. We shall discuss a charactcsrisation of the "Fourier transform" of certain subspaces of LP(c) due to P.C. Trombi and V.S. Varadarajan [T~v] , whose proof has been ele-

.

gantly simplified by J.-L. Clerc [~le] We shall also consider the Fourier transfo m s of geueral L'-functions,

the "Kunze-Stein phenomenon", discovered

by R.A. Kunze and E.M. Stein [KS~]for SL(2,R)

and (after various generalisa-

tions based on uniformly bounded representations (v-i.)) by M. Cowling [Col]

.

established in general

It may be worthy of note that D. Poguntke, following up

a suggestion of M. Duflo, has recently found analogous phenomena for some solvable groups, and that M. Picardello independently proved the same results. In the solvable case, it is essential to use analytic continuations which act isometrically on ~'-s~aces rather than uniformly bounded representations, for uniformly bounded representations of solvable groups are equivalent to unitary representations. 1 Let us consider the convolution algebra L (GI, armed with the involution

-, defined

by the formula f'(g)

1 f E L (G)

= f-(g-l)

1

.

1

We shall denote by L (K\G/K) the subalgebra of L (G) of K-biinvariant functions, i.e. those functions f such that g E G , k, k' E K .

f(g) = f(kgkl) In this section, we shall denote by

p

the measure on G, supported on K, given

by integration against the normalised Haar measure of K.

The map P, defined

by the formula

.

1 Pf=,,+f*,, f E L (GI 1 1 is a non-norm-increasing projection of L (G) onto L (K\G/K), whose restriction 1 to the subspace L (K\G/K) is the identity map. 1 It is quite easy to produce pathological examples in L (K\G/K), because this is a commutative Banach algebra with a well-defined spectrum. lifts these examples to the whole group, using the projection P. essential pcint behind the L ~ ( G ) results which we discuss here.

Then one

This is the On the other

hand, this approach is not fine enough to yield the Kunze-Stein theorem. PROPOSITION 6.1.

1 The convolution algebra L (K\G/K) is commutative.

1 To show that L (K\G/K) is an algebra is easy: one notes that 1 1 1 f in L (G) lies in L (K\G/K) if and only if f = Pf. Then, if f, f' E L (K\G/K) Proof.

,,*'f * f l * , , = , , * , , * f

*,,*,,4f1r,,*,,

=,,*f*,,*,,*fl*,, =f*fl

,

because

is an idempotent measure. 1 To show that L (K\Gj'K) is commutative is no more difficult.

We recall

that any g in G may be written in the form kak', with k, k t in K and a in A.

.

KgK = KaK = KwawK = ~ a - = l ~ g - l ~ 1 This i m p l i e s t h a t i f f E L (K\G/K), then f i s e q u a l t o i t s r e f l e c t i o n f'. 1 (f * f')" f'* f' f , f ' E L (GI 1 (compare with Lemma 1.8), and so, f o r f and f ' i n L (K\G/K).

Thus

-

f

*

*

f ' = (f

f')"

5

f'"

*

f

= f'

f

Now

.

1 Both t h e a l g e b r a of r a d i a l f u n c t i o n s on V and t h e a l g e b r a L (K\G/K) t r e a t e d i n a u n i f i e d manner by usi& t h e t h e o r y of Gelfand p a i r s .

can be

A. Korgnyi

f i r s t noticed t h i s f a c t , which e x p l a i n s t h e s i m i l a r i t i e s between t h e proofs of Lemma 1.8 and P r o p o s i t i o n 6.1. When one d e a l s with a commutative Banach a l g e b r a , one looks f o r i t s spec-

trum.

I n t h i s c a s e , we a r e lead t o t h e theory of s p h e r i c a l f u n c t i o n s .

c r i b e t h e s e a s follows: i n t h e spaces H and n 1 ~. ,5 1,-5 a r e i d e n t i c a l l y 1.

n

-

of t h e r e p r e s e n t a t i o n s 19-5 and f -whose r e s t r i c t i o n s t o K 5 -5 The s p h e r i c a l f u n c t i o n $ - i s g i v e n by t h e formula

195 t h e r e a r e unique f u n c t i o n s f

and H

We des-

It i s not hard t o show t h a t , i f we d e f i n e t h e s p h e r i c a l transform

?

of f i n

C (K\G/K) by t h e r u l e

then

(f

*

f ' ) ^ ( ~ )= f ( 5 ) z ' ( 5 )

.

This follows from t h e analogous m u l t i p l i c a t i v e formula f o r t h e "Fourier t r a n s forms" s

195

(£) :

.

(f) = J G d g f ( g ) n (g) f E c~(G> 135 195 a c t i s o m e t r i c a l l y w i t h t h e Banach norms 1.1 (K), when The r e p r e s e n t a t i o n s rr 1,5 P l / p = 5 / 2 r + 112, a i d so i f 6 E T, t h e tube where 5 E [-r, r ] , then n

Therefore t h e s p h e r i c a l transform extends t o a m u l t i p l i c a t i v e l i n e a r f u n c t i o n a l 1 o n L (K\G/K), a s long a s 5 E T. We n o t e t h a t t h e d u a l i t y between n and 1,5 n implies t h a t 19-5

-

+
1,

W and c or Sp(W),

,> .

or Sp2,,

,>

is the

We will denote it variously as

or SP~~(F), or simply SP,

generally keeping the designation as short as is consistent with precision of reference. Define a two-step nilpotent group H(W),

the Heisenberg

attached to W by the recipe:

as set, and has group law (w,t) (w' ,tl) = ( w i d ,

t+tl+($)

<w,w'>

)

with w, w' € W and t, t' € F. The group H(W) = H has a natural locally compact topology, and we may consider its unitary representation theory. It is very well understood. Observe that the center commutator subgroup of H. on Z(H)

Z(H)

of H

is also the

Therefore all representations of R

trivial

factor to the abelian group

and are thus identified to one-dimensional characters of W. Let p then be an irreducible unitary representation of H which is non-trivial on

Z(H)

.

Then the restricted representation

must be a multiple of a single unitary character character of

p

.

X(p),

Z(H)

the central

The basic result about p is the Stonevon Neumann

Theorem, which says that p is determined up to unitary equivalence by X (p)

.

In other words, given a non-trivial character X

of

Z (H) , there

is a unique irreducible

p

, up to unitary equivalence, such that

x = X(P). For a general locally compact group, let G denote the unitary dual of G, the collection of equivalence classes of irreducible representations

of G. -Then according to our remarks just above,

where 1 here is the trivial character of

Z(H).

It is convenient to introduce coordinates on description (1.2) of H, the center

z(H)-

is identifiable with

character

G.

X1 of F, called a

Z(H)

*

Z(H)

.

In the

is identified to F, so that

We choose once and for all a non-trivial

basic character

of F.

Then an arbitrary

character of F has the form

Write F

F.

-

(0)= F~

for the multiplicative group of non-zero elements of

Then with the identifications just made we can write

We can denote the element of

HI

with central character

Here is a realization of the representation subspace of W

Xt by

Ptpt. Let Y be the

spanned by the fils of the standard basis

X be the span of the eils. We can realize

(1.1), and let

pt on the space L ' ( Y ) .

The formulas for elements of H acting on f € L2 (Y)

are

The symplectic group

Sp(W)

a c t s on

H(W)

by automorphisms i n t h e

obvious way:

Whenever one h a s a group

G a c t i n g by automorphisms on a l o c a l l y

H, one has a l s o an a c t i o n of

compact group

i,denoted

on

G

Ad

*,

by

the recipe

p

i s f i x e d by

p E

If

f o r each

g E G.

G, it means

*

is u n i t a r i l y equivalent t o

Ad g ( p )

Hence t h e r e i s a u n i t a r y o p e r a t o r

U

g'

uniquely

defined up t o a s c a l a r m u l t i p l e , such t h a t g E G , h fH.

Ug P ( h ) ~ g l= p(g(h)) Since

U

g

i s well-defined up t o s c a l a r s , t h e transformation i t induces p

on t h e p r o j e c t i v e space a s s o c i a t e d t o t h e space of period.

Also, it i s easy t o s e e t h a t t h e map

up t o s c a l a r m u l t i p l e s . representation.

Hence t h e map

g

g + Ug

One can then show (see e.g.

-P

U

g

i s well-defined i s multiplicative

i s called a projective

[My];

t h e g e n e r a l proof i s

d i f f i c u l t but under c e r t a i n simplifying assumptions which apply h e r e , it

i s r e l a t i v e l y easy) t h a t t h e r e i s a c e n t r a l extension bonafide u n i t a r y r e p r e s e n t a t i o n

*

U

of

-

G of G, and a ," G on t h e space of p , such t h a t

-,

g to denote both an element of G and its image in

where we have used

G. We will generally abuse notation in this way.

If G is perfect

" ,

(equal to its own commutator subgroup) then G may also be taken to be w

perfect, and it is in this case uniquely defined, and so is U. We may specialize these remarks to the action of Sp(W) Since Sp acts trivially on Z(H)

*

are fixed by Ad Sp.

on H(W).

.

we see the representations

pt 6. H

Since Sp is perfect, there is a unique perfect

w

central extension Sp, and a unitary representation, which we will denote o , of Sp, such that t

We will call

at

(or the projective representation of Sp that gives -,

rise to it) an oscillator representation of Sp. Nominally

o

t

depends

on t, but in fact this dependence is rather weak, and there are only a finite number of mutually non-equivalent typical one of them simply by

at's.

We will denote a

o.

w

Of course, the group Sp could conceivably depend on t. However, it is a theorem of Shale [Sh] for F = R

and Weil Rid

otherwise that it " ,

does not, and furthermore, except for the case F = 6 , when Sp = Sp, the " ,

group Sp is a two-fold cover of Sp, so that we have a diagram

Under

a, the group Z2 is represented by

21.

In any irreducible

-,

representation of Sp, the group Z2 will either act by

+1, or just by 1.

In the latter case, the representation factors to define a representation of Sp.

It is not easy to give formulas for However, on a certain subgroup of with the formulas (1.4).

,.

for all g 6 S"p.

Sp, formulas can be given consistent

Let Pm(W) = Pm be the subgroup of Sp(W)

that preserves X, the span of the ei's.

where Nm(W) = Nm

mt(g)

We have

is the subgroup of Pm that leaves X pointwise

fixed. Furthermore, we can identify Nm with the space of symmetric bilinear forms on Y,

by the rule

..,

..,

We let Pm denote the inverse image in Sp of Pm, and similarly for

EL(Y).

However, Nm may be lifted in a unique manner to a subgroup of

S"p, and we will continue to denote this subgroup of

Ep

also by Nm.

We

can then write, for f 6 L ' ( Y )

where y

is a factor of absolute value 1, and det g denotes the Y

determinant of g acting on Y, and ] , Chap. 1, 83) value on F ( [ w ~ Z

I I

indicates the standard absolute

.

The multiplicative group F~ also acts by automorphisns on H by the rule

For this action we see that

Since this action of

fl

on H obviously commutes with the action of Sp,

we conclude, by the naturality of the extension process, that

is parametrized by the

Hence the set of oscillator representations

.

finite set P / F ~ ~ Furthermore, the set of

is closed under taking contragredients. t * Given a representation P of a group G on a Hilbert space ff , let a denote the contragredient representation of G on the dual space

* X = X t

-t

,it is clear that

The representations

:P = P-t for

H*.

Since

A

Pt c A, and hence

have some important hereditary properties t 1 which we will now detail. Let W = W (8 w2 be a decomposition of W into two orthogonal subspaces.

where

8-

w

Then it is obvious that

denotes the antidiagonal of

t ( ~ ~ )x t ( ~ ~ * ) F

below, if x denotes some object attached to W

F. Here and

in the discussion above,

then x1 and x2 denote the similar objects attached to From (1.13) it is clear that

x

w1

and

w2.

1v 2 Pt * Pt @ Pt

(1.14)

(outer tensor product)

Further, we have embeddings

which may be lifted to maps of

gp.

Then (outer tensor product)

Finally we come to one of the most remarkable properties of

a.

Let W be a symplectic vector space as usual, and let V be a vector space on which an inner product (a symmetric, non-degenerate, bilinear form) is defined. Put

Both W and V may be identified with their dual spaces by means of the bilinear forms defined on them. This leads to isomorphisms

We may use one of these alternative forms for The space

wV

wV

when convenient.

is naturally a symplectic vector space, with symplectic

form given by the tensor product of the forms on W and V.

Let O(V)

denote the isometry group of the given inner product on V.

There are

obvious embeddings of Sp(W)

and O(V)

into

sp(wV),

= 0

and the images of

the two groups clearly commute. In fact, it is not hard to see that O(V) is the centralizer of Sp(W)

in Sp(W v), and vice versa.

Thus the pair

(Sp(W)

, ON))

form what I have called a dual pair in sp(wV).

Consider an oscillator representation

&(w)

and ;(V)

Note that

aV t

denote the inverse images in

&(w)

h(wV).Let

of

&(w v)

of

Sp(W)

and O(V).

as here defined may not be the same as what was defined

earlier. When it is necessary to distinguish them, we will denote the present one by

gP(~)V.

The difference between the two is fairly mild. "

According to the Shale-Weil description of Sp(W),

we have

"

"

SP(W)~

N

Sp(W)

if dim V is odd;

"

SP(W)~ N Sp (W)

if dim V is even.

x if2

Using formulas (1.11) and (1.15) it is not difficult to see that the restriction

-

v atlsp(w)

is a tensor product of dim V oscillator jvi)iiCl

representations. Explicitly, if

is an orthogonal basis for V,

and the inner product of vi with itself is ti, then

We will call

The groups a dual pair in

"

V"

atl~p(V)

S~(W)

&(wV).

the and

representation of Sp(W)

;(v)

associated to

1.

commute with one another and so form

The reduction of

:a

on either group would

be provided by the following V" a (O(V)) t other's commutants, in the sense of von Neumann algebras. Conjecture:

The groups

V" at(Sp(W))

and

generate each

Although proving this conjecture is at present probably more a matter of hard work than insight, a proof has not been written down. However, for certain V, relatively direct proofs are available [Hl], [HZ]. Theorem 1.1: O(V)

The above conjecture is true if 2 dim V

is compact (i.e., anisotropic).

5

dim W, or if

Furthermore, in the case that

2 dim V 5 dim W,

&(w)

.-. the image of the parabolic subgroup P,(W)

already generates the eommutant of

U;(~(V)).

of

The N -rank of representations of m

2.

Sp.

In this section we introduce a simple-minded general method for studying representations, and we apply it in a particular way to the study of representations of

gp(w)

for W a symplectic vector space. The

rigidity of the representation theory of the Heisenberg group makes the application fruitful. In the next section we will prove some auxiliary results which show that the considerations of this section are less ad hoc --

than they perhaps seem at first. An analysis similar to that given

here applies to other classical groups.

The exceptional groups are

significantly different. Let G be a separable locally compact group, and let H 5 G be a closed subgroup. We will assume G and H are type I.

If the

representation theory of H is quite well understood while that of. G is relatively mysterious, we might attempt to study representations of G by restricting them to H.

In this connection, we make three definitions.

Take a representation

p €

H.

p l ~ , its restriction to

is type I, we know by the direct integral theory [DX], [~k],

Since H

that

and consider

is defined up to unitary equivalence by a projection valued L

measure on H.

We will call this projection-valued measure, and the

unitary equivalence class it defines, the g-spectrum of H-spectrum is obviously a unitary invariant of

p

p.

The

and therefore provides

L

a potential means of classifying

p € G. We note that in fact we may

define the H-spectrum for any representation of G, not only for irreducible ones. Useful information about p might follow from knowledge about

p

considerably cruder than its exact H-spectrum. For example, the dual space L

H has the structure of a To topological space defined by the Fell

topology [Fl].

The (closed) support of the H-spectrum of p

will be

called the geometric E-spectrum. A still cruder piece of information about is the following. Given a representation o n a positive integer or

-

, denote

Recall that two representations a if

a*~

and

a*~'

the direct sum of n copies of

and

0'

are called quasi-equivalent

H is quasi-equivalent to some sub-

representation of the regular representation of G. say

a.

are equivalent. We will say that the representation

of G is g-regular if

p

of H, we let n a , for

Otherwise we will

H is &-singular.

An obvious class of candidates for mysterious groups G are the semisimple groups. A class of candidates for subgroups whose representation theory is well-understood are the unipotent radicals of parabolic subgroups. We will consider the case G = Sp(W) in (1.6),

the unipotent radical of the stabilizer of the span of the eils,

with the eils as in formula (1.1). 2* S (Y),

Nm

and H = Nm where Nm is as defined

where Y

As noted in (1.7),

is the span of the fils. Since. X and Y are

in duality via the form c

, z , we may

also write

2 with S (X) denoting the second symmetric power of X.

s2(x)

we have

The dual space of

.

is s2*(x), the space of symmetric bilinear forms on X. We may 2* 2* identify S (X) with Nm, by associating to p 6 S (X) the character X

defined by

B

xp(n)

(2.1) Here

= xl(B(n>>

n € Nm

3

2 S (X)

.

X1 is the basic character of F chosen in 51. We have Nm

5 Pm 5 Sp. Consider the Nm-spectrum of a representation A

o

of . , P

This is a projection-valued measure on Nm.

For a Bore1 set

n (U) denote the associated projection. Since n comes u u * from a representation of Pm, it allows Ad Pm as a group of automorphisms.

U

Gm, let

Specifically, we have the formula

-

via Ad* is identified via (2.1) to the m 2* Thus the Ad*% orbits in Nm are on S (X).

The action of Pm on natural action of n(X)

naturally parametrized by the isomorphism classes of symmetric bilinear forms over F of rank less than or equal to dim X.

In particular, since

F is not of characteristic 2, the number of orbits is finite. If

0

a symmetric bilinear form on X, let

p

.

0

We define the %of Each

B

B

B

is

denote the Ad*pm-orbit through

to be the rank of

fj

.

is an analytic variety over F, and a such it carries a

(I

B

well-defined measure class, which is locally represented by Haar measure in any local

coordinate system. Although the

(I

B

do not generally

carry A~*P m-invariant measures, the canonical measure class just described

*

for each orbit is invariant under Ad Pm. It follows from the transformation law (2.2) that the restriction to an orbit O C im must be absolutely u Bcontinuous with respect to the canonical measure class on

of the spectral measure n

.

to 0

restriction of n

B

0

OB the must be of uniform multiplicity. Summarizing

this discussion yields Proposition 2.1:

Given a representation

of Pm, the Nm-

spectrum of o is determined by the multiplicities n(a,p) restricted to

0

for each

B

* Ad Pm-orbit

The multiplicities n(a,$) will say the orbit supported on the

OF occurs in

of n (5

m

OB C Nm.

are non-negative integers or g

OP which occur in

if n(0,B)

.

,0.

+ m .

We will say

We is

Given a r e p r e s e n t a t i o n

o

of

Sp,

we may r e s t r i c t i t t o

to

as to

Sp.

D e f i n i t i o n 2.2: of in

0 0

Given a r e p r e s e n t a t i o n

i s t h e maximum of t h e ranks of t h e

.

We w i l l say t h a t

occurring i n

On

* Ad Pm

% .

of

Sp(W),

the

.

orbits i n

-

.

t

0

i s of pure Nu-rank

0

have rank

Example 2.3:

Nm

Nm-rank

." --

occurring

if a l l orbits

4

F, d e f i n e t h e rank one symmetric b i l i n e a r form

The formula (1.8) shows t h a t t h e o s c i l l a t o r r e p r e s e n t a t i o n

is supported on t h e o r b i t

i s of pure

and

F u r t h e r , they apply j u s t a s w e l l

a l l t h e above n o t i o n s may then be applied.

Sp

Pm,

ps

N -rank 1.

s =

with

0

1

-(T)t.

at

of

~p

In particular,

m

This example shows t h e concept of rank i s n o n - t r i v i a l .

We w i l l

e s t a b l i s h some b a s i c p r o p e r t i e s of t h e Nm-spectrum, and p a r t i c u l a r l y of Nm-rank. Given s e t s

Lemma 2.4: An o r b i t

Ul,

Let

U2

c cm,

o1

and

a2 be two r e p r e s e n t a t i o n s of

a

Op 5 Nm occurs i n

o1

occurring r e s p e c t i v e l y i n

Proof:

we use t h e conventional n o t a t i o n

ai

@

o2

fm x fm.

such t h a t

Then t h e

'm.

i f and only i f t h e r e a r e o r b i t s 0

B

i s open i n

ol

F i r s t consider t h e o u t e r t e n s o r product

r e p r e s e n t a t i o n of

,..

Nm x Nm-spectrum of

i s t h e d i r e c t product, i n t h e obvious sense, of t h e

" @

o2 a s a

4@4

N -spectra of

m

O1

and

u2.

Explicitly, f o r s e t s

U1.

1

U2

5 Nm,

we have

ol

Taking t h e i n n e r t e n s o r product of

o2 amounts t o

and

r e s t r i c t i n g t h e o u t e r t e n s o r product t o t h e diagonal. N -spectrum of t h e i n n e r t e n s o r product m

for

U

In particular, the

w i l l be given by

ul @ u2

5 io,where

is t h e d u a l of t h e d i a g o n a l map

Since t h e s p e c t r a l measure o f

ul

or

i s t h e sum of i t s

u2

r e s t r i c t i o n s t o t h e v a r i o u s o r b i t s , we may a s w e l l assume f o r purposes of t h i s lemma t h a t

oi

i s supported on a s i n g l e o r b i t

that

If

d i s j o i n t from

B1 so clearly

" OB2' TI

If

0

OB is an o r b i t d i s j o i n t from

01

@

02

OB i s contained but not open i n

+

0

82

,

then

6*-1 (OP)

be

( O ) = O

B

O!3,

+

OP,'

then

6*-'

(oe

1

Opl

w i l l be a s u b v a r i e t y of p o s i t i v e codimension i n

x

Op2

.

and w i l l

-

have zero measure with r e s p e c t t o t h e canonical measure c l a s s on

0

p1

x

.

0 82

Hence again

Lemma 2.5: X.

Recall

n 01

p1

Let

02

@

(0

p2

and

be two symmetric b i l i n e a r forms on

m = dim X.

a ) A l l o r b i t s which a r e open i n

+

min(m, rank

pi.

r a d i c a l of

pi

p1

+

factored t o

X/(R

1

n

R ) 2

x / R ~ . Hence

of rank equal t o

rank

p1 +

0

rank

$1

+0

have rank equal t o

p2

p1 + rank p2.

p2) = rank

p1 + p2

Then t h e r a d i c a l of

factored t o

Proof:

0

rank p2)

b) Suppose rank (pl

p1 + p2

0.

=

B

is

Let R1

Ri

n R2,

be t h e and

i s isomorphic t o t h e d i r e c t sum of t h e

+0

contains only a s i n g l e o r b i t

82

p2.

P a r t a ) i s obvious s i n c e t h e condition t h a t

pi +

have rank l e s s than t h e maximum p o s s i b l e , a s s p e c i f i e d i n a ) , i s a nont r i v i a l polynomial condition on Suppose

p1 + p2

t h e r a d i c a l of

p1 + p2. rank

(pi,$;)

are a s i n b).

0

6

Pl

Clearly

*s2 %

fl R2

i s contained i n

On t h e o t h e r hand we have t h e standard formula

p1 + dim

R

1

= m = rank

p2 + dim

R2

Hence

+ dim R2 - m - rank( P1 + P2)

dim(% fl R2) ? dim R1 = m

By dimension counting, then, we s e e

p1 + p2.

We may d i v i d e by

R1 fl R2 = {O)

, so

that

R1

R2

R1 Il R2

= m

-

(rank

p1 +

rank f12)

i s indeed t h e r a d i c a l of

and reduce t o t h e c a s e when

rank(pl

+ p2)

=

In this case we see that X = Since F i

@

% ~3R2

+ rank p2

= dim X

is a direct sum decomposition of X.

is complementary to R2, the restricted form

degenerate. R1

rank p1

Similarly

p1 IR2

p21~l

is non-

is non-degenerate. Thus we see that

R2 is an orthogonal direct sum decomposition for the form

and exhibits as isomorphic the direct-sumof the

pi

p1

factored to

+,p2, X/Ri.

This proves the lemma. *

Lemma 2.6:

Let

ol and

(a1 g~o2)

a2 be two representations of Po. Then

Nm-rank

b)

If o1 and o2 are of pure rank, then so is

C)

If each of

=

min(m, Nm-rank ol

+ Nm-rank 02);

a)

ol @I u2 ;

ol and o2 is supported on a single M*P~ orbit

A

in Nm, and if the sum of the Nm-ranks of the al @ u 2 is also supported on a single

oi is at most m, then

* Ad Pm-orbit.

Proof: These statements are immediate from the two preceding lemmas. We need also to understand how Nm-rank behaves under restriction m to smaller symplectic groups. Let {ei,fi)i=l be the standard symplectic basis of formula (1.1).

Let Wk denote the span of

for i 5 k. Thus dim Wk = 2k and W = Wm.

Yk = Y

n Wk.

Pk(Wk)

=

Pm(W)

subgroup of

Let Pk(Wk)

n

Sp(Wk).

Sp(W)

Set Xk = X

be the stabilizer of Xk

n

Wk

in Sp(Wk).

Here we are considering Sp(Wk)

{ei,fi) and Then

to be the

I

leaving Wk, the orthogonal subspace to Wk, spanned

by ei and fi for i > k, pointwise fixed. Let Nk(Wk)

denote the

Then Nk(Wk) = Nm(W) n Sp(Wk). Also, as in 2 formula (1.7), we have Nk(Wk) = S (Xk). Thus we have a diagram

unipotent radical of Pk(Wk).

The t o p

The v e r t i c a l isomorphisms a r e given by d u a l i z i n g formula (1.7).

h o r i z o n t a l map i s j u s t i n c l u s i o n , and t h e bottom h o r i z o n t a l map i s t h e symmetric square of t h e i n c l u s i o n t h a t diagram (2.3)

commutes.

It i s c l e a r from formula (1.7)

% 5 X.

It follows t h a t t h e d u a l diagram

I n diagram (2.4) t h e t o p map i s j u s t r e s t r i c t i o n of a

a l s o commutes.

c h a r a c t e r from N (W) m

to

and t h e bottom map i s given by

Nk(Wk),

r e s t r i c t i o n of b i l i n e a r forms from

X

to

Xk. " ,

Lemma 2.7: the restriction

Let

a be a r e p r e s e n t a t i o n of

alik(wk).

OBI 5 $w,)

Ad*pk(wk) - o r b i t

u and such t h a t

diagram (2.4)

$' € S occur i n

* (Xk).

Opt

and consider

I n order t h a t the

a

k

* Ad Pk(W)-orbit

and s u f f i c i e n t t h a t t h e r e i s an occurs i n

2

Let

P,(W),

0

B

i t i s necessary

in

i s open i n

Gm(w) which with

*

i

as in

.

Proof:

This i s analogous t o lemma 2.4.

Let

n

0

be t h e

Nm(X)" ,

spectrum of Then f o r a s e t

a.

Let

denote t h e

U' 5 Gk(wk)

Nk(Wk)-spectrum of

we have

A s i n lemma 2.4, we may assume f o r purposes of t h i s proof t h a t

supported on a s i n g l e

olpk(wk).

* Ad Pm(W)-orbit

a is

O8 Then we see if i s n o t contained i n Ad*pk(wk) - o r b i t , and Oe *-1 i f ( O ) , obviously i ( 0 ,) i s d i s j o i n t from so 8 i*-l Opt i ( 0 ), but i s n o t open, then Opt) = 0. I f B (OBI)"$ Opt 5 ik(wk)

i s an

'

c *

op

.

OP , and t h e r e f o r e has measure

is a s u b v a r i e t y of p o s i t i v e codimension i n zero f o r t h e c a n o n i c a l measure c l a s s . ( n ) (0

a k

B

0

Hence i n t h i s c a s e t o o we have

= 0.

a)

2* $ t S (X), and

Consider

lemma 2.8: If

i s open i n

O$'

* i (Op),

then

rank $ ' = min(k, rank If

b)

R fl

%,

r a n k ( $ 1%) = rank $,

where

R

i s t h e r a d i c a l of

n a t u r a l l y isomorphic t o

$

$1

t h e n t h e r a d i c a l of

$, and

1 f~a c t o r e d t o

condition t h a t

$

1%

$ factored t o %/(R fl

This is analogous t o lemma 2.5.

Proof:

is

X/R

is

%).

P a r t a ) h o l d s because t h e

have rank l e s s than t h a t s p e c i f i e d i s a n o n - t r i v i a l

pll%

E Op.

polynomial c o n d i t i o n on of

p ' c s2*(%)

For p a r t b) observe t h a t t h e r a d i c a l

$ lxk has dimension

But c l e a r l y be equal.

R

n%

Pixk,

i s contained i n t h e r a d i c a l of

Dividing out by

R

n%

X

=

%8R

they must

reduces u s t o t h e c a s e when

rank $ = rank($ 1%)

It i s then c l e a r t h a t

SO

= k

i s a d i r e c t sum decomposition, and

p a r t b) of t h e lemma follows. Lemma 2.9: olik(wk) = a'.

Let

Pm(W), and consider

Then

a)

Nk(Wk)-rank

b)

If

NkfWk) -rank.

.-,

o. be a r e p r e s e n t a t i o n of

(0')

o i s of pure

= min(k, Nm(W) -rank ( a ) )

NmJm(W)-rank, then

o'

i s of pure

Nm(X)-rank of o i s no more than

c) I f the on a s i n g l e

M*P

* Ad Pk(Wk)-orbit Proof:

m

(W)-orbit

. Nk(Wk).

in

in

Gm(w), t h e n

0'

k, and

o i s supported

i s supported on a s i n g l e

T h i s i s a n immediate consequence of t h e preceding two

lemmas. We may now prove our f i r s t main r e s u l t concerning t h e spectrum of r e p r e s e n t a t i o n s of Let o

Theorem 2.10: H i l b e r t space

H

* Ad Pm(W)-orbit

.

Let

Sp(W).

be a u n i t a r y r e p r e s e n t a t i o n of

no

. 0, 5 Nm,

be t h e

Proof: dim W = 2, and

i3 < m,

rank

Nm(W)-spectrum

of

Sp(W)

o

.

on a

For each

set

s o t h a t t h e s p e c t r a l measure of Then f o r

Nm(W)-

*

Nm a c t i n g on

Hp

t h e subspace

.-.

i s i n v a r i a n t under

We w i l l prove t h i s by i n d u c t i o n on

we a r e d e a l i n g with

i s concentrated on

H,

.

o(Sp)

dim W = 2m.

1

S L ~ ( F ) . The o n l y p o s s i b l e r a n k s a r e

i s t h e one-point o r b i t c o n s i s t i n g

0, and t h e only o r b i t of rank 0

HO,

The corresponding subspace,

fixed vectors.

I n t h i s c a s e , t h e theorem f o l l o w s from [HM] which s a y s

H0

i n v a r i a n t under

Pm.

Since

m > 1.

Sm

It i s c l e a r t h a t t h e

.-.

Sp,

not i n

consider t h e p a r a b o l i c

*

Pn,

$

,

$,

are a l l t o prove t h e

H,

.

In particular,

c o n s i s t i n g of t r a n s f o r m a t i o n s i n

which p r e s e r v e

X1,

t h e l i n e through

i f we can show

H,

i s i n v a r i a n t under

el.

1-

a s specified, there a r e

which p r e s e r v e

Pl(W) = P1

N

SLZ(F).

H

i s a maximal subgroup of

r e s u l t i t w i l l s u f f i c e t o show t h a t f o r elements of

c o n s i s t s of t h e

i n f a c t c o n s i s t s of t h e f i x e d v e c t o r s f o r a l l of

From now on, we t a k e

$ '

For

af the origin.

that

H

Since m when

7

rank

1, P1 # Pm.

p

K

Sp Hence

m, t h e theorem

w i l l follow. To begin, c o n s i d e r t h e space by [HM] we know t h a t

Ifo

of

Nm-fixed v e c t o r s

H0

=

W'.

where

Ho

away.

.

{O)

We review t h e s t r u c t u r e of t h e group

w1 L=

Sp.

Ho, and we may a s w e l l throw

Thus from now on we w i l l assume

W

*

c o n s i s t s of f i x e d v e c t o r s f o r a l l of

H0

Hence t h e theorem i s t r u e f o r

subspace of

Again

Recall

P1.

orthogonal t o t h e p l a n e spanned by

el

is the

W;

and

fl.

Write

We have t h e decomposition

N1 = N1(W)

i s t h e unipotent r a d i c a l of

PI.

i n such f a s h i o n t h a t t h e a c t i o n by conjugation of t h e a c t i o n s d e s c r i b e d i n $1. of t h e s e groups i n

We l e t

Sp, except t h a t ,

f a s h i o n t o a subgroup of

*

P1,

Furthermore

Sp(W1)

and

FX become

e t c . , denote t h e i n v e r s e images

since

*

Sp, we w i l l l e t

may be l i f t e d i n unique

N1

N1

denote t h e l i f t e d group a l s o .

We must c l a r i f y one t e c h n i c a l p o i n t concerning t h e s e l i f t e d groups. Since t h e k e r n e l of t h e p r o j e c t i o n map F =

,.

Sp

-+

Sp

is

Z2,

(except when

a, which we w i l l n o t e x p l i c i t l y t a k e i n t o account) t h e same w i l l be

t r u e f o r any of t h e s e groups.

The subgroup

S ~ ( W ' ) . F5 ~ Sp(W)

I n p a r t i c u l a r , we have e x a c t sequences

i s a c t u a l l y a d i r e c t product

Sp(W1) x FX.

Hence we may combine t h e f i r s t two sequences above and map them t o t h e t h i r d .

From t h i s diagram, we s e e t h a t t h e k e r n e l of t h e middle v e r t i c a l map i s t h e diagonal subgroup

A(Z2 x X2).

t h a t a representation

p of

I n p a r t i c u l a r , f o r l a t e r use, we n o t e r.

r e p r e s e n t a t i o n of

Sp(wq) x FX w i l l f a c t o r t o d e f i n e a

(Sp(W1) -FX)-

i f and only i f

d e f i n e p r e c i s e l y t h e same r e p r e s e n t a t i o n of Return t o c o n s i d e r a t i o n of Consider t h e r e s t r i c t i o n

01

0.

ZN1.

Let

ker jl

piker

and

j2

5. denote t h e c e n t e r of

ZN1

Since t h e f i x e d v e c t o r s of

Nm

N1.

have

been eliminated, we know from lemma 2.6 ( o r a g a i n from [HM]) t h a t t h e r e a r e no f i x e d v e c t o r s f o r N1

= H(W')

occurring i n

Z N ~in

H.

Thus t h e only r e p r e s e n t a t i o n s of

c r l ~a ~ r e the representations

pt

provided

f o r by t h e Stone-von Neumann Theorem, and described by equations (1.4). Thus we may d e s c r i b e t h e r e p r e s e n t a t i o n s of a n a l y s i s there. ~ d '* F

Ni

=:

is {+1}-

i = 1,2,3,4.

According t o formula (1.10),

{?I}-. N1.

The

We may extend

(pi

by expanding on t h e

t h e i s o t r o p y group of

pt

under

i n 4 p o s s i b l e ways t o t h e group

Let us denote t h e 4 extensions by

i pt

for

i pt may b e obtained from one another by t e n s o r i n g

w i t h c h a r a c t e r s of {f 11-.

where

pt

.P,1

Thus we have

is a c h a r a c t e r of

{+1}-.

From Mackey's g e n e r a l theory [My] we know t h a t t h e induced r e p r e s e n t a t i o n s

-x of F .N1 are irreducible, and constitute all irreducible representations "X

of F .N1 which are non-trivial on

ZN~. Thus we

We may extend each representation

have the description

"

-rit

to

'X

(Sp(W1) x F )-N1

"

by means of the oscillator representation of Sp(W1).

"

Since Sp(W1)

is

a perfect group, these extensions are unique. We will continue to denote i them by rt. If we extend the characters cpi of ( 5 ) to characters -x (p l i of F , then from the compatibility of induction and tensor product we see that

Precisely 2 of the 4 characters of of the projection

{z}

+

{fl}, and

{c}will be trivial on the kernel 2 will not. Hence from the

compatibility criterion noted above, precisely 2 of the " " x from (Sp(W1) x F )-N1 to yield representations of P1.

-

these are

where

cp

r1 and t

2

rt.

7 ;

will factor

We may assume

Note then that we can write

is a character of FX

"X

(more precisely, a character of F

which factors to FX) which is non-trivial on {fl}. (The character 'X 2 (p will factor to FX from F because zt must also satisfy the compatibility criterion.)

i T ~ , i = 1,2, are obtained by starting with Pt' " "x extending suitably to N;, inducing to F -N1, then extending again to P1Alternatively, we can start with pt, extend via the oscillator Summarizing, the

representation to Sp(Wt)-N1, ways, to

({kl}

extend again, in the 2 possible compatible

. s~(w'))~-N~, then finally inducing up to F1.

In any

case, we can see that

1

where we have labeled the oscillator representation by m-1 =

I

dim W1,

to indicate with what space it is associated. In fact, with hindsight, we may note the representations

i t

T

om t mof iP(w) decomposes into two irreducible components' u? and u t' It is not difficult to verify that these representations remain irreducible, are already familiar to us.

Indeed, the oscillator representation

and are inequivalent, under restriction to

,. PI.

It is also easy to see

they have the appropriate restriction to N1, so that in some order mi- " mi wt Ipl and wt Ipl are equivalent to the T t' Continuing with Mackey's theory we know that any representation u

-

"

of P1 which contains no fixed vectors for

ZN1

has the form

.

i vt are appropriate representations of ip(w').fX Although i" will be of concern to us, it is not essential, because only vtlSp(w')

where the

rnO

we note that the

: v

may be taken of the following form. Let

be a character of

gX

which does not factor to FX. Then we may write

i pt

where

is a r e p r e s e n t a t i o n of

( s p (w') SF')-,

and

Sp(W1), viewed a s a q u o t i e n t of

Sp

i s a r e p r e s e n t a t i o n of

p i

on which

&(W1) x

Fx

.

.Sp(W) ,

In p a r t i c u l a r , t h e r e p r e s e n t a t i o n o of concerned can, on r e s t r i c t i o n t o formula (2.8). t

of

s a t i s f i e s t h e c o m p a t i b i l i t y c o n d i t i o n and s o f a c t o r s t o

(Sp(W1)" ) ' F .

vi

@ q0

p:i

a c t s by minus t h e i d e n t i t y , s o t h a t t h e r e p r e s e n t a t i o n

ker j1

Let

PI,

w i t h which we a r e

be decomposed i n t h e manner of

ri b e r e a l i z e d on a H i l b e r t space t

.

be r e a l i z e d on a B i l b e r t space

Y,:

and l e t

Corresponding t o (2.8) we have

t h e decomposition

of t h e space Set

of

H

Nm(W)

o.

n

Sp(W1) = Nm-l(W')

NA-~

t h e decomposition (2.9) by considering t h e For each

ff:

opt

~d*~i-~-orbit

in

i n analogy w i t h (2.5).

Then

We may f u r t h e r r e f i n e

= NA-l.

s p e c t r a of t h e

fftp1

d e f i n e t h e subspace

ff:

i s t h e sum of t h e

i t'

v

of

ffi

t$

,,

so that

from (2.9) we g e t

The group

Nm(W) i s a subgroup of i vt

Nm-spectra of t h e r e p r e s e n t a t i o n s normalizer 2 S (X,),

P1

n Em

of

in

Nm

P1.

t h e a c t i o n by conjugation on

-P1,

so we may consider t h e

and

i ' c ~Denote . by

When

Nm

Nm maps

is identified t o Q

n o t t o a l l of

b u t t o t h e p a r a b o l i c subgroup

Ql

Thus a given

fjm w i l l decompose i n t o s e v e r a l

Precisely, i f

*Ad Pm $ €

orbit in

s**(x),

of

the

Q

GL(Xm) preserving t h e l i n e

we may w r i t e

GL(Xm),

*Ad Q

X1. orbits

where the

are Q1 orbits in the GL(Xm) O$ ,t describable as follows. For t # 0,

ogSt=(8' Some of the

t s

Op,t

2*

(XI:

8'

0 , and

€

B

orbit

Op

and are

2 $'(elsel) = s t, for some s

€1~1

p.

may be empty if t is not represented by

Further 2* Op,o = {$' € S (X):

p'€ 0

B

and pf(el,el) = 0, but

$'(elsei) # 0 for some i} 2* Op,oo= { $ ' € S (X); $ ' €

The

Op and

pl(el,ei) * 0 for all i s m }

OBSt for t # 0 are open in 0 , while the union

B

Og

is a closed subvariety of

Op,o

Op,oo

of positive codimension. i t m and from example at,

From the equivalence, noted above, of the representations *

with the restrictions to P1 of the components of

T

i 2.3, we see that the Nm-spectrum of is concentrated on s single T~ 2* GL(X)-orbit in S (X), the orbit of the form ps in the notation of 1 example 2.3, with s = -(-)t. Evidently for t # 0, the only non-empty 2

Q1-orbit i

T~

0

$,st

contained in

as being supported on

Oes*.

p'

8s

is

.

on X fl W'

n W1)

orbit of p',

we have

€

.

Thus we may regard

i ~ - S2*~ (X n W') .

to a form p($')

by letting X1 be in the radical of p(p'). the GL(X

OP

ffip, for $ '

Consider the subspace We may extend the form

0

on all of X

Then clearly if

( $ 9

5s

Equally c l e a r l y we s e e t h a t t h e

&,

* Ad Ql

is t h e Take

~(p',t)= Y

on

X

Q1

on

*

orbit

' p ( ~ ' ) ,OO

n W'),

$ ' E s2*(x

Nm-spectrum of t h e a c t i o n of

and

t €

5 Nm

'

Define a form

.'F

by

Then it i s n o t hard t o convince y o u r s e l f t h a t t h e sum of th'e o p ( p l ) , ~ ~and

in

Ql

orbits

is

s2*(x)

Reasoning e x a c t l y a s i n lemma 2.4, we can t h e r e f o r e conclude t h a t t h e spectrum o f t h e ~d*;-orbit

Nm

a c t i o n on

.

oY($',s),s

i

Htp,

i

@

Yt

i s concentrated on t h e

Taking i n t o account t h e decomposition (2.11),

we s e e by comparing Nm-spectra t h a t , with t h e

IfB from formula (2.5),

w e may w r i t e

It i s obvious t h a t rank Hence i f rank

f3 c m, t h e n rank

B'

c m-1.

i s i n v a r i a n t under a 1 1 of

assume t h a t (2.12) e x h i b i t s

y(B1, s ) = rank

ffB

..

a s a Pl-module.

$'+ 1

Thus by i n d u c t i o n we may Zp(wv). But then formula

As we noted above, t h i s e s t a b l i s h e s t h e

theorem. Corollary 2.11:

If

(&(w))',

a

then a has pure Nm-rank.

If

S.n

Nm-rank (a) c m, then a is concentrated on a single Ad Pm orbit in Proof:

Suppose the Nm-rank of a is &

of rank less than 8

og

& rank $ c m.

If an orbit

0, then in particular,

Thus theorem 2.10 says that the spectral projection

corresponding to 0

yields a non-trivial

8

0,

occurs in

.

8.

contradicting the irreducibility of a

Zp(w)

.

subrepresentation of

If

C c m,

and two orbits

occurred in a , then both rank c m and O82 < m. Hence theorem 2.10 yields two non-trivial, mutually orthogonal

O$1 and rank

e2

, again contradicting irreducibility.

hence proper, subrepresentations of o

" ,

Corollary 2.12:

If a is a representation of Sp2,

of pure

Nm-rank C > 1, then

where the

vt are representations of pure Nm-l-rank C

m-1 are oscillator representations of at Proof:

- 1,

and the

" ,

'"2 (m-1) '

This is immediate from the decomposition (2.12) and formula

(2.7). " ,

rank &

Corollary 2.13:

If a is a representation of Sp2,

,

then

and k

5 C

,

algp2(m-k)

of pure

is a (finite) sum of

representations of the form m-k

"e "em-k

where form

p

"8

is the Weil representation of

associated to the

jp2(,k) " ,

of rank k, and v

Nm-k-rank C

@

- k.

B

is a representation of SP~(,,~)

of pure

This follows by i n d u c t i o n from Corollary 2.12 and t h e

Proof: formula (1.15).

Corollary 2.14: Nm-rank

C < m,

then

even, except when Proof:

o

If

o

factors t o

Sp2,,,(F)

Then

o factors t o

Sp2m

if

S P ~ ( , - ~ ) . Look a t t h e decomposition

,.

of

m-1

S P ~ ( , - ~ ) . We know t h a t t h e o s c i l l a t o r r e p r e s e n t a t i o n s ,.

of rank 1 and do n o t f a c t o r t o m-1 w t

and

Sp2(m-1),

SP~(,-~).

ot

are

Considering t h e r e s t r i c t i o n s of

t o t h e k e r n e l of t h e p r o j e c t i o n from

,.

Sp2(m-l)

to

t h e r e s u l t follows.

Corollary 2.15: N -rank, m

or

For

Sp2,

c o n s i s t s of r e p r e s e n t a t i o n s of even pure

N -rank m. m

Proof:

This is immediate from c o r o l l a r i e s 2.13 and 2.11.

C 5 m, l e t

A

(EP(~))C denote t h e subset of

of r e p r e s e n t a t i o n s of pure

Nm-rank

( ~ P ( w ) ) ; denote t h e subset of whose

is

C

By i n d u c t i o n we can assume t h e r e s u l t i s t r u e f o r t h e r e p r e s e n t a t i o n s

(2.13).

vt

i f and only i f

i s some m u l t i p l e of t h e t r i v i a l

o

C P 1.

Hence consider

factors to

vt

SpZm(F) of pure

F = a.

I f C = 0, then by [HM]

representation.

.

i s a r e p r e s e n t a t i o n of

C

.

( i p (w))"

N -spectrum i s concentrated on t h e

m

For a form

(gp(w))

A

consisting

2* € S (X),

let

.

c p n s i s t i n g of r e p r e s e n t a t i o n s A ~ * P o~r b i t

Op

5 Nm.

Corollary 2.11 t e l l s us we have a d i s j o i n t union V

A

(2.14)

rank $ < m

The o s c i l l a t o r r e p r e s e n t a t i o n s a r e examples of rank 1 r e p r e s e n t a t i o n s . Lemma 2.6 t o g e t h e r with formula (1.8)

of

,.

Sp

a s s o c i a t e d t o t h e form

$

t e l l s us t h a t t h e Weil r e p r e s e n t a t i o n

decomposes i n t o r e p r e s e n t a t i o n s belonging

" A

to Spe,

where

$

=

.

1 -(T)p

(This slight discrepancy is an artifact

of our conventions and could be eliminated. See example 2.3.) particular, none of the detail in 8 4 .

In

" A

Spp are empty. We will study them in more

We finish this section with an observation about the rank

of the most familiar type of representation, tempered representations. In order for a representation of an abelian group to be quasiequivalent to a subrepresentation of the regular representation, its spectral measure must be absolutely continuous with respect to Haar measure on the Pontrjagin dual. the

* Ad Pm(X)

orbits in

im(x)

Since the canonical measure classes on are absolutely continuous with respect

to Haar measure only for the open orbits, which are of rank m, we have Proposition 2.16:

A representation

cJ

of

Sp is N~~J)-regular,

in the sense defined at the beginning of this section, if and only if it is of pure Nm-rank m. Proposition 2.17:

All irreducible tempered representations of

"

Sp2m are of pure Nm-rank m. Proof: By the preceding proposition, it will suffice to prove tempered representations are Nm-regular.

In fact, we will show

something much more general. Proposition 2.18:

If G

is a reductive group and N

ZG

is a

L

unipotent subgroup, and

p € G

is tempered, then p

is N-regular.

It will be convenient to postpone the proof of this until $7.

3:

N -rank and r e g u l a r i t y

I n t h i s s e c t i o n we d i g r e s s s l i g h t l y from our development o f t h e p r o p e r t i e s of

Nm-rank t o put it i n a more g e n e r a l s e t t i n g .

be a n a r b i t r a r y p a r a b o l i c , and l e t W e w i l l r e l a t e t h e n o t i o n of

N-regularity t o

Xk b e t h e span of t h e standard b a s i s v e c t o r s 1

< k2
, the

r e s t r i c t i o n of t h e form
,

we know t h a t

r e p r e s e n t a t i o n s of

Sp(W1)

& J ( w ~ )is c o n j u g a t e i n

formula (1.15) t e l l s u s t h a t

o1ip(w1)

We know t h e o s c i l l a t o r

i s a l s o a sum of o s c i l l a t o r r e p r e s e n t a t i o n s .

-

olip(~i)

form a f i n i t e s e t

{at}

parametrized by

FX/FX2. We a l s o know ( i t i s a very s p e c i a l c a s e of theorem 1.1) t h a t each

t h e r e a l i z a t i o n of

cot

on

t

f(-y)

- f(y).

w

And

f ( y ) = -f(y).

t

transformation of for

s-1t

W

il(w1).

functions

w i l l be t h e odd f u n c t i o n s :

{+1} where h e r e and i t s negative.

{fl}

f

N1(W1)

(1.8),

fl(wl)

.

Indeed,

wt

has

In

t h e space

such t h a t f u n c t i o n s such t h a t

a r e seen by i n s p e c t i o n

indicates t h e identity w'

t

spectra.

and

Since

a r e p a i w i s e inequivalent

Furthermore, it is f a i r l y easy t o s e e t h a t each

i r r e d u c i b l e on

p.

(9,Y)-bounded f o r some compact subgroup K c G ,

i s s t r o n g l y mixing o r s t r o n g l y

then

9

or

@ C L P ( ~ ) . This is because t h e a l g e b r a i c sum of t h e K-isotypic

subspaces

HP

according a s

L',

Q E C,,(G)

H.

( t h e space of K - f i n i t e v e c t o r s ) i s dense i n

(9.Y)- boundedness

Also

need only be checked f o r a dense subspace i n

H,

by

e s t i m a t e (6.4). g)

A d i r e c t sum of r e p r e s e n t a t i o n s s a t i s f y i n g any of c o n d i t i o n s

(6.7) a),b),c)

The only c o n d i t i o n t h a t is n o t completely

from t h e formula (6.11).

obviously maintained under d i r e c t sums i s observe t h a t , w i t h n o t a t i o n a s i n (6.11), and s i n c e

This follows

o r (6.8) a g a i n s a t i s f i e s t h e same condition.

2

1 1 ~ 1 1=~ 1 1 ~ ~ 1 1+

(9.Y)-boundedness. if

2 llx211 , one h a s

by t h e Schwartz i n e q u a l i t y .

x € f/ then P'

The p r e s e r v a t i o n of

For t h i s , xi E (ffi)w,

(@,I)-boundedness under

d i r e c t sums f o l l o w s . h)

It i s obvious t h a t t h e p r o p e r t y of being s t r o n g l y mixing,

a b s o l u t e l y continuous, o r

(Q,Y)-bounded i s i n h e r i t e d by s u b r e p r e s e n t a t i o n s ,

However, t h e p r o p e r t y of being s t r o n g l y r e g u l a r r e p r e s e n t a t i o n of Cc(G)

G

on

5 L ~ ( G ) i s dense, and

q

supported by formula (6.9). (e.g.,

2 L (G)

LP

is not.

is strongly

~ fo ~ r ,u,v ~ € Cc(G)

For example, t h e

1 L , since is compactly

non-integrable d i s c r e t e s e r i e s ) which a r e n o t s t r o n g l y i)

Given

x,y 6

H, i f

t h e n t h e same w i l l b e t r u e f o r

1 f i € L (G)

by formula (6.5).

2 L (G)

But t h e r e a r e s u b r e p r e s e n t a t i o n s of

belongs t o p(fl)x

and

c~(G) or

p(f2)y

1

L

.

L~(G),

f o r any

Hence t o check s t r o n g mixing o r s t r o n g

~ ~ - n e sofs

, it

p

subspace of

i s enough t o check i t f o r

which g e n e r a t e ff

H

trreducible, then

x,y

a s G-module.

In particular i f

w i l l be s t r o n g l y mixing o r scrongly

p

one m a t r i x c o e f f i c i e n t i s i n

C,,(G)

or

L'(G)

comes t o e s s e n t i a l l y t h e same t h i n g ; more p l e a s a n t t o work with.

D

L-

p

i f only

(9,Y)-boundedness which

i t i s l e s s p l e a s a n t t o d e f i n e , but

We w i l l s a y

p

i s modified

(@,Y)-bounded

i f t h e same e s t i m a t e s on

hold, except t h a t when both x and (Px,~ belong t o a given K-isotypic space, we only r e q u i r e t h e e s t i m a t e of d e f i n i t i o n (6.8) t o hold when for

x

and

y

x=y.

is

respectively.

There i s a s l i g h t m o d i f i c a t i o n of

j)

belonging t o a

y

Of course, t h i s would make no s e n s e

belonging t o d i f f e r e n t K-isotypic spaces.

For any

x,y

we t h e have e a s i l y v e r i f i e d i d e n t i t y

qx, y (here

i =

subspace

dz.) ff

CL

.

-

- 'PX-Y ,x-y

'P*,x+y

Suppose

x

and

y

+

'Px+iy,x+iy

-

qx-iy,x-iy

a r e u n i t v e c t o r s i n t h e K-isotypic

Then i f we assume modified

(O,Y) -boundedness, we get

Using t h e p a r a l l e l o g r a m law

1*12 and t h e assumption

But c l e a r l y i f a l l vectors. (@,

+ Ilx-Yll

2

= 2 ( 11x112

+ llY112)

llxll = llyll = 1, we g e t

(@,Y)-boundedness holds f o r u n i t v e c t o r s , i t h o l d s f o r Hence we s e e t h a t modified

flI)-boundedness

.

(9,Y)-boundedness implies

We need some f u r t h e r f a c t s about d e f i n i t i o n s (6.7) and (6.8) a r e more involved than t h e above, though still n o t d i f f i c u l t .

that

These

concern r e s u l t s on r e s t r i c t i o n and i n d u c t i o n , products of groups and t e n s o r products of r e p r e s e n t a t i o n s , and t h e F e l l topology. Consider two r e p r e s e n t a t i o n s

H1 ff 3

HZ.

and =

ffl 8 HZ.

and

4

Form t h e tensor product

x and

y

of

ff3

is

p1 63 p2, we f i n d

p C3 p2 1

formed from f i n i t e rank t e n s o r s a r e

sums of products of m a t r i x c o e f f i c i e n t s of P r o p o s i t i o n 6.1: a )

b)

a c t i n g on

of t h e form (6.13) and u s e (6.12) t o compute t h e

Thus matrix c o e f f i c i e n t s of

pl 8 p2

b p2

a c t i n g on spaces

i s spanned by t h e f i n i t e rank t e n s o r s

ffg

matrix c o e f f i c i e n t

then

p2

The d e f i n i t i o n of t h e i n n e r product on

A dense subspace of

I f we t a k e

pl

If either

pl

P1 or

and p2

P2-

i s s t r o n g l y mixing,

is a l s o .

If either

pl

or

p2

i s a b s o l u t e l y continuous, t h e n s o i s

P1 8 P2c) pl b p2

If

pl

i s s t r o n g l y LP

is s t r o n g l y

r

L

where

and

4

is strongly

Lq,

then

There i s no r e s u l t analogous t o t h e s e f o r

Remark:

(+,y)-

boundedness because t a k i n g t e n s o r p r o d u c t s mixes up t h e K-types s o much. However, we w i l l s e e i n p r o p o s i t i o n 6.2 t h a t

(9,Y)-boundedness works

w e l l i n s i t u a t i o n s where t h e above c o n c e p t s do poorly. Proof:

Statement a ) i s completely obvious from formula ( 6 . 1 4 ) ~

and s t a t e m e n t c ) i s a l s o , because of t h e well-known f a c t s governing t h e products of

L~

f u n c t i o n s [DS].

consider t h e c a s e when 2

that

til = L (G).

space

L (G; H2)

Z

fi

is the (right) regular representation, so L~(G@ ) H2

The space of

For s t a t e m e n t b ) , i t w i l l s u f f i c e t o

can a l s o be regarded a s t h e

Hz-valued f u n c t i o n s on

G

w i t h s q u a r e i n t e g r a b l e norm.

Then t h e t e n s o r p r o d u c t a c t i o n becomes

Define a map

A

It is c l e a r t h a t

Thus

on

A

L ~ ( G ; H ~by)

is unitary.

We compute

d e f i n e s a u n i t a r y equivalence between

A

proving b)

R 8

4

(dim P ~ ) R ,

.

The behavior of

(@,I)-boundedness under t e n s o r p r o d u c t s i s more

d i f f i c u l t t o explicate.

To do s o r e q u i r e s some p r e l i m i n a r y work.

A topology h a s been d e f i n e d on t h e u n i t a r y d u a l

[ ~ l ] . If then

and

{pn)

{p ) n

i s a sequence of r e p r e s e n t a t i o n s i n

converges t o

, if

of

E,

and

given a m a t r i x c o e f f i c i e n t

G

o

by F e l l

is fixed

'Px9~

of

a , there are matrix coefficients

V

n

of pn which converge to Xn'n' uniformly on compact sets. A related concept is that of weak cp

(Px,~ containment. A representation a is weakly contained in a representation p

if the matrix coefficients of a can be uniformly approximated on

compacta by matrix coefficients of

p.

In particular, to an arbitrary

representation p of G, we may assign a set supp p Z all

a € G

which are weakly contained in p

closed in G. Further, if G

.

1

G

consisting of

Clearly supp p is

is of type I, as we will assume, then 1

supp p is exactly the support of the projection-valued measure on G defining p

up to unitary equivalence.

K. Let

+

-

Let G be a locally compact group with compact subgroup

Lemma 6.2:

be a function on G and Y

a function on K.

a

The subset of G

a)

consisting of representations which are

,.

(Q,Y)-bounded is closed in G.

modified

A representation p of G is modified

b)

only if all

a € supp p

are modified

(@,Y)-bounded

if and

(Q,Y)-bounded. L

{pn) be a sequence of representations in G

Proof: Let a

converging to

a € G.

Suppose the pn are

vector x in the space tl of choose vectors xn

a

in the space

have the same length as x and

.

(Q,Y) -bounded. Fix a

Then we know from [Fl] that we can

Hn of

pn such that all the xn

a x , x ' + '

is the uniform-on-compacta limit

.

Pn a of the Select p, v € K, and choose unit vectors '+'xn,xn x C HCL and y € Hv Then, assuming p # v , the vectors x and y

.

will be orthogonal, so llx+yl12 = 2. such

Thus we can find vectors vn €

Pn converges uniformly on compacta to IPvn,vn 1 Let ep, ev be the central idempotents in L (K) corresponding

llvnl12 = 2, and

a 'Px+y,x+y

'

%

to p and v respectively. Then from formula (6.5) we see

e

0

Theref o r e

'Px,~

1L

*

u

*

e*=

u (Px,~

v

'P+~,*

i s t h e uniform on compacta l i m i t of

where

Clearly

un

and

Hence t h e product

by d e f i n i t i o n of

wn

a r e orthogonal, and

11 uJl

i s a t most 1, so t h a t

IlwJl

(@,Y)-boundedness.

I n t h e l i m i t , remembering t h a t

llxll = Ilyll = 1 by choice, we g e t

Since it is c l e a r l y enough t o v e r i f y t h e c o n d i t i o n f o r

o does s a t i s f y t h a t c o n d i t i o n , a t l e a s t

on u n i t v e c t o r s , we s e e t h a t when

I I$

Y

.

(@,Y)-boundedness

The proof when

v

= p

i s completely s t r a i g h t f o r w a r d and

i s omitted. E s s e n t i a l l y t h e same argument shows t h a t f o r a n a r b i t r a r y r e p r e s e n t a t i o n p of

p

is.

ff

of

G, a l l

0

in

supp p 5

It remains t o prove t h e converse. p

2

Let

and consider t h e matrix c o e f f i c i e n t

a r e modified x

(Q,Y)-bounded i f

belong t o t h e space

'Px,xS The d i r e c t i n t e g r a l

i s a unif orm-on-compact l i m i t of sums

q~:,~

theory [Nk] implies t h a t of the form

where t h e that above.

. 2 .Zllvmll .

am range through

1 1 ~ 1 1 ~is

supp p

a l i m i t of

Select K-types p and

v

Comparing values a t 1, we conclude W e now perform t h e same t r i c k a s

and choose u n i t v e c t o r s

.

Approximate t h e matrix c o e f f i c i e n t y € Hv qIx+~,* a m form (6.16). Then ( P ~ , ~ i s approximated by sums

x € ff

P ' by sums of t h e

where

Here a s above

e

P Then we again have

since

urn and

wm

and

e

a r e t h e projections of

subspaces of t h e space of "Iumll Therefore, by

a r e t h e c e n t r a l idempotents f o r p

om.

llvmll 5

1

am

%rn,wm (g) 1

I n t h e l i m i t , t h e modified

p

-

and

Therefore

1

(@,Y)-boundedness of t h e

la

vm onto

C

I.

2

2 llvmll

supp p, one has

a 6 2

( C IIvmII

(@,I)-boundedness of

Y(P) Y(v) @(P) p

follows.

and

v

v

.

- isotypic

Again l e t

G

b e a l o c a l l y compact group with compact subgroup

i s a uniformly largq subgroup of

Following F e l l [F3] once more, we s a y K G

i f there is a function M

m u l t i p l i c i t y of

m u l t i p l i c i t y bound f o r

in

K

such t h a t f o r any

p i s a t most

I ? in

~l f

2,

on

K.

M(d.

a

6,

(

We c a l l M

the a

To have uniformly l a r g e compact subgroups

G.

is a v e r y s t r o n g c o n d i t i o n f o r a group t o s a t i s f y .

Such groups a r e i n

p a r t i c u l a r t y p e I. P r o p o s i t i o n 6.3:

Let

and

G1

w i t h uniformly l a r g e compact subgroups m u l t i p l i c i t y bound f o r G1 x G2

in

such t h a t

functions (al x

Ki

ai

on

+2 , Y)-bounded

pl @ F~ € (K1

x

Remark:

K1

and

yi

on

and

Suppose

Gi.

i s modified Gi

be two l o c a l l y compact groups

G2

Kg.

Let

i s a r e p r e s e n t a t i o n of

p

(ai, yi)-bounded

A

Ki.

Then

a s a r e p r e s e n t a t i o n of

.

p

G

1

be t h e

Mi

for suitable

i s modified

x G2,

where f o r

K ~A ) we d e f i n e

A r e s u l t l i k e t h i s completely f a i l s f o r t h e s t r o n g mixing,

absolute continuity, o r strong

p r o p e r t i e s of r e p r e s e n t a t i o n s .

L'

example consider t h e j o i n t l e f t and r i g h t a c t i o n of a s a r e p r e s e n t a t i o n of

R x IR.

2

IR on L (R), taken

This a c t i o n i s s t r o n g l y

f a c t o r , b u t t h e d i a g o n a l subgroup a c t s t r i v i a l l y .

For

L'

on

each

Hence t h e m a t r i x

c o e f f i c i e n t s of t h i s a c t i o n a r e constant on c o s e t s of t h e d i a g o n a l subgroup, so t h e r e p r e s e n t a t i o n of Proof: irreducible f o r

B x R

i s n o t even s t r o n g l y mixing.

By lemma 6.2 i t i s enough t o consider t h e c a s e when G1 x G2.

Since t h e

is

having uniformly l a r g e compact

Gi,

subgroups, a r e of t y p e I we may f a c t o r

p

p

i n t o a t e n s o r product:

Select

Ki

types

hi

and

a r e t y p i c a l elements of p

UP and l i k e w i s e f o r

x =

most

=

ff

A

.

,

= (Hl)

I f pi

i s r e a l i z e d on

Hi,

then

and (Hz)

@

'5

CI2

,

.

ff

A t y p i c a l element of

where

Then

(K1 x K2)

ffl 8 H2

i s r e a l i z e d on

vi.

ai € (ff )

and

l'5

ff

can be represented i n t h e form

P

zai8pi

pi

min(dim(ff )

€

(ff )

) 5

CI2

.

The number of suwnands i s a t

min(Mi(pi)

dim pi)

Furthermore, t h e s p e c t r a l theorem t e l l s u s t h a t we may s o s e l e c t t h e summands s o t h a t t h e

pi

ai

a r e mutually orthogonal i n

a r e mutually orthogonal i n

(ff )

P2

.

and t h e Vl We then have t h e r e l a t i o n

where each norm i s taken i n t h e a p p r o p r i a t e space. of

ffv

can be w r i t t e n i n analogous f a s h i o n :

With t h i s n o t a t i o n , we may compute

(ff )

A t y p i c a l element

y

Since pl and

p2 are assumed modified

(+i,Yi)-bounded, we get the

estimate

Here the factor 4 comes from remark j) above. From the equation (6.17) and the Schwartz inequality, we can estimate

Plugging this into (6.18) gives the relation for with Y as specified in formula (6.16).

x

+2,Y)-boundedness,

Actually, it eives a slightly

sharper estimate, since in (6.16) we have replaced the minimum of

% by the geometric mean, to make the formula more symmetric.

Mk(%)dim

It seems appropriate in this general discussion to make some observations about the behavior of the properties of definition (6.7) under induction and restriction. Proposition 6.4:

Let G be a unimodular locally comapct group

and let H be a unimodular closed subgroup. a)

If p is strongly mixing, absolutely continuous, or strongly 'L

as a representation of G, then

p l ~ has the same properties relative

to H. b)

If

o is a representation of H

that is strongly mixing, or

absolutely continuous, then the representation p =

indG H

has the same properties relative to G. Proof: It is obvious that if p is strongly mixing, then is also.

p l ~

For absolute continuity, it suffices to consider the case when

p

2 L (G);

h

b u t t h e n it i s c l e a r t h a t

and i n p a r t i c u l a r i s a b s o l u t e l y continuous. strongly

m

Let

and l e t

F i n a l l y suppose

2

L (HI, is

p

m a t r i x c o e f f i c i e n t of

p

.

denote a t y p i c a l element i n a n i c e s e t of c o s e t r e p r e s e n t a t i v e s

f o r t h e q u o t i e n t space on

L'

q X s y be a n

i s a m u l t i p l e of

plH

H\G,

and l e t

dm

denote t h e G-onvariant measure

Then

H\G.

S i n c e t h e integrand i s everywhere p o s i t i v e and t h e t o t a l i n t e g r a l i s f i n i t e , t h e i n n e r i n t e g r a l must be f i n i t e f o r almost a l l

m b y Fubini.

Since a

s e t whose complement h a s measure zero i s everywhere dense, we s e e t h a t qx,p(m)~

belongs t o

L'(H)

G.

The s e t of such p a i r s

of

p

.

x

for and

m

a r b i t r a r i l y close t o the i d e n t i t y i n

p(m)y

a r e c l e a r l y dense i n t h e space

T h i s concludes p a r t a ) of t h e p r o p o s i t i o n .

Consider now a r e p r e s e n t a t i o n p .of

a

of

H.

G

induced from a r e p r e s e n t a t i o n

Because t h e r e p r e s e n t a t i o n induced from a d i r e c t sum i s t h e

zum of t h e r e p r e s e n t a t i o n s induced from t h e summands, t o deduce a b s o l u t e c o n t i n u i t y of

p from t h a t of

But then obviously

p

2

a

, it

w i l l suffice t o take

o

2

N

L (H).

2 L (G), which i s a b s o l u t e l y continuous.

To prove t h e o t h e r a s s e r t i o n s of p a r t b) of t h e p r o p o s i t i o n , we must compute some m a t r i x c o e f f i c i e n t s of a compact s e t of

H x C

C

5

G

p.

such t h a t t h e map

(h,c)

We w i l l suppose we can f i n d +

onto a neighborhood of t h e i d e n t i t y i n

hc) G.

defines an injection For a l l groups

w i t h which we s h a l l d e a l , t h e e x i s t e n c e of such a l o c a l c r o s s - s e c t i o n t o the

H

c o s e t s w i l l be obvious.

Given

x

i n t h e space of

a

, define

w .

x

in t h e space of

p

by

The s e t of such f u n c t i o n s , f o r v a r i a b l e space of p

where If

dc

y(cg)

as

G-module.

x

and

C, c l e a r l y g e n e r a t e t h e

Let u s compute

i s t h e measure on

C

p u l l e d back from i t s image i n

H\G.

Z 0, t h e we may w r i t e cg = h c '

h€H, c l € C

Thus

If cgc at

g +

'-1

-

-

in

G, c l e a r l y

which a r e i n

on

H, s o w i l l

H

c g c go t o

-

-+ =

in

vanish a t

Q;,:

is s t r o n g l y mixing, s o i s

'-1

H.

in

-

G

a l s o , and so those

Hence i f on

G.

QX,Y

vanishes

I n o t h e r words, i f o

p.

We conclude t h i s s e c t i o n w i t h a t r i c k of Cowling [Cg] showing how a n e s t i m a t e f o r m a t r i x c o e f f i c i e n t s of K-invariants v e c t o r s can be parleyed i n t o a n e s t i m a t e f o r more g e n e r a l v e c t o r s . Theorem 6.5: compact subgroup

K.

(Cowling) Let p

Let

G

be a l o c a l l y compact group with

be a r e p r e s e n t a t i o n of

G, and l e t

*

p c3 p

be t h e t e n s o r product of and

y

f u n c t i o n Q on

G.

Suppose t h a t i f

Proof:

Then

is

p

Let

p be r e a l i z e d on t h e H i l b e r t space

may be r e a l i z e d on t h e space

H.S.

H

.

Then

of Hilbert-Schmidt o p e r a t o r s

H, via the action g EG, T

The i n n e r product on

H.S.

*

tr

S,T

E H.S.

i s t h e standard t r a c e functional.

Given

x,y i n

H

, we

It i s t h e n easy t o check t h a t

and

E H.S.

i s given by

(S,T) = t r ( S T ) where

x

* p b p , then

2 dim p)-bounded.

* p 8 p on

with its contragredient.

a r e two K-fixed v e c t o r s i n t h e s p a c e of

f o r some K-bi-invariant (Q1",

p

can form t h e dyad

Elby

E H,

by t h e r e c i p e

f o r an operator

on

T

Take a v e c t o r

H

x

.

in

In particular, for

ff

.

Let

..., xd

xl,

b a s i s f o r t h e l i n e a r span of t h e K-orbit of xl = of

11~il-lx.

men

s

=

a

qS,S

.

x

in

, we

have

be an orthonormal

H

.

We may assume

i s orthogonal p r o j e c t i o n onto t h e span

E

Xi'Xi p(K)x, and s o i s i n v a r i a n t under

(6.18) a p p l i e s t o

x,y,u,v € ff

*

p 8 p (K).

Therefore inequality

On t h e o t h e r hand,we see

dimension of t h e span of t h e K-orbit

p(K)(x).

(S,S) = d, the

Also

Therefore we conclude (6.19) Choose r e p r e s e n t a t i o n s and

y €

ff

.

Observe t h a t

means of a matrix algebra of span of

dli2

5

p(K) (x)

p

1

L (K)

rank

i s a t most

has dimension a t most

and

(dim p)2

~ *Ir2 x ~ ~

.

v E K.

.

dim p

(dim v)

2

Choose v e c t o r s

a c t s on the span of

(dim

+

~

p(K) (x)

.

P

by

Hence the dimension of t h e

Similarly t h e span of 2

x €

p(K) (x+y)

Hence (6.19) s p e c i a l i z e s t o

la1

of F. Let

denote the absolute value of a

homomorphism from A

.

la1

Then

is a

the positive real numbers. For s € R+x ,

to R*,

set

+

As = {a € A :

1.

(a) a s for all

a €

z+}

+

+

Evidently As is a subsemigroup in A when s 2 1. We call A1 positive Weyl chamber in A.

the

We will suppose we have the decompositions

of G into

G

=

KB

+ G = KA1 K

(Iwasawa decomposition) (Cartan decomposition)

The Iwasawa realization is always achievable with appropriate choice of

K. The Cartan decomposition is not. However, it is achievable for groups over IR or U

(Lie groups), and for many groups over nondrchimedean

fields, in particular for Sp. In general, it almost holds, and our arguments can be modified to cope with the general case. However, we will assume the Cartan decomposition in the form (7.1) for simplicity. Let E be the basic zonal spherical function for K defined by Harish-Chandra PC11, [HC3]. Precisely, 3 is the matrix coefficient of the K-fixed vector in the representation of G induced from the trivial representation of B. More explicitly, 3 is produced as follows. Let 6B denote the modular function of B, so that if deb is a leftinvariant Haar measure on B, one has

Then drb = 6B(b) dt(b)

is a right-invariant Haar measure on

Let rb. be a unitary character of B.

Consider the space

H;

B. of smooth

301 (6.20)

((dim P)

I%+Y,*l-

2

+ (dim v)

2 112 )

t11x11~ + llyl12)

I f we u s e t h e p o l a r i z a t i o n i d e n t i t y of remark j) above, we can conclude

I

(6.21)

Since

dim

p

5_

5

((dim p12

+

+

(dim

11~11~)

1 we have 2 dim p dim v 1 ((dim p)

Hence, i f we choose

x

and

y

so t h a t

2

+

2 112 (dim v) )

llxll =

Ilyll,

e s t i m a t e (6.21)

says

This is' p r e c i s e l y t h e e s t i m a t e f o r

(@

'I2,

2 dim ~1)-boundedness of

But i t c l e a r l y s u f f i c e s t o prove such a n e s t i m a t e when we may a c h i e v e t h i s by simply m u l t i p l y i n g Thus theorem 6.5 i s proved.

x

or

y

llxll =

llyll,

p

.

for

by a s c a l a r f a c t o r .

7: Asymptotics of matrix coefficients for semisimple groups In this section, we use the general concepts of $6 to study matrix

.,

coefficients of representations of Sp. For abelian groups the asymptotic properties of matrix coefficients of representations are relatively delicate analytic properties. For example for abelian G, L2(G)

is

resolved into a direct integral of characters, each of which individually is only L-.

However, some things are known which suggest the situation

is rather different for semisimple groups.

For example, Harish-Chandrals

theory of the Plancherel formula for semisimple groups shows that for 2 semisimple G, the regular representation on L (G) is resolved into representations which are strongly L ~ * (c.f. theorem 7.1).

At the other

end of the spectrum, there is ~azhdan's result [Kn] that if G has split rank at least 2, then the identity representation of G is isolated in G. These facts suggest that for semisimple groups the asymptotic properties of matrix coefficients reflect something relatively robust about

.

the representations from which they come, and are related to the topology of the unitary dual G. The goal here is to study this phenomenon systematically, especially in the exemplary case of symplectic groups. Our first result in this direction is valid for general semisimple groups. Let G be a semisimple group over the local field F, and let

K be a maximal compact subgroup of G. We will assume K is "good" in the following sense. Let B 2 G be a minimal parabolic subgroup. Let A

ZB

be a maximal split torus, and let N f B be the unipotent

radical of B. to a collection

The action Ad A of A on N 2+

of positive roots of A.

character of A, a homomorphism from A

by conjugation gives rise Each root a is a rational

to FX, the multiplicative group

functions

f

on

satisfying

G

(bg) = 6;j2

f

Then

G

OD

a c t s on

ffJ,

Y (b)f (g)

b EB, g EG.

by r i g h t t r a n s l a t i o n s .

d e f i n e s a G-invariant inner product on

.

The inner product

The completion

*

of

ff

i n t h e a s s o c i a t e d H i l b e r t space norm i s t h e space of t h e u n i t a r y G indB $

representation

Jr

on

tions

,

t h e (normalized) induced r e p r e s e n t a t i o n from

For t h e moment we w i l l a b b r e v i a t e i t t o

ind

$ a r e c o l l e c t i v e l y termed t h e u n i t a r y p r i n c i p a l s e r i e s .

The

ind

\Ir

principal series. t r i v i a l on

*

f 0 = fO.

B

n

4'

such t h a t

(b) ~ ( b )

I f we then compute t h e matrix c o e f f i c i e n t

$

is

ipto

b € B , k € K of

ind $

with

f O , we f i n d

= j

K

Harish-Chandra's

where

ind $

They w i l l c o n t a i n a unique K-invariant f u n c t i o n

fO(bk) = 6:12

respect t o

The representa-

which contain a K-fixed v e c t o r a r e c a l l e d t h e s p h e r i c a l These w i l l c o n s i s t of t h e

K.

ind

It w i l l b e given by t h e formula

(7.2)

(7.4)

JI.

B.

function

X

fO(kb)dk

JI

i s given by

%: =

1 900

1 h e r e denotes t h e t r i v i a l r e p r e s e n t a t i o n of

B.

It i s then c l e a r

from (7.3) that (7.5) Harish-Chandra [HCl][Sllhas proven some basic facts about the asymptotic behavior of €iB

8. We will recall them.

The modular function

is related to the positive roots of the torus A 5 B by

where m(a)

is a positive integer, the "multiplicity" of a

By virtue of the Cartan decomposition (7.1), is determined by its restriction to .A: (7.7)

~,6;~'~(a)

5

e (a)

for some positive constants cl and

.

the function

B

Harish-Chandra has shown that -1/2+€

5 c2k) €iB

c~(E), for any

(a)

E > 0.

a

c Al+ If we

write Haar measure in terms of the Cartan decomposition, then we have [Hnl [wrl

r;h;re

~(a) is a positive function on :A

for some constant d2, and constant dl(t)

satisfying

which is positive for t > 1.

It follows from formulas (7.7) to (7.9) that the representations ind $ are strongly L2".

It will follow from our first result that all

representations in the support in strongly L*".

of the regular representation are

Estimates like that of Theorem 7.1 are found frequently

in the work of Harish-Chandra DCq and work based on his [Ar], [TV], [V 1. However the simple dependence of estimate (7.10) on the auxiliary parameters, e.g., the different tempered irreducible representations and the K-types, is essential to us and is not readily dug out of that literature. Also the methods of theorem 7.1 are different from those of Harish-Chandra. Theorem 7.1:

Let G be semisimple and K L G the maximal

compact subgroup specified above. Let

2

Then p is (E, (dim k) )-bounded.

representation of G. p

p be an absolutely continuous

In particular,

.

is strongly L2+€ Proof:

Since (+,Y)-boundedness is inherited by subrepresentations

and preserved under taking direct sums, it will suffice to prove the 2 theorem when p N L (G). From formula (6.9) we see that this amounts to 2 showing that if u and v are in L (G), and u transforms under left translations by Kbya multiple of an irreducible representation

of

*

K, and v

transforms by another

IU * v* I 5 where

v €

IIuIJ21

K, then

1 ~ 1 1dim ~ p2 dim v2

E

,

2 llull indicates the L -norm of u, and similarly for v. We will establish inequality (7.10) in three steps. We will first

prove it for K-biinvariant functions. Then we will establish a weaker analogue of (7.10) for functions which also transform under right translations by K according to a multiple of an irreducible representation. Finally we will reduce estimate (7.10) to this weaker version. The case of (7.10) when u and v are K-bi-invariant follows directly from the Plancherel formula of Harish-Chandra BCI]DC4 (for Lie groups) and MacDonald mca (for p-adic groups) for K-bi-invariant functions.

A simplified proof of Barish-Chandra's theorem is given in [Rg]. These G theorems say that the representation indK 1 of G decomposes into a direct integral over the unitary spherical principal series. The K-biinvariant functions in L2(G) form exactly the space of K-fixed vectors G * in indK 1. so a function u * v , with u and v K-bi-invariant in 2 G L (G), is just a matrix coefficient of indK 1 with respect to 2 K-fixed vectors.

where dp(9)

Therefore the Plancherel Theorem says

is Plancherel measure and p(u)

"spherical transforms" of u and v.

and p(v)

are the

One has

2 where llull is the L~ norm of u C L (G).

Equation (7.11) and (7.12)

combine with estimate (7.4) to yield estimate (7.10) when u and v are K-bi-invariant

.

Kext, suppose u transforms to the left under K according some multiple of an irreducible representation

u,

and transforms to the

right under K by a multiple of some other representation

u'.

Similarly

suppose v transforms to the right and to the left under K by multiples *

of v

and

v ' CK.

Consider the restriction of u KgK. Via the mapping a function u'

on K

a function on K

x

(kl,k2) x

K.

+

to a given (K,K) double coset

klg k2, ki € K, we may pull u back to

By our assumptions about u, we know that as

K, u' will belong to the minimal ideal associated to

the representation

pQ

realized on a space

J

p'

.

of K

x

K.

Suppose

y C3 p' = p"

Then there is an operator T on

J

is such that

tr

where and

11 lj2

K x K.

t o be

denotes t h e usual t r a c e f u n c t i o n on denote a s usual t h e supremum and

End(J).

norms f o r functions on

L~

(Here i t i s understood t h a t t h e measure of 1.)

Let

11 11 2,J

i s normalized

K x K

denote t h e Hilbert-Schmidt norm on

The formula (7.13) w i l l be recognized a s defining Hilbert-Schmidt inner product of

$'(x)

11 11-

Let

with

by taking t h e

u'(x)

* T , the

End(J).

a d j o i n t of

T.

By t h e Schwartz i n e q u a l i t y , we have

On t h e o t h e r hand, t h e Schur Orthogonality r e l a t i o n s t e l l us

(7.15)

IIT1I2,

~ ~ =u (dim ~ ~ v1')-1' 22

J

Combining (7.14) and (7.15) y i e l d s

Return t o consideration of t h e functions (7.17)

u(g) = max

and define Iu(g)

1

*

v

similarly.

5 u(g), and t h a t

t lu(klg

k2)

1

u

: kid

and

v

on

G.

Define

K)

It i s c l e a r from i t s d e f i n i t i o n t h a t

6(g)

i s K-bi-invariant.

From t h e i n t e g r a t i o n

formula (7.8) we f i n d

s

(dim p12(dim

v'12 Z

=

(dim

dim

/

+A (a) 1 u' (k1gk2) I 'da dk1dk2

KxKxAl

1u12 dg

G

Analogous estimates apply to v.

Therefore using estimate (7.10) for

K-bi-invariant functions and estimate (7.18) gives us

(Actually, here)

u

*

v*

0

=

p' # v ' ;

if

. 2 Finally consider u, v E L (G),

left under K by a multiple of of

but that is not important

v €

. K.

p E

such that u transforms to the

2,

and v

transforms by a multiple

By an obvious approximation argument, to prove inequality

(7.10) it is enough to prove it when u has compact support. Let 2 H1 C L (G) be the closed span of left translates of u, and let vl be the projection of v

* * u * vl = u * v .

into

ffl.

Then

Aence we may as well assume v C

denote the subspace of L~(G)

.

H1.

Let L~(G; p*)

consisting of functions which transform

to the right under K by a multiple of contragredient to

Ilvll12 5 llvl12, and

p*, the representation

Define

2

Since u has compact support, it is in L (G), so T is a bounded operiitor. By inspection of the formula for w

* u* , we

see that the

kernel of T is the space of functions orthogonal to all left translates of u. lemma

By definition of HI, we see T is injective. [ ~ a ]therefore

gives us an isometric embedding

The general Schur's

H1

S:

*

2 L (G; p )

+

which i n t e r t w i n e s t h e l e f t a c t i o n of

G

on t h e s e two spaces.

(This i s

e s s e n t i a l l y a n i n s t a n c e of Frobenius R e c i p r o c i t y . ) Because of t h e i n t e r p r e t a t i o n of

u

* v*

a s a matrix coefficient,

we w i l l have

But

S(u)

* p .

and

S(v)

transform t o t h e r i g h t according t o a m u l t i p l e of

Therefore t h e e s t i m a t e (7.19) i s a p p l i c a b l e .

Remembering t h a t

S

i s i s o m e t r i c we g e t

* V*I

lu But we may assume t h a t by

(dim

dim p 5 dim v

dim v12

i n general.

5 llul12 llv112 (dim p)

.

3

dim v

8

Then r e p l a c i n g

(dim p13 dim v

f o r purposes of synrmetry, we o b t a i n e s t i m a t e (7.10)

This concludes Theorem 7.1.

We can immediately p a r l a y theorem 7.1 i n t o a n e s t i m a t e f o r m a t r i x c o e f f i c i e n t s of s t r o n g l y C o r o l l a r y 7.2: and suppose ( Ellm,

(dim p) )

Proof: product

Let

p 5 2m 2

If

LP

represjentations, f o r any

p be a s t r o n g l y

f o r some i n t e g e r bounded

(@ p)m of

p

LP

is strongly

Let

ff

select vectors t e n s o r powers of

be t h e space of

x € H

and

x

y.

CL and

.

r e p r e s e n t a t i o n of p

G,

is

p

y E ffv Then

with

p 5 2m, then t h e m-fold t e n s o r

L2, hence a b s o l u t e l y continuous, by

remark d ) of $6 and p r o p o s i t i o n 6.1. (@ pIm.

Then

m.

-

.

is strongly

p

L'

p c

Therefore Theorem 7.1 a p p l i e s t o

.

For r e p r e s e n t a t i o n s

.

Let

x'

and

y'

p, v €

E,

denote t h e m-th

We may decompose the m-th tensor power of y into irreducible components:

.

pi E K

for appropriate projection of x ' Decompose

and multiplicities ai.

into the

Let x i denote the m pi -th isotypic component of (By)

.

(8v ) ~ similarly into a sum of

the component of y'

v

j

€

f,

and let y'

in the v -isotypic subspace of j

j

(O H~)~. Then

inequality (7.10) gives us

The Schwartz inequality gives x

(dim pi)

2 5

(2 IIxil]

2 112 )

(2 (dim pi

But

since the x i are orthogonal. Furthermore

C dim pi 5 dim(am p) Therefore

=

(dim p)m

be

4 112 1)

Similar estimates hold for y. the estimate defining

Combining (7.20), (7.21) and (7.22) yields 2 (dim 1) )-boundedness, so the corollary is

proved. Combining corollary 7.2 with Harish-Chandra's estimate (7.7) and estimate (7.8), and applying lemma 6.2 we obtain the following result. Corollary 7.3:

For any p, let

(ElP

denote the subset of

consisting of representations which are strongly LP. integer m

(6)2mtE

=

?

1, the closure of

n

(e)q

( ~ 1 ) ~in~ 6

Then for any

is contained in

.

q > 2m Remarks: a)

These corollaries illustrate a dramatic difference

between semisimple harmonic analysis and abelian, or more generally, amenable harmonic analysis. One can also show (c.f. Theorem 8.4 ) that there is a p
1/2/

= nla,(~)

5

Furthermore, i f

n

u

F"

1-1'2~

lai(T)I

-1/2

u(yl,.

.., Y ~ ) V ( ~ ~ ( T )..Y,an(T)yn)dyl.. ~.. ..

u ( ~ ( T ) ' ~ Y ~..$(TI . Ilu 11,

-1 yn)v(Yl,-

.dyn

,Yn)dYl-

llvlll

i s c o n s t a n t i n a neighborhood of

0

in

and

Y,

v

i s supported i n t h i s neighborhood, which we w i l l assume i n v a r i a n t bqt (A;)-',

and

0

= u

, and

v 2 0, t h e n i n e q u a l i t y (8.5)

And i n any c a s e we

a c t u a l l y an e q u a l i t y .

is

have a n asymptotic formula

Comparing t h e s e f a c t s w i t h t h e i n t e g r a t i o n formula (7.8) and t h e formula (8.3) with

we can come t o t h e following conclusion (c.f

tjB,

. [HM],

proposition6.4). Proposition 8.1: strongly

L~~~

Remark:

, but

The o s c i l l a t o r r e p r e s e n t a t i o n s

a r e not strongly

For a p p r o p r i a t e

4 L

are

at

.

t , t h e r e w i l l be a v e c t o r

uo

in

L~(Y)

- dYn

, . ,

which i s an eigenvector f o r vector

u

When

F

i s non-archimedean, t h i s

can be arranged t o be t h e c h a r a c t e r i s t i c f u n c t i o n of t h e

l a t t i c e spanned by t h e

When

wt(K).

F = R,

fi's.

the vector

For t h i s v e c t o r , one has

u

can be made t o be t h e Gaussian f u n c t i o n

Then one can compute t h a t

F'roposition 8.1 shows t h a t t h e

j u s t m i s s having t h e decay

cot

Nevertheless, we can

necessary f o r a s h a r p a p p l i c a t i o n of c o r o l l a r y 7.2. g e t t h e e s t i m a t e c o r o l l a r y 7.2

f a i l s t o y i e l d h e r e by another means.

*

f a c t , it i s n o t hard t o do by analyzing

ot 63 ot

.

In

However, t h e following

argument w i l l g i v e u s a good, though n o t t h e b e s t p o s s i b l e , r e s u l t , and i s what we need f o r l a t e r developments. Theorem 8.2: r e p r e s e n t a t i o n s of

If

p

SpZm ( i . e . ,

q u a d r a t i c form of degree Proof:

a Weil r e p r e s e n t a t i o n a s s o c i a t e d t o a

m), then

p

is

( E l l 2 , 2 dim p)-bounded.

By theorems 6.5 and 7.1, it w i l l s u f f i c e t o show t h a t t h e

K-f ixed v e c t o r s i n representation.

i s an m-fold t e n s o r product of o s c i l l a t o r

p @

d

g e n e r a t e a n a b s o l u t e l y continuous

We w i l l only g i v e t h e proof f o r p-adic groups ( i . e . ,

non-Archimedean base f i e l d s ) .

The proof f o r

i n s p i r i t b u t more t e c h n i c a l l y involved.

F =R

or

U

is s i m i l a r

for

We know from, e.g., r e p r e s e n t a t i o n of

[Hl] I1 $ 3 , t h a t

s p e c i f i e d above f o r in

(

non-archimedean,

F

a n orthogonal b a s i s f o r

R

module w i l l c l e a r l y be i n v a r i a n t by

f

@

F

~ and ~ l, e t

b a s i s of @

eits

Lo

and

Lo, where

A

R module i n

a r e dense

2 L (V).

Let

x

I f we t h i n k of

of

Sp.

V

has t h e form

F ~ . The c h a r a c t e r i s t i c

* Q z s y, f o r

z C

H , embeds H

w i l l have shown t h a t t h e

into

a c t s on

V

LL(sp).

2

L (V)

H

we

2 K L (X) is a b s o l u t e l y l e t u s observe t h a t

Fm, and commutes w i t h

Sp.

From

GLm(F) permutes them

Hence i t w i l l s u f f i c e t o d e a l w i t h a s i n g l e l a t t i c e .

N a t u r a l l y we w i l l choose t h e s t a n d a r d l a t t i c e work with.

and l e t

x under t h e a c t i o n

Moreover

t h e form of t h e K-invariant l a t t i c e s , we s e e t h a t transitively.

A @ Lo.

Then c l e a r l y

Sp module generated by

v i a i t s a c t i o n on

L*(v)~,

2 y C L ( x ) ~ such t h a t t h e map

continuous, and t h e theorem would be proved. mm(F)

A0 @ Lo,

generated by

Suppose we can f i n d a f u n c t i o n

This

FZm generated by t h e s t a n d a r d

denote t h e c h a r a c t e r i s t i c f u n c t i o n of

denote t h e closed subspace of

K

as

V

F i x a K-invariant l a t t i c e

2 L (V)

The

R module i t spans.

functions of t h e s e l a t t i c e s thus a l s o form a spanning s e t f o r although not an orthogonal one.

as

o r b i t s form

K

f i t s , then a K-invariant module i n

i s any

Sph

R denote t h e r i n g of i n t e g e r s

K.

b e t h e R-module i n

K

orbits

K

2 L ( v ) ~ ,t h e K-fixed v e c t o r s i n

Then each K-orbit i s determfned by t h e

F.

with

~ = ~V. ) With ~

t h e open

o r b i t s a r e described i n [HI] I, $11. L e t

A

F

Hence t h e c h a r a c t e r i s t i c f u n c t i o n s of t h e open

V.

of

2 2m m L ((F ) ),

SpZm, and can b e r e a l i z e d on

a c t i n g v i a i t s diagonal l i n e a r a c t i o n on

factors t o a

p @ p*

Let

x

i s t h e m-fold

i s t h e m-fo%+-tensor

be t h e c h a r a c t e r i s t i c f u n c t i o n of t e n s o r product of

2

L (F

2m

)

L:

m

c= R

m Lo.

@ Lo

to

Observe t h a t

with i t s e l f , and t h a t

product of t h e c h a r a c t e r i s t i c f u n c t i o n of

Lo

x

with

itself. m

We w i l l a l s o c o n s t r u c t our f u n c t i o n

different functions i n LO c F"'.

f u n c t i o n of q

-1

= In

1

2 2m K L (F )

Let

.

Let

n

Lo.

R.

Normalize Haar measure on

11 11

i s t h e norm i n

Let

A

and l e t

b(R/nR) = q

Let P~~

4

2 2m L (F )

ai

denote t h e a c t i o n of

function

4(T)(u)

spanned by

v

is the

be t h e c h a r a c t e r i s t i c so that

Lo

has

{ai(T)ei,

The r e s u l t i s

I n t h e same way we f i n d

The q u a n t i t y

(

,)

i s t h e i n n e r product. Sp, defined a t t h e beginning

be t h e r a t i o n a l c h a r a c t e r s of formula (8.1). on

Sp

2 2m L (F ). Then f o r

T € A, t h e

i s t h e c h a r a c t e r i s t i c f u n c t i o n of t h e l a t t i c e -1 ai(T) f

t h e volume of t h e i n t e r s e c t i o n compute.

and

b e our standard Cartan subgroup of

of t h i s s e c t i o n , and l e t Let

Thus

R,

1. Then

measure

where

denote t h e c h a r a c t e r i s t i c

u

.

c a r d i n a l i t y of t h e r e s i d u e c l a s s f i e l d of TI

a s a t e n s o r product of

n be a prime element of

be t h e a b s o l u t e value of

f u n c t i o n of

y

I.

The i n n e r product

Lo fl TLO.

(us 4 ( T ) u )

This volume i s not hard t o

TLO,

is

I

-1 is equal to 1 or to q , according to whether lai(T) 1 or -1 takes on only the lai(T)/ = 1. Therefcre the quotient '%,u Vu,v 2m values '*q for 0 5 j m; and it takes on the value q only on

K. Therefore, if we set z = u j

- qmtjv,

one of the functions

vanishes at any point of S P ~ ~ - K .Hence the product of the vanishes everywhere but on K.

then the product of the of

* p@ p .

cp (j)

But if

~(j) is just the matrix coefficient

Qx,~

is just a multiple of the characteristic

Since

function of K, we see that y has the desired properties. This proves theorem 8.2.

for

Recall that W c W is the subspace spanned by the ei and fi Ci5 C We want to study the relation between (Q,Y)-boundedness

.

on Sp(W)

and on Sp(WC).

intersection K

n

We will take as compact subgroup of WC the

Sp(WC) = K(WC)

compact subgroup of Sp(W)

where K is the standard maximal

specified above. Let

spherical function for Sp(WC)

E(W )

e

be ~arish-Chandra's

with respect to K(WC).

Proposition 8.3:

Let p be a representation of gp(~), and 1 PliP(~C) is (E(w~)~, Y(W C))-bounded for fi = 2 suppose that the reciprocal of some integer s, and some function y(W8) on K ( w & ) ~ . Then for some

E > 0,

the representation p is itself

bounded, for some function

ye

on K, where

(e 9 ' , ~ E

1-

[XI

where

p ~ g p ( ~ k i) s s t r o n g l y Proof: in

TP,

etc.

, then

is strongly

p

m = bE

+

r

In particular i f where

L~~

f o r non-negative i n t e g e r s

We w i l l decompose

Vi

of dimension

Vi

and

U

L

2s

qT = 1.

-'s,

For convenience i n t h e proof we suppress a l l Write

r -= E.

with

x.

denotes t h e l a r g e s t i n t e g e r l e s s than

28,

i n t o a d i r e c t sum of

W

and another space of dimension

w i l l be spanned by c e r t a i n of t h e p a i r s

b

b

2r.

as

r,

and

subspaces Each of t h e

e.,f J j

belonging t o

t h e standard b a s i s , b u t we w i l l n o t s p e c i f y u n t i l l a t e r which p a i r s ej,fj

W

belong t o which spaces.

=(

@ Vi) i

dU

into

Sp(W).

then

B(Vi)

I n any c a s e t h e decomposition

induces an embedding of

I f we s e t

B(Vi) = B fl Sp(Vi),

is a minimal p a r a b o l i c subgroup of

i s a s p l i t Cartan subgroup of

and

K(U)

in

Sp(W)

K(Vi) = K

to

SP(WC).

Sp(U)

ejls

Sp(vi)

and

A(Vi)

We have

U.

and do l i k e w i s e f o r

s a t i s f y c o n d i t i o n s (7.1).

permutation of t h e i s taken t o

SP(Vi)

A(Vi) = A fl Sp(Vi),

B ( v ~ ) . Also

S i m i l a r n o t a t i o n s and remarks apply t o

Set

and

U.

Note t h a t each

Then t h e

Sp(Vi)

K(Vi)

i s conjugate

I n f a c t , t h e conjugation may be accomplished by a and t h e

f .'s, 3

B(W8), and s i m i l a r l y f o r

and i n such a way t h a t

A(Vi)

and

K(Vi).

i s conjugate i n s i m i l a r f a s h i o n t o t h e subgroup

by a conjugation w i t h analogous p r o p e r t i e s .

B(Vi)

Also t h e group

Sp(Wr)

of

Sp(W8)

Let

E(Vi)

with respect t o

d e n o t e Harish-Chandra's

K(Vi).

Define

E(U)

spherical function f o r

similarly.

By t h e conjugacy

Sp(Vi), we s e e t h a t t h e hypotheses of t h e p r o p o s i t i o n

p r o p e r t i e s of t h e imply t h a t

p l ~ p ( ~ i )i s

f u n c t i o n on

K ( V ~ ) " o b t a i n e d from t h e f u n c t i o n

conjugation.

Sp(Vi)

( ~ ( 7 1 Y(Vi))-bounded, ~ ) ~ ~

It w i l l a l s o hold t h a t

bounded f o r some f u n c t i o n

where

y(W8)

p l ~ p ( l J ) is

Y(Vi)

on

is the

K ( W ~ ) " by

( x ( u ) ~ , Y(U))-

It w i l l be convenient t o d e l a y s l i g h t l y

Y(U).

t h e d e r i v a t i o n of t h i s e s t i m a t e . I t i s known [wr],

[gn], [ L ~ ] t h a t t h e subgroup

i s uniformly l a r g e i n t h e s e n s e of $6. us that the r e s t r i c t i o n

and

Y

((

n

Sp(Vi))

i

T h e r e f o r e p r o p o s i t i o n 6.3 t e l l s x Sp (U)

is

(@,Y)-bounded, where

on

@'

Sp(W)

by t h e r e c i p e

.

T h i s d e f i n i t i o n makes s e n s e by v i r t u e of i n c l u s i o n (8.9) is

Sp(W)

i s whatever i t t u r n s o u t t o be.

Define a f u n c t i o n

p

of

K

1 (*I, p )-bounded f o r a n a p p r o p r i a t e f u n c t i o n

Y1

I claim t h a t

k.

on

6

Indeed, s e l e c t

~l

of t h e s p a c e of

K, and l e t

c p.

x

belong t o t h e

The r e s t r i c t i o n of

p - i s o t y p i c component

( ll K(Vi)) x K(U) i decompose i n t o a sum of f i n i t e l y many i r r e d u c i b l e r e p r e s e n t a t i o n s

of t h e s m a l l e r group.

Let

s e l e c t a n o t h e r K-type

v

component. (

Let

v

,

K(VI)) x K(U) , and l e t

i

+. A1

we have

be t h e

and a v e c t o r

to

F~-component of y

i n the

decompose i n t o r e p r e s e n t a t i o n s

n

T €

xi

p

y

j

be t h e

x.

will pi

Similarly,

v-isotypic vj

vj-component

on r e s t r i c t i o n t o of

y.

Then f o r

The l a s t s t e p follows because t h e

x,

and s i m i l a r l y f o r t h e

Since

(klTk2)

Now observe t h a t f o r

ki € K,

'

s t i l l i s i n the

p(kl)x

t h e same norm a s

%,Y

y j

a r e mutually orthogonal and sum t o

xi

p-isotypic component of

x , and s i m i l a t l y f o r

p

,

and has

y, we g e t estimate (8.12) f o r

a s well a s f o r

necessary t o e s t a b l i s h

(TI. But t h i s i s p r e c i s e l y t h e estimate %,Y 1 1 (9 , Y ) -boundedness, with

To f i n i s h proving t h e proposition, i t remains t o s p e c i f y how t h e

ej

and

f

j t h e function

a r e d i s t r i b u t e d among t h e 9l

t o t h e function

9

Vi

.

and

U,

and then t o r e l a t e

The idea i s t o perform t h e 9l

d i s t r i b u t i o n t o maximize t h e compatibility between

and

S

.

Our

recipe is (8.13)

Vi = s p a n { e j , f j :

j = bk+ i

j = m

U = span ( e j y f j : j = m -

-

for

Osk

(b+l)k+ i (b+l)k

for

for

5 8-r,

and

8 5 k 5 r)

0 5 k c r)

.

R e c a l l t h e i n e q u a l i t i e s (7.7) r e l a t i n g t h e f u n c t i o n modular f u n c t i o n describing

6B f o r

the

and t o

Sp(Vi)

there is a constant

(Here r e c a l l

B.

of

Sp(W). Sp(U).

t o the

%

R e c a l l a l s o formula (8.3) e x p l i c i t l y These formulas a p p l y m u t a t i s mutandis t o Combining them we s e e t h a t f o r any

c(c)

such t h a t on

A:

E 7

0,

,

s $ = 1.)

A t t h i s p o i n t we can demonstrate t h e asymptotic e s t i m a t e we claimed for

p l ~ p ( ~ )W . e s e e from t h e same formulas used f o r i n e q u a l i t y (8.14)

that i f a representation

p

of

Sp(Wt)

i s @ ( w ~ ) ' , Y')-bounded,

+ A fl

K(Wt)-finite m a t r i x c o e f f i c i e n t s , r e s t r i c t e d t o

then the

Sp(Wr), decay f a s t e r

1

than

f o r any

E

> 0.

Thus

p l ~ P ( ~ r i) s s t r o n g l y

L*'

where

Hence c o r o l l a r y 7.2 provides t h e d e s i r e d e s t i m a t e f o r We need t o compare t h e product of t h e

lai]

plsp(wr). i n s q u a r e b r a c k e t s on

t h e r i g h t hand s i d e of (8.14) w i t h t h e product d e f i n i n g t h e exponent w i t h which write

j+l = (b+l)k

+ i,

lam-j with

I

contributes t o i 5 b,

6lI2 B

6B.

is

We s e e t h a t j+l.

I f we

t h e n t h e exponent w i t h which

c o n t r i b u t e s t o t h e r i g h t hand s i d e of (8.14) i s a t l e a s t

M-1.

lam-j

Thus t h e

1

exponent of of

lail

lail

in

6B i s never more than

i n t h e product of (8.14).

i n t h e two f u n c t i o n s with exponents (b+l)8

i s s t r i c t l y l a r g e r than

+ A1

on

/ail

, we

some c o n s t a n t

b+l

times t h e exponent

Moreover, t h e f a c t o r and

rn

8

m, and s i n c e

respectively. lal/

appezrs

Since

dominates a l l t h e

s e e t h a t f o r a l l s u f f i c i e n t l y small

d

lall

E : , 0,

there is

such t h a t

From e s t i m a t e (8.15) t h e statement of t h e p r o p o s i t i o n is immediate. Remark: C

divides

m

Specifically i f then

I n s p e c t i o n of t h e proof of p r o p o s i t i o n 8.3 shows t h a t i f e x a c t l y , then e s t i m a t e (8.8) i s g r e a t e r than

$

can be changed t o

Y

replace

y

+E

by

y'

;i n s t e a d

y' = (8/ms).

-c ,

for

1

E

can be improved.

of e x a c t l y e q u a l t o i t ,

Or if

f o r any

Y

$ =

then we can

> 0.

We a r e now i n a p o s i t i o n t o r e l a t e rank t o asymptotic decay of matrix coefficients. parabolic

Recall t h a t

Pm(W) which preserves

Nm(W) Xm,

i s t h e u n i p o t e n t r a d i c a l of t h e

t h e maximal i s o t r o p i c subspace

spanned by t h e

ei.

We r e c a l l t h e n o t i o n of

s t u d i e d i n $2.

We w i l l prove two main r e s u l t s .

t o asymptotic decay of matrix c o e f f i c i e n t s .

N -rank of r e p r e s e n t a t i o n s m

One r e s u l t r e l a t e s rank

It s a y s t h a t t h e l a r g e r t h e

rank of a r e p r e s e n t a t i o n , t h e f a s t e r its m a t r i x c o e f f i c i e n t s tend t o decay. The second r e s u l t , based on t h e f i r s t , r e l a t e s rank t o t h e topology i n

ipsUnfortunately, desired

-

vengeance.

our c o n t r o l on asymptotics s t i l l l e a v e s much t o be

t h e remark b) following c o r o l l a r y 7.3 a p p l i e s h e r e w i t h a Consequently, t h e s e f i n a l r e s u l t s a r e f a r from b e s t p o s s i b l e .

However, they do i l l u s t r a t e t h e phenomenon a t question.

Theorem 8.4:

p

a)

m > 1, t h e n

If

[XI b)

c)

p

If

.

L4&'

m z 2, then a l l

If

Sp(W8)

Lq

where

m > 3, and t h e i n t e g e r

1 representations

Nm-rank

+-rank

2

P

is

8

satisfies

% .

Sp(W8)-regular;

8 5

and when

m > 2

% , then i f

that is, the restriction

P a r t s a ) and b) of t h i s theorem t o g e t h e r imply t h a t when

.

1, a l l n o n - t r i v i a l i r r e d u c i b l e r e p r e s e n t a t i o n s of

)

representations

i s a b s o l u t e l y continuous.

Remark:

L2mC~

x.

2m+€

r z 2 8, t h e r e p r e s e n t a t i o n

,.

is strongly

m > 1, t h e n a l l

Additionally, i f

a r e strongly

m >

of pure

a g a i n denotes t h e g r e a t e s t i n t e g e r n o t l a r g e r than

a r e strongly

PI

.Sp(W)

be a r e p r e s e n t a t i o n of

r i m.

Nm-rank

where

Let

m > 2

Ep2,

t h e only r e p r e s e n t a t i o n s which a r e n o t

L~~

a r e t h e components of t h e o s c i l l a t o r r e p r e s e n t a t i o n s . every r e p r e s e l l t a t i o n of

SpZm i s s t r o n g l y

t h e only r e p r e s e n t a t i o n s which a r e not s t r o n g l y

L~~

are

L

4m+€

,

(indeed Also f o r

and i f

m > 3,

a r e t h e rank 2

r e p r e s e n t a t i o n s described i n $ 5 . Proof:

m > 1, t h e n theorem 4.2 i m p l i e s t h e only rank 1

If

r e p r e s e n t a t i o n s a r e t h e components of o s c i l l a t o r r e p r e s e n t a t i o n s , and these a r e strongly

L4mte

we may argue s i m i l a r l y . r e a s o n a s follows. 2.13 implies representations.

If

For rank 2 r e p r e s e n t a t i o n s ,

O r , independently of c l a s s i f i c a t i o n , we may

m > 3,

plsp(w2) Thus

by p r o p o s i t i o n 8.1.

and

p

i s of pure rank 2, then c o r o l l a r y

i s a sum of two-fold products of o s c i l l a t o r

I

p sP(w1)

is

(E (w2) 'I2, 2 dim p)-bounded.

Then a s l i g h t a d a p t a t i o n of t h e argument of p r o p o s i t i o n 8.3 shows

p

is

With t h e s e remarks, we consider p a r t b) of t h e theorem

strongly proven.

Next consider p a r t c ) of t h e theorem. to

Sp(W ) E

of a r e p r e s e n t a t i o n

of

p

Consider t h e r e s t r i c t i o n

Sp(W)

of pure rank

a 8 .c where

r e p r e s e n t a t i o n s , involving r e p r e s e n t a t i o n of

E

implies

either

7

is a t e n s o r product of o s c i l l a t o r

7

min(r, m- 8 )

Sp(WE) of rank

-8

m

p . 2 4.

is a

Our r e s t r i c t i o n on

r > 2.8 by assumption, we s e e t h a t

Since

2.8

o s c i l l a t o r representations, or

o s c i l l a t o r r e p r e s e n t a t i o n s , and rank a > 1.

c a s e , our r e s t r i c t i o n on and 6.1 imply

a

f a c t o r s , and

max(0, r+ 8-m).

i s a product of more than

of e x a c t l y 28

E > 2.

t e l l s us

m

According

i s a f i n i t e sum of

t o c o r o l l a r y 2.13, t h e r e s t r i c t i o n r e p r e s e n t a t i o n s of

r.

I n the l a t t e r

Hence p r o p o s i t i o n s 8.1

i s a b s o l u t e l y continuous, and p r o p o s i t i o n 6 . 1 t h e n

z

a p p l i e s a g a i n and s a y s

o 8

7

i s a b s o l u t e l y continuous.

Hence p a r t c )

of t h e theorem is t r u e . Finally consider part a ) .

i s a t most

m

2

restriction

.

Then

m

pl sp(wr)

representations.

-

Consider t h e c a s e when

r = rank

r 2 r , s o t h a t a g a i n by c o r o l l a r y 2.13,

p

the

i s a sum of r - f o l d t e n s o r products of o s c i l l a t o r

Thus theorem 8.2 i m p l i e s

p i sp(Wr)

is

( E ( w ~ ) " ~ , 2 dim d-bounded, and t h e e s t i m a t e (8.16) follows from p r o p o s i t i o n 8.3. This argument extends a l s o t o t h e c a s e c a s e c o r o l l a r y 2.13 s a y s t h a t

a8

T

where

z

r e p r e s e n t a t i o n s and

Is an

pl Sp(Wr)

(r-1)-fold

o has rank

o s c i l l a t o r representations. again give t h e r e s u l t .

2r = el. For i n t h i s

is a sum of r e p r e s e n t a t i o n s

t e n s o r product of o s c i l l a t o r

1. Hence

a i s a sum of components of

Therefore theorem 8.2 and p r o p o s i t i o n 8 . 3

The case when 2r 2 ui+2

Thus 2m' = m

if m

is easier. Set

is even, and

2m' = m-tl

application of corollary 2.13 says that representations of the form

(3

cg

is odd. Another

plSp(Wm,)

where

7

if m

7

product of oscillator representations, and

is an

is a sum of (m-m')-fold

tensor 2

u has rank r-(m-m')

2.

Again applying propositions 6.1 and 8.1, and part b) of this theorem, we pl~p(~~,) is strongly L4. Then corollary 7.2 and pro-

conclude that

position 8.1 give the desired conclusion. This proves part a) of the theorem. We close with a result that shows that repreeentations of small rank cannot be obtained as limits of complementary series. Compare ~uflo's [Df] computation of the unitary dual of SP~.~(&). "

symmetric bilinear form on Xm. Let associated to

p

in $2.

-

Let

p

A

(SP)~ be the subset of

A

be a

(3~)~

(SP)~

is both open and The subset (SP)~ of 2m closed if rank p 5 - 3 for Proof: It is completely clear that the union of the (ip); Theorem 8.5:

*-

rank

p

less than some given bound is closed in

simply by looking at Nm-spectra.

" A (Sp)

.

It is also clear that

(Sp);,

relatively open in the union of the

with rank

This follows (Sp);

8' g

rank

is

p.

Hence

to prove the theorem, it will suffice to show that no element p

6 (ip);

is a limit of representations of larger rank. Alternatively,

it will suffice to show that if irreducible, of

.Sp, ,

and

CJ

IJ

is a representation, not necessarily

has pure Nm-rank r > rank p, then

not contain weakly any representation

p

of Nm-rank equal to rank

does

IJ

p

.

*

By c o n s i d e r a t i o n of t h e a c t i o n of t h e c e n t e r of under

p

then t h e rank of

, we

s e e (by c o r o l l a r y 2.14) t h a t i f

Nm-ranks of IJ

and of

product

Define a n i n t e g e r

28 5

Nm-rank of

2m , so 3

o

8

p @w

.

28,

F = C.

o

.

, the

o

be an

tensor

--

Since t h e olsp(w8)

Therefore c o r o l l a r y 7.2 i m p l i e s t h a t any

s a y s t h i s r e p r e s e n t a t i o n i s only

t h e weak c l o s u r e of

p

t h a t r e s u l t implies t h a t

o must be

But c o r o l l a r y 2.13 s a y s

i s not

2 @(W8), (dim p) ) -

P ~ S ~ ( W i s ~a ) (28-1)Then p r o p o s i t i o n 4.1

L ~ +where ~

2 (X(W8), (dim p) )-bounded,

and

p

cannot be i n

This concludes theorem 8.5, except i n t h e c a s e

A s l i g h t refinement of t h e argument covers t h a t c a s e too.

We

omit d e t a i l s . Remark:

This r e s u l t i m p l i e s f o r example t h a t holomorphic

r e p r e s e n t a t i o n s of

Sp2,(R)

,

Hence, by t h i s device,

f o l d tensor product of o s c i l l a t o r r e p r e s e n t a t i o n s .

plsp(w8)

Also, l e t

i s odd, and t h a t i t i s a t most

r e p r e s e n t a t i o n i n t h e weak c l o s u r e of

Hence

p

by

i s g r e a t e r than

Sp(W8).

.

theorem 8.4, p a r t c ) i s a p p l i c a b l e .

i s a b s o l u t e l y continuous.

bounded on

p

o weakly c o n t a i n s

w i l l weakly c o n t a i n p

and

have t h e same p a r i t y , s o t h a t t h e

Then i f

we may assume t h a t t h e rank of

o

under

o weakly c o n t a i n s

i s a t l e a s t 2 more than t h e rank of

o s c i l l a t o r representation.

Then

p

Sp

of s u f f i c i e n t l y small rank a r e i s o l a t e d i n

1.

I n conclusion, I would l i k e t o thank P r o f e s s o r Michael Atiyah and t h e Mathematical I n s t i t u t e a t Oxford U n i v e r s i t y f o r a v e r y p l e a s a n t s t a y i n May-June 1978, during which time some of t h e i d e a s developed h e r e germinated.

Also, thanks a r e due t o Mrs. Me1 DelVecchio f o r a n

e x c e l l e n t and e x p e d i t i o u s job of typing.

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*

[Dx] J. Dixmier, Les C -algebres et leurs representations, GauthierVillars, 1964, Paris. [Df] M. Duflo, ~e~rgsentations unitaires irrgductibles des groupes simples complexes de rang deux, preprint. [DS] N. Dunford and J. Schwartz, Linear operators, Interscience 1958-1971, New York. [Fr] T. Farmer, On the reduction of certain degenerate principal series representations of Sp(n,C), Pac. J. Math. 84, No.2(1979), 291-303. [Fl] J. Fell, The dual spaces of c*-algebras. T.A.M.S., 364-403.

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-

[Zl]

G. Zuckerman

o r a l communication.

[22]

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CEN TRO INTERNAZI ONALE MATEMATICO ESTIVO

( c , I . M . E .1

SOME A P P L I C A T I O N S OF GELFAND P A I R S I N CLASSICAL

AKALYSIS

ADAM KORAKYI

SOME APPLICATIONS OF GELFAND PAIRS I N CLASSICAL ANALYSIS Adam Koranyi (Washington U n i v e r s i t y )

Introduction

Let

G

be a unimodular Lie group and

known,

(G,K)

right

K - i n v a r i a n t f u n c t i o n s on

K a compact subgroup.

As well

i s c a l l e d a Gelfand p a i r i f t h e c o n v o l u t i o n a l g e b r a o f l e f t - a n d G

i s commutative; t h i s i s e q u i v a l e n t t o

saying t h a t e v e r y i r r e d u c i b l e r e p r e s e n t a t i o n of 2 G on L (G/K)

K

occurs a t most once i n t h e

.

regular representation of

The most important c l a s s i c a l example i s t h e c a s e where n i a n symmetric s p a c e ; t h e f a c t t h a t

(G,K)

G/K

i n t h e r e p r e s e n t a t i o n theory of non-compact semisimple Lie groups. t h e c a s e o f a compact group

U

, the

i s a Rieman-

i s a Gelfand p a i r i s then c r u c i a l

observation t h a t

(U X U

Even i n

, diag

U X U)

i s a Gelfand p a i r l e a d s t o t h e most i l l u m i n a t i n g way t o prove t h e Peter-Weyl theorem. The o t h e r main c l a s s i c a l example, a s well-known, where

G = K X

A

i s the case

(G, K)

i s a s e m i d i r e c t product w i t h a n Abelian normal subgroup A ;

t h i s g i v e s t h e b e s t framework f o r t h e harmonic a n a l y s i s o f r a d i a l f u n c t i o n s on

R"

.

I n t h e s e l e c t u r e s I w i l l d e s c r i b e two f u r t h e r examples t h a t have a r i s e n n a t u r a l l y i n t h e c o n t e x t of some q u e s t i o n s of c l a s s i c a l a n a l y s i s . these

G

I n one of

i s a s e m i d i r e c t product o f a compact group and a s p e c i a l type of

n i l p o t e n t group, and t h e purpose i t i s used f o r i s t h e s t u d y of c e r t a i n analogues of r a d i a l f u n c t i o n s on t h e n i l p o t e n t group.

I n t h e o t h e r example we

w i l l consider some o f t h e most w e l l - s t u d i e d compact groups, b u t i n a c o n t e x t

which involves a s l i g h t extension of the n o t i o n of a Gelfand p a i r . Both examples t o be discussed o r i g i n a t e from the same source:

the theory

of s i n g u l a r i n t e g r a l s on c e r t a i n homogeneous v e c t o r bundles developed i n [ l o ] . Let us remark i n passing t h a t i n [ l o ] t h i s theory is not put i n t h e language of v e c t o r bundles, b u t i t goes through w i t h o u t any change i n t h i s s e t t i n g . fact, l e t

Bi

x E X

the f i b r e over denoted t i o n s of

where

Bq

denoted

E(i)

and t h e a c t i o n of

The l i n e a r o p e r a t o r s

A

g

G

on

mapping s e c t i o n s of

B1

E

In

, with Bi t o sec-

can be w r i t t e n , a t l e a s t symbolically, a s

S(x, y )

measure. A

.

ui(g)

X = G/K

be homogeneous v e c t o r bundles over

( i = 1,2)

i s a l i n e a r transformation

and dy i s a

G-invariant

w i l l be a d i s t r i b u t i o n - v a l u e d k e r n e l . )

(In general, of course, S

w i l l commute with the a c t i o n

E l -. E~ Y x

T of

G

on s e c t i o n s given by

i f and only i f

-1

S(gx,gy) = 0 2 ( g ) S ( x , ~ ) o l ( g ) x,y E X

for a l l

,g

E G

.

Introducing t h e f u n c t i o n

k(g) = S ( g , e ) and denoting by

o

the base p o i n t i n

(Af)(g. 0 ) = (with

du

o1

and

,A

can a l s o be w r i t t e n a s

SG~ ~ ( u ) * ( ~ - ~ g ) o ~ ( u ) - ~ f ( ~ ) d u

denoting Haar measure).

[ l o ] except t h a t

G/K

This i s e x a c t l y the same formula a s i n

ag have a more s p e c i a l meaning there.

However,

the r e s u l t s and proofs remain v a l i d under t h e p r e s e n t i n t e r p r e t a t i o n .

G/K has a

one has a complete corresponding theory of operators.

So, i f

G-invariant pseudometric s a t i s f y i n g the c o n d i t i o n s l i s t e d i n [ l o ] , G-equivariant s i n g u l a r i n t e g r a l

We should a l s o mention t h a t i f t h e homogeneous v e c t o r bundles given i n the form

G

x Kvi ( i

= 1,2)

, where

vi

the

Bi

are

a r e finite-dimensional

K-modules, and t h e s e c t i o n s a r e i d e n t i f i e d with f u n c t i o n s

v: G -. V

satisfying

(so the a c t i o n o f

G

on s e c t i o n s becomes simply l e f t t r a n s l a t i o n ) ,

then t h e

G-equivariant o p e r a t o r s appear i n t h e form

w i t h a k e r n e l such t h a t

for a l l

kl,k2

E

K ;g

kO(g):

E

G

v1 4 v 2

is linear for a l l

g

and

E G

.

The r e s u l t s t o be discussed i n t h e p r e s e n t l e c t u r e s a r i s e from t h e attempt t o use t h e harmonic a n a l y s i s of

G

e q u i v a r i a n t s i n g u l a r i n t e g r a l o p e r a t o r s , i. e.,

and

K

i n t h e study o f

G-

t o look a t t h e s e o p e r a t o r s "on

the Fourier transform side", where they appear a s mu1t i p l i e r operators. Of course t h i s kind o f harmonic a n a l y s i s i s most convenient t o use i n s i t u a t i o n s where every i r r e d u c i b l e r e p r e s e n t a t i o n of t h e group occurs a t most once.

G

I n the p r e s e n t case t h i s means t h a t we a r e considering v e c t o r bundles 2 L

-

x KV where t h e n a t u r a l r e p r e s e n t a t i o n o f K on t h e space o f a l l

s e c t i o n s c o n t a i n s every i r r e d u c i b l e r e p r e s e n t a t i o n a t most once.

This is t h e

e x t e n s i o n of t h e n o t i o n of Gelfand p a i r r e f e r r e d t o above; i n the c a s e where V = C

K

and t h e a c t i o n of

n o t i o n of a Gelfand p a i r .

on

V

is t r i v i a l ,

I n our examples i t w i l l even be true, although we

w i l l not make use of t h i s f a c t , t h a t f o r i r r e d u c i b l e r e p r e s e n t a t i o n s of call

(G, K)

it c o i n c i d e s w i t h the u s u a l

K

irreducible

occur a t most once i n

a "strong Gelfand p a i r " i n such a case.

8 1.

Vector-valued f u n c t i o n s on spheres

K-module

G

x KV

.

V

,all

One could

In our f i r s t example we consider

L = L ~ ( s " - ' , R ~ ) (n

n

space of

R -valued f u n c t i o n s on the u n i t sphere o f

qn

>_ 5) ,

.

the

L ~ -

A s customary, we

w i l l r e f e r t o these a s vector-valued functions, although i t i s f u n c t o r i a l l y more c o r r e c t and more i l l u m i n a t i n g t o t h i n k of them a s covector-valued functions, i. e., Let

d i f f e r e n t i a l forms.

G = SO(n)

, acting

Rn

on

be the s t a b i l i z e r of the p o i n t

i n the u s u a l way, and l e t

e

= (0,.

.., 0 , 1 ) .

regarded a s s e c t i o n s of the t r i v i a l bundle

-

a (g): ( x ' , v )

,

f E L

(g

.xl,g.

.

v)

Sn-I X R~

So the a c t i o n o f

G

with

L

1)

can be

a c t i n g by

on the s e c t i o n s ,

G

-

K r SO(n

The elements of

i. e.,

on

i s given by

The subspace

H c L

formed by the boundary v a l u e s of Riesz

of g r a d i e n t s of harmonic f u n c t i o n s i n the u n i t b a l l , The orthogonal p r o j e c t i o n

P: L -. H

s i n g u l a r i n t e g r a l operator,

i s clearly

systems, i. e. G-invariant.

was shown i n [ l l ] t 0 be a G-equivariant

bounded i n every

L'

were proved i n [ l o ] about t h e "Riesz transform",

(p

>

i.e.,

. ~ s s o c i a t e st o the normal component of every element of

1) ; similar r e s u l t s the map

H

R

that

t h e corresponding

t a n g e n t i a l component. Here we wish t o d e s c r i b e the main r e s u l t s of [12] concerning the harmonic a n a l y s i s of the underlying v e c t o r bundle a n d t o mention some of t h e main a p p l i cations. of

sO(n)

I , . .

.0

Let us denote by

D ~ ' O

resp.

Dr'l

the i r r e d u c i b l e r e p r e s e n t a t i o n s

whose maximal weight i n the usual p a r a m e t r i z a t i o n [2] i s (r, l , O ,

rep.

.0 .

harmonic polynomials of degree the r e p r e s e n t a t i o n

D ~ ' O

r

Wr

Let on

denote t h e space of homogeneous

R~ ; a s w e l l known, G

a c t s on

Wr

by

.

F i r s t of a l l , we c l e a r l y have the orthogonal sum decomposition

Here f (x')

LTan

i s the s e t of " t a n g e n t i a l v e c t o r f i e l d s " ,

x' = 0

product on

for a l l

. her

R ~ )

f ( x l ) = cp(x')xl

x'

E

n- 1 S

i.e.,

all

f

such t h a t

( t h e d o t h e r e denotes t h e n a t u r a l i n n e r

i s the "normal v e c t o r f i e l d s " ,

with some scalar-valued

2 n-1 cp 5 L (S ,

i.e.,

.

such t h a t

I t i s then obvious

that

Go:

being the space

v E Wr\ , and

iv(x')x'l

the representation

carrying

$,0 To decompose

i n t o i r r e d u c i b l e subspaces we have t o observe t h a t i t

$an

i s e x a c t l y t h e homogeneous v e c t o r bundle K

of

=

-

SO(n

1)

on

n- 1

R

t h a t the r e p r e s e n t a t i o n s of restriction to SO(n

-

1)

.

on

and

Dr'O

Dr"

t h a t the hypothesis tions. )

AS

for

G

n- 1

with the natural action

So t h e Frobenius r e c i p r o c i t y theorem implies occurring i n

LTan

a r e e x a c t l y those whose

c o n t a i n s t h e n a t u r a l r e p r e s e n t a t i o n (of type

K

those r e p r e s e n t a t i o n s

i.e.,

.

G x KR

By t h e Branching Theorem (cf. (ml,..

.,mP)

[2])

Dly0) of

t h i s gives exactly

f o r which

, w i t h m u l t i p l i c i t y one f o r each n 2 5 is used; t h e c a s e s n 0,

n

E N

, and

i s a constant making sure t h a t

can

= 1 .

Given a b i r a d i a l f u n c t i o n

Sf\, n .

f

on

N

, we

w i l l write

P(k,n)

(This i s r e a l l y only p a r t of the Gelfand transform,

for

but i t i s the

only p a r t t h a t w i l l occur i n the Plancherel formula; the remaining p a r t can anyway be obtained by taking l i m i t s of the biradial,

one can a l s o w r i t e

( t h e i n t e g r a l is independent of

u E S2)

.

.

?(~,n) )

Since

f

is

Sf.,

I t i s now easy t o prove the Plancherel formula f o r the Gelfand transform. Given a b i r a d i a l f u n c t i o n

f

Our l a s t expression f o r coordinates i n the

where

c

, we

write

then takes t h e form, a f t e r introducing p o l a r

X-variable,

i s a p o s i t i v e constant.

(Even though i t would p r e s e n t no d i f f i c u l t y ,

we s t o p keeping t r a c k of t h e c o n s t a n t s and j u s t w r i t e since

{e-x12 xa12 L:(~)

/

, c ' , c"

c

i s a complete orthonormal system on

. ) Now,

,

( 0 , ~ ) we

have by Parseva 1 ' s formula,

Multiplying by

, integrating

in

and using the Plancherel theorem f o r

A R~

,

then r e i n t r o d u c i n g the v a r i a b l e

i n the

Y-variable,

X

we g e t

which i s the d e s i r e d r e s u l t . Now we d e s c r i b e a few a p p l i c a t i o n s of t h e s e r e s u l t s . (i)

We f i r s t consider a s i n g u l a r i n t e g r a l operator on

N

i n t h e sense

k

i s biradial

of [ l o ] ,

*.

with the p r i n c i p a l value taken with r e s p e c t t o t h e gauge / e x p (X

+ Y)/ =

4

(bl~(

+ 4 1 I 2~)

Suppose t h a t the k e r n e l

+ tY))

besides the usual c o n d i t i o n s of homogeneity, k ( e x p ( t 1 I 2 ~

=

-q-E = t

k(exp(X

the gauge.

+ T ) ) , and

of having mean zero on "spheres" with r e s p e c t t o

Biradia.1 k e r n e l s occur n a t u r a l l y :

the k e r n e l s considered in

[ l o , 661 and i n [ 7 ] , a s w e l l a s some o c c u r r i n g i n t h e work of Knapp-Stein on intertwining operators

[a],

a r e of t h i s type.

To prove the c o n t i n u i t y o f

T

2 L (N)

in

,

the u s u a l method found by Knapp and S t e i n [8] and a l s o used i n

[ l o ] makes use of a l e m of M. t h e s i s on

k

.

If

k

Cotlar, and r e q u i r e s a s t r o n g smoothness hypo-

i s biradial,

proof based on F o u r i e r a n a l y s i s . f;(h,n)

i s bounded.

t h i s condition can be relaxed, and the

In fact,

the problem reduces t o showing t h a t

Now a change of v a r i a b l e shows t h a t t h e homogeneity

c o n d i t i o n of v a r i a b l e shows t h a t the homogeneity c o n d i t i o n on $ ( ~ , n ) i s independent of

),

, and

value c o n d i t i o n guarantees t h a t (ii)

another computation shows t h a t the mean i s bounded a s a f u n c t i o n of

n

a s well.

[ 6 ] about Hermitian hyperbolic space and extended by Cygan

[3] t o a l l non-compact symmetric spaces

x E X (n

of rank one.

X

i s a bounded harmonic f u n c t i o n on

F

, then

x E X

e x i s t s f o r some proof,

means t h a t

Another a p p l i c a t i o n i s t o g i v e a s i m p l i f i e d proof of a r e s u l t of

Hulanicki-Ricci that i f

k

X

and i f

The r e s u l t says limn,

f (no x )

i t e x i s t s , and has the same value,

for

h e r e denotes t h e g e n e r i c element of the Iwasawa group

i n a nutshell,

*

F(na x) = f

c o n s i s t s of w r i t i n g

, noticing

P (n) X

that

Px

N)

all

.

i s b i r a d i a l , checking t h a t

gx(k,n)

i s nowhere zero, and f i n a l l y applying the Wiener Tauberian Theorem s i m p l i f i c a t i o n i s on one hand t h a t t h e r e i s no t i o n and making case-by-case

The

a s a Poisson i n t e g r a l ,

F

The

need of using the c l a s s i f i c a -

computations, on the o t h e r t h a t one works

d i r e c t l y with t h e s p h e r i c a l f u n c t i o n s and does not have t o go through a n e x p l i c i t d e s c r i p t i o n of the r e p r e s e n t a t i o n s of (iii)

.

N

on

An

.

MN-invariant Riemannian m e t r i c ( a c t u a l l y a family of such m e t r i c s ) can be d e f i n e d by l e t t i n g the l e n g t h of the tangent v e c t o r

N

1Xl2

N

One can use the p r e s e n t i d e a s i n t h e study of p o t e n t i a l theory on

+

8, = A1

,

c - ~ I Y ~(c~

+

c L+

.

> 0)

.

X

Al

The corresponding Laplace-Beltrami operator i s

appears then a s a l i m i t i n g case when

Taking the case of a f i x e d

c

>

0

,

c -. 0

.

.)

one can, s i m i l a r l y a s i n the case of a

Riemannian symmetric space, consider weakly harmonic f u n c t i o n s (i.e.,

N

be

(The p a r t i c u l a r l y i n t e r e s t i n g p o t e n t i a l theory f o r the

s u b e l l i p t i c operator

that

+Y

and s t r o n g l y harmonic f u n c t i o n s ( i . e . ,

A f

such

%f = 0 )

on 1 I n t h e analogous s i t u a t i o n on a symmetric space a well-known theorem of

A f

=

0)

=

Furstenberg s t a t e s thateveryboundedweaklyharmonic f u n c t i o n i s stronglyharmonic.

A praof of F u r s t e n b e r g ' s theorem given by ~ u i v a r ' c ha p p l i e s t o the c a s e of N a s w e l l ( t h e e s s e n t i a l p o i n t being a g a i n t h a t

(MN,M)

i s a Gelfand p a i r ) .

Applying

t h e c l a s s i c a l L i o u v i l l e theorem twice, one s e e s e a s i l y t h a t a bounded s t r o n g l y harmonic f u n c t i o n on following r e s u l t :

N

must be a constant.

So we have obtained t h e

every bounded weakly harmonic f u n c t i o n on

N

i s a constant.

References [I]

L. Ahlfors, A s i n g u l a r i n t e g r a l e q u a t i o n connected w i t h q u a s i c o n f o r m a l

[2]

mappings i n space, Enseignement Math. 3 (1978), 225-236. nd H. Boerner, D a r s t e l l u n g e n von Gruppen, 2 ed., S p r i n g e r 1976.

[3]

J. Cygan, A t a n g e n t i a l convergence f o r bounded harmonic f u n c t i o n s on a

[4]

D. G e l l e r , F o u r i e r a n a l y s i s on t h e H e i s e n b e r g group> Proc. ~ a t ' Acad. l

rank one symmetric space, t o a p p e a r . Sci., [5]

USA

2

(1977) 1328-1331.

S. Helgason, D i f f e r e n t i a l Geometry and Symmetric Spaces,

Academic P r e s s ,

New York 1969.

[6]

A. H u l a n i c k i and F. R i c c i , A Tauberian theorem and t a n g e n t i a l convern gence f o r bounded harmonic f u n c t i o n s o n b a l l s i n C , t o a p p e a r i n Inv. Math.

[7]

A. Kaplan and R. Putz, Boundary b e h a v i o r o f harmonic forms o n a r a n k one symmetric space, Trans.

[8]

A. Knapp and E.M. Ann. of Math.

Amer. Math. Soc.

231

(1977),

369-384.

S t e i n , I n t e r t w i n i n g o p e r a t o r s f o r semisimple groups,

93

(1971), 489-578.

[9]

A. ~ o r h n y i , F o u r i e r a n a l y s i s o f b i r a d i a l f u n c t i o n s on c e r t a i n n i l p o t e n t

[lo]

A. ~ o r d n y iand S.

groups,

t o appear.

vQgi,

S i n g u l a r i n t e g r a l s o n homogeneous s p a c e s and

some problems o f c l a s s i c a l a n a u Ann. Scuola Norm. 25 [ 111

Sup. P i s a

(1971), 576-648.

,

Cauchy-Szegel i n t e g r a l s f o r systems of harmonic f u n c t i o n s

Ann. Scuola Norm. Sup. P i s a

2

(1972),

181-196.

,[ to

[I21

appear. [13]

[14]

B. Kostant, On t h e e x i s t e n c e and i r r e d u c i b i l i t y of c e r t a i n s e r i e s o f representations,

i n "Lie groups,

reprcsentations",

Budapest 1971.

Summer s c h o o l on group

M. Reimann, A r o t a t i o n - i n v a r i a n t d i f f e r e n t i a l e q u a t i o n f o r v e c t o r

f i e l d s , t o appear. [15]

G.

Schiffmann,

5 i n "Anaprincipale,yse

harmonique s u r l e s groupes de Lie", Ma t h e m a t i c s #739,

S p r i n g e r 1979.

pp. 460-510,

L e c t u r e Notes i n

CEIi TRO I N TERKAZIONALE MATEMATICO E S T I V O (c.I.M.E.)

EIGENFUNCTION

EXPAhTSIONS ON S E M I S I M P L E L I E GROUPS

V. VARADARAJAK

EIGENFUNCTION EXPANSIONS ON SEMISIMPIB LIE WOWS

V. Varadarajan Department of Mathematics University of California Los Angeles, CA 90024

1. Representations of t h e Principal Series.

Harish Chandra's Plancherel

formula. 1.

In these l e c t u r e s it w i l l be my aim t o discuss some aspects of t h e problem

of obtaining an e x p l i c i t Plancherel formula f a r a CoMeCted r e d semisimple Lie group with f i n i t e center, and the close connection of t h i s problem with t h e theory of eigenfunction expansions on the group.

The c e n t r a l r e s u l t s are those

of Harish Chandra, and it i s impossible t o give anything more than a p a r t i a l o u t l i n e of h i s monumental work t h a t began i n t h e early

' 50's and has Spanned

almost t h r e e decades. For a given l o c a l l y compact group which i s separable and unimodular, t h e f'undamental problein i s t h a t of decmposing i t s regular representation i n t o i r r e d u c i b l e constituents. classical.

I f t h e group i s cammutative or compact t h i s i s quite

However, apart from some general existence theorems (see f o r

instance Segal [ l ] ) , t h e r e i s no systematic development of harmonic analysis on general l o c a l l y compact groups.

The category of l o c a l l y compact groups (even

separable and unimodular ) i s so extensive and the structure of i t s individual members so varied t h a t it has s o f a r proved impossible t o develop analysis on these @;roupsbeyond a few general theorems.

For Lie groups the s i t u a t i o n i s

much b e t t e r , and among these t h e semisimple groups (both r e a l and occupy a central position.

P-adic)

W e know t h e i r s t r u c t u r e i n great d e t a i l and a r e

able t o use t h i s knawledge i n formulating and solving t h e questions of harmonic analysis i n a significant manner. Although our i n t e r e s t i s essentially oonly i n t h e semisimple @-oups we consider a smewhat wider class of groups f o r a variety of reasons. example, m

For

w theorems i n t h e subject are proved by induction on t h e dimension

of t h e group via a descent principle that t r a n s f e r s t h e problem from the given group t o a Levi factor of one of i t s parabolic subgroups; these Levi factors are i n general neither semisimple nor connected, ,even i f t h e ambient group i s .

Furthermore, i n number theoretic applications, the groups whose

representations are important are often t h e r e a l points of a reductive algebraic group defined over

Q.

These and other reasons suggest t h a t it w i l l be con-

venient t o work with a c l a s s of reductive Lie groups which are not necessarily connected.

Following Harish Chandra we s h a l l work with groups

G with t h e

f o l l m i n g properties : (i)

G i s reductive ( i . e . ,

9,

t h e Lie algebra of

is a r e a l

G,

reductive Lie algebra) (ii)

[ G : Go] < m where

Go

i s t h e connected cauponent of

G containing

the i d e n t i t y (iii)

(iv)

If

G1

G1

i s closed i n

If

Gc

t i o n of

i s t h e analytic subgroup of G

then

and has f i n i t e center

i s t h e (complex) adjoint group of g),

g = [g,g],

G defined by

sc

(= the complexifica-

then A ~ ( Gc) Gc

These are the groups of t h e so-called Harish Chandra class

#;

for a more

detailed discussion of t h e i r properties, see Varadarajan simple r e a l Lie groups with f i n i t e center are i n

R such t h a t

group defined over

G( R) i s of class

Cartan involution 9

of

c.

G

i s an algebraic

(6

G of c l a s s

t h e fixed point s e t of

G, w i l l be denoted by

for t h e Lie algebra of a suffix

G;

if

C o ~ e C t e dsemi-

is irreducible and reductive, then

G(C)

Frau now on we f i x a group

#.

ccanpact subgroup of

#;

111.

K.

#

8, which i s a maximal

We s h a l l write

g

(resp. 1 )

Complexifications w i l l be indicated by

(resp. K).

K meets a l l connected cmponents of

One knows t h a t

and a

G.

A

G be t h e s e t of equivalence classes of irreducible unitary repre

Let

sentations of

G.

A

If

i s well known t h a t

w E G

and

B

i s a representation i n t h e c l a s s

has a character, namely t h e d i s t r i b u t i o n

B

(f E c ~ ( G ) ) . This d i s t r i b u t i o n depends only on C

moreover

determines

O w

w

all inner automorphisms of

8

t i o n for the algebra

f

H

tr(Tr(f ) )

and i s w r i t t e n as

w

it

w,

eU;

uniquely, and is an invariant (= invariant under G) d i s t r i b u t i o n on

G which is an eigendistribu-

of a l l bi-invariant d i f f e r e n t i a l operators on

G.

By

an e x p l i c i t Plancherel formula i s meant an "expansion" of t h e Dirac measure G a t t h e i d e n t i t y element a s an i n t e g r a l of the

on

Here of

p

Ow

i s a nonnegative measure on for

p-almost all w

Plancherel measure f o r

$;

8

Ow:

and we r e m i r e e x p l i c i t descriptions

as well as of

p.

The measure

i s called t h e

C1

G.

It is a remarkable f a c t t h a t i f

G

i s nontrivial i n t h e sense t h a t

G1

i s not compact, t h e "support" of the Plancherel measure is not t h e whole of

h

G.

In f a c t Harish Chandra discovered t h a t one can introduce a notion of temperedness of -

distributions on

classes, i . e . , classes

w

in

of

A

w E G

6/2t

G,

and t h a t i f

f o r which Ow

Et

is tempered, then

are called exceptional.

t r i v i a l representation of

is t h e subset of all tempered

If

n

A

P(o\G~= ) 0. The

G is nontrivial, the

G is exceptional or, what i s t h e same thing, Haar

measure on

G

i s not a tempered distribution.

The meaning and significance

of t h e exceptional representations i s one of t h e outstanding puzzles of t h e harmonic analysis of semisimple groups.

2.

Let us now proceed t o an e x p l i c i t statement of Harish Chaulra's Planeherel To do t h i s we need a description of t h e irreducible representations

formula.

t h a t w i l l enter the Plancherel formula. F i r s t of all we have t h e discrete s e r i e s A

belongs t o

Gd

Gd

G = S L ( ~R), ,

G = ~ 0 ( 1 , 2 k+ l ) , A , , cmpact, G = Gd;

L (G) has a d i r e c t summand t h a t belongs t o

G = sp(n,

A

G

G is

G is semisimple and

f,

we have

by t h e o r b i t s under t h e Weyl group

of the l a t t i c e of i n t e g r a l elements i n G i s not compact but

If

i s a CSA f= Cartan subalgebra) of

Hermann Weylls p a r ~ m e t r i z a t i o nof

when

G = ~ 0 ( 1 , 2 k ) ; nonexamples a r e

R),

i n t h i s case, assuming further t h a t

IJ c 1 = g

U.

rk(G) = r k ( ~ ) ;

G = any connected cmplex semisimple group.

simply connected, i f

w ( ~a),

L ~ ( G ) , or equivalently, i f and only if

t o be nonempty it i s necessary and s u f f i c i e n t t h a t

examples are

of

2

t h e regular representation of For

A class

by d e f i n i t i o n i f and only i f t h e matrix coefficients of t h e

representations of the c l a s s are i n

A

A

15

Gd c Gt.

(-1)12.

In t h e generrrl case

r k ( ~ )= r k ( ~ ) , Harish Chandra's theary of t h e

discrete series a . l l . 0 ' ~ us~ t o proceed i n an a.lmost ccanpletely analogous fashion. Fur instance, l e t

G be a connected r e a l form of a simply connected complex

semisimple group with with Lie algebra b c i . (-1)'12

b*;

Let IL' with C

rk(G) = r k ( ~ ) ; l e t

B

We write IL f o r t h e l a t t i c e of i n t e g r a l elements in

L i s canonically ismurphic t o t h e character

be the s e t of regular elements of IL,

(a,X)

K be a CSG (= Cartan subgroup)

#

0 f o r each root

a of

(

s, be).

C

B/B where B i s t h e narmalizer of B i n G, Chandra' s theory gives a unique b i j e c t i o n

i.e.,

If W

A

group

B

of

t h e s e t of dl X

B. €

L

W = W(G,B) is t h e group

operates on lL'.

Harish

X

such t h a t if

E

E' and w = w(X) t h e corresponding c l a s s of

Z

HI

@,(am

Here q = where

P

1

4x1

= (-1Iqsgn

A

Gd,

~ ( s )~ex ( H )

exp H regular)

(H E b,

s'&_m

and A(exp H) = &p(e a(H)/2

~ & U ( G / K )W(X) , = bP(a,X)

is a fixed positive system of roots of

formula is independent of t h e choice of

P.

(9c, bc);

- ,-dH)/2)

of course t h e

For an a r b i t r a r y

G in

# with

t h e parametrization is a l i t t l e more subtle; f o r instance,

r k ( ~ )= rk(K),

even f o r semisimple

G,

it m a y happen t h a t

G

i s t h e r e a l form of a complex

group whose character l a t t i c e does not contain p where t h e sum of positive roots.

i s as usual half

p

To see what t h e formula i s i n t h e general case we

note t h a t the usual formula

can we rewritten as

which has t h e advantage t h a t

as we^ as

exp H w e (sp-p)(H) (even Ad(B)). characters of

E

I-+ s [b*]

SsP-P

_p H

i s i n the root l a t t i c e and s o I+

( -1)1/2 b*

of

- ea('))

h p ( l

P W

B

b*(-

as before and put

B

B*

sp

-

s [b* p.

i s t h e dimension

there i s a unique element

+

~ ( b * ) = l o g b*

defined by

of irreducible

d(b*)

H) = d(b*)e (1 '

is then defined by

on B*

i s t h e character of

If

corresponding t o b*,

such t h a t

a r e functions on B

and t h e s e t

W = w(~J/B) operates on B*.

B;

positive system

s ,b*

-P

We introduce a CSG B c K

of t h e representation of

P = log b*

sP

(H E b); p.

1 = sb*

B*'

we f i x s.

The a f f i n e action where

i s then t h e set of

b* E B*

with u ( x ( ~ * ) )# 0;

it is s t a b l e under t h e above a f f i n e action of

W,

and we have a unique b i j e c t i o n

such t h a t i f

b*

for a l l regular

E

B*'

b

E

and

B

w = w(b*)

is t h e corresponding class i n

(regular means as usual t h a t

$.,.(b)

A

Gd,

1 f o r a l l roots

a). The d i s t r i b u t i o n obtained by a n i t t i n g t h e s i g n factors i n t h e above expression i s denoted by

eb*.

It i s possible t o characterize it as t h e

unique tempered invariant eigendistribution f o r regular points

b

E

B

8

whose values a t t h e

are given by

I n all these statements we are t r e a t i n g t h e characters as point functions on G.

This i s of course permissible i n view of t h e celebrated r e g u l a r i t y theorem

8 on

of Harish Chandra which assests t h a t an invariant eigendistribution f o r S

i s a locally summable f'unction which is analytic on t h e s e t of regular

elements. The matrix coefficients of the representations i n d i s c r e t e classes s a t i s f y orthogonality r e l a t i o n s t h a t imitate those i n t h e theory of compact groups. A

Gd; rY a representation from t h e c l a s s

More precisely, l e t

(J

a E l j e r t space

then there i s a rnunber

degree of

w,

H;

E

such t h a t f o r a l l

Note t h a t the value of

d(w)

d(w) > 0,

cp, cpl, Jr, Jrl

E

w

acting i n

c u e d t h e formal

H,

depends on t h e normalization of

dx.

If

OL~(G)

i s the d i s c r e t e p a t of the regular representation, it i s t h e Rilbert space span of t h e matrix coefficients of t h e d i s c r e t e classes; and i f

0

E

orthogonal projection L*(G) +OL~(G), we have, fur some constant

i s the c >0

a d

f o c~(G),

all

(?(Ic)= f(x-l)). formula for

G.

This i s obviously t h e "discrete part" of the Plancherel

In our case, there is a constant

c > 0 such t h a t (b*

d(w(b*)) = c lm(~(b*))ld(b*)

E

B*' )

There i s no need t o t r e a t t h i s expansion i n more d e t a i l since we s h a l l subsume

it under a more general Plancherel f a m u l a presently. A given group

G of class

# does not alwa~rshave a d i s c r e t e series.

To construct t h e s e r i e s of representations of such groups t h a t enter the

kt

Plancherel formula we proceed as follows. under

8; we can then write

compact subgroup of a

E

%.

A

and

A =

+

Aqk,

$ =A n

is a w c t a r group with

There e x i s t parabolic subgroups (psgrps)

positions are of t h e farm P = %N The group M

i s then of class

particular,

r k ( ~ )= rk(%)

subgroup of

M

A

e Md

where

A be a CSG of

and

v E

and

where

%

$=K

where

% = Lie

i s the maximal

B(a) = a-I

(cf. Varaaarajan E l ] ,

#,

for a l l

Part 11, § 6).

i s a caupact CSG of M;

ll M = K

algebra of

stable

P whose Langlads decau-

KP

fixed by t h e Cartan involution

* %

K

G,

= AR,

in

is the maximal compact

elM.

A

Thus Ma

$.

If

then one can s t a r t with

t h e representation

where

o i s a representation of

inverse of

exp : aR +

%,

M i n the class

w

and l o g :

and obtain a representation of

% +%

i s the

G by inducing from

P. Note t h a t since G/P does not admit a

G i n v a r i a n t measure we must use

P t o u n i t a r i e s of

t h e so-called unitary induction t h a t takes u n i t a r i e s of Let us write

G.

f a r the unitary representation of

Tp Y

t

G thus obtained.

It can be shown t h a t its c l a s s i s independent of t h e choice of f'urther be proved t h a t i f far a n r o o t s

O

WJ

B of

V

normalizer of

A

E

(g,aR),

in

G,

A

W(G,A)= Z/A where

If

,

9

(v,B)

#

0

& t us w r i t e

3 is

the

it i s easy t o see t h a t W(G/A) operates i n a

as well as

natural manner on Md

i s even irreducible.

$,w,Y

It can

P.

is regular i n t h e sense t h a t

rp

for t h e character of

v

* aR

;a:

and one has t h e symmetry

The procedure outlined above associates with each conjugacy c l a s s of CSG1s of

A

G,

G a parametrized subset of

a c t u a l l y of

3

Gt; i n t h e case when

r k ( ~ )= r k ( ~ ) , exactly one of these conjugacy classes consists of compact CSGts, and the associated s e r i e s of representations is

A

Gd.

We note t h a t i n

a l l cases there are only f i n i t e l y many conjugacy classes of CSGts.

From t h e

representation theoretic point of view Harish Chandra' s Plancherel forraula a s s e r t s t h a t the regular representation of

i::

,=lg.al

G can be decanposed as a d i r e c t

of the representations described above and further t h a t t h e measure

i~i-colvedi n t h i s decomposition i s mutually absolutely continuous with respect t o t h e Lebesgue measure

dv

theorem (cf. Harish Chandra [ 8 Theorem 27.3. ).

O,

be a minimal ps@;rp contained i n the

c > 0,

let

A;(Q

Eo

>

:t )

0, m

( ~ )(H E aO). F ~ X

2 0 such that for all

be the conic s e t of all

h

E

C~(A;) f o r which

!3 ( l o g h)

t po(log h).

Q

Then 3 C > 0, m 2 0

>0

El

such t h a t

far a u h

A;(Q:~).

E

To i l l u s t r a t e t h e power of t h i s method of studying t h e asymptotics we mention the characterization due t o Harish Chandra of eigenfunctions i n L ~ ( G ) .It i s a consequence of the f a c t t h a t the above error estimates a r e square integrable. Theorem.

Fix

(a)

f

2

(b)

G has compact center am^

(c)

f

f

0

i n /A(G: T).

Then t h e following are equivalent:

(G: 2)

E

E

f

Q -- o

f

for

ps*s

Q f G

C(G:T).

It is also natural t o ask whether one can not only define t h e constant

t e r m along Q but associate an e n t i r e verturbative expansion along Q.

This

question i s not completely s e t t l e d but it has been a f r u i t f u l l i n e of invest i g a t i o n (cf

. Harish Chandra [111,

Trombi-Varadarajan [I], Trombi [I], [21, [3],

Eguchi [ 1 1 etc.). For a detailed treatment of the ideas of t h i s lecture see Vaxadarajan Harish Chmdra [ 3

6 . Wave packets

I.

i n Schwaxtz space

1. The next step i n doing haxmnic analysis i s to investigate t h e decay

properties a t i n f i n i t y on tempered representations of representations H,

To

G

of wave packets of matrix coefficients of G depending on a continuous parameter

8.

The

axe ~ n e r a J 2 . yassumed t o act i n a single Hilbert space

and t o possess infinitesimal characters; it is d s o convenient t o assume

1 I,

that the restrictions

TeIK are admissible and do not depend on 8.

i s a f i n i t e subset of

A

family of

K

5-sphericdL

and

$

If

F

i s defined a s i n $5.1, then we have t h e

functions

which are eigenflmctions f o r

8;

and one m a y begin t h e study of the wave

packets

Let P = MAN be a psgrp which i s cuspidal, l e t

A

w E

Md

and l e t T p w Y

( V E a*)

be t h e family of representations introduced i n $1.

realization a

of

i n a Hilbert space

,

We choose a

~ ( a ) and denote by H t h e Hilbert

space of all (equivalence classes o f ) functions

such t h a t (i)

f o r each

k1

E

5 = K n M,

f o r almost all k~ K. (ii)

llfl12

= JK lf(k) 12ak < m.

Right translations by elements of K

in H.

fact, if

K define a unitary representation

By Frobenius reciprocity it i s c l e a r t h a t b

TK of

rK is admissible; in

A

E

K,

Let us now f i x

V E

a*

and introduce the space

@(v) of equivalence classes

such t h a t f o r each

p = man

f o r almost all x.

The space

elements of tion to

G.

Since

P,

E

@ ( v ) is s t a b l e under ri&t translations by

G = PK,

any

q e @(v)

K and so, i f

then t h e subspace space with

11-11

$(v)

of

@(v)

as its norm.

The above action of

G.

sp,

representation and is i n f a c t

i s a unitary isonlorphism of may transfer

G,

rr P,w,v

%

to

h

F cK

HF,

E

a*,

@(v) leaves

,

k,v,F

are

with

H t h a t takes

Bp , ~ , v l ,

T ~ '

and l e t

U

K. be t h e f i n i t e dimensional Hilbert

K as i n $5.1;

regarded as a bimodule f u r

F(x) = %sp,o,Vcx)E~

(X E

K.

Define

G).

J

T -spherical eigenfunctions f o r

F

a2(v)

t h i s is a unitary

denote the corresponding double representat ion of

$ The

v

on

I: and thus guarantee t h a t f o r these representat-

space of endomorphisms of TF

If

G

is a H i l b n t

As t h e map

is t h e i r r e s t r i c t i o n t o

Fix a finite set

let

a2(v)

ilqly < m

of all 'p with

s t a b l e and defines a representation of

ions of

is determined by its r e s t r i c -

8,

and f o r each

V E

a*,

they are tempered. Let us now describe t h e figenhomomorphism t o which

$w, V,F belongs.

we

We s e l e c t a

and L =

8-stable CSG L of

Lqi

( r e c a l l P = MAN

G such t h a t i s cuspidal).

w = ~ ( b * ) f o r some irreducible character

where

a r b i t r a r y but f i x e d p o s i t i v e system).

i n a l a t t i c e i n (-3-)1/2~;

when

b* E

t h e action of an element of with subspaces of

I

3f

.

I

k = l o g b*

+ pI

(m, 1 I) (with respect t o some

The element

W(M,A ).

Then

Let

.':L

i s regular and varies

v a r i e s ; f o r fixed

w

M

In the parametrization of $1

i s h a l f t h e sum of p o s i t i v e r o o t s of

pI

is a compact CSG of

LI

w

it i s unique upto

We a l s o i d e n t i f y c:I

and

I&

=

:0

X + i v i s a well defined element of 1 *

A

=

E

ac,

and we have t h e following. h

a A.

For

z c

8, v

*

We axe e s p e c i a l l y i n t e r e s t e d i n t h e case

*.

v c a

For t h i s we have the

following r e s u l t . Lama B.

(a)

h

€

(-1)1/21:

i s regular i n t h e sense t h a t

a of

for each imaginary r o o t

(b)

A + iv

(c)

Suppose

determined upto conjugacy.

c-r

(

E

0

~c ) . l

(-1)li21*

v

#

@,A)

a*

*

v E a

if

X + i v is regular.

and

More precisely, l e t

I,

Then

( j = 1,2)

I

is

be two

J

&stable CSA1s of

g,

A. J

c (-1)l/~1;,

and suppose t h a t

I

and

I$

are

regular; if

f o r all

z

E

8,

then one can f i n d

k c K

The point is t h a t the condition on

such t h a t

I

and

I$

governed a p r i o r i only by t h e complex adjoint group.

kl

= 1 2.

expressed by ( c ) i s This lemma shows t h a t

t h e regular p a r t s of t h e s p e c t r a coming from t h e various s e r i e s of represen-

t a t i o n s are d i s j o i n t . gonal decmposition of

It is the foundation on which one can build an ortho2 L (G)

i n terms of t h e wave packets associated with

the various series.

I had remarked t h a t f o r fixed they s a t l f y the weak inequality.

V E

*,

a

the

$w,v,F

are tempered, i.e.,

Actually they do much more; t h e constants

involved i n these estimates grow a t most polynomially on wehave,fma;U

f o r suitable

*

v s a , x E G

C = CF > 0, r = rF 2 0;

and furthermore, such estimates are

v a l i d f m the derivatives (with respect t o x

as we vary

V

More precisely,

V.

as well as

i n the complex dcmain, the growth i n

v

v)

also.

Finally,

i s also well behaved;

we have estimates of the form

f o r a;U x 2.

E

G, v E .a:

Motivated by t h e above considerations we s h a l l introduce the theory of

wave packets in a very general context.

Actually we are not as general as we

snould be; we have assumed throu@out t h a t the double representation of involved is f i n i t e dimensional.

K

Ultimately one should vary it and an elegant

way t o do t h i s i s t o consider possibly i n f i n i t e dimensional double represent a t i o n s systematically frcnn the very beginning, as i s done by Harish Chandra. I decided t o keep t o the simpler framework since the main ideas may be under-

stood well enough already i n t h a t context. Let

9 be a 8-stable

h=gne by

8.

where

CSA of

g = 1 CI3 e

W e fix h s (-1)1/29;

g;

as usual we put

gI

=

i s the Cartan decomposition of and assume it is regular, i . e . ,

9

n

1,

g determined

( o r , X ) f 0 f o r each imaginary root

(1)

Ct

of

(gc, be).

We write

and a unitary double represen-

We f i x a f i n i t e dimensional Hilbert Space U t a t i o n T of

K on it. By an eigenfunction of type

II(X)

we mean a

function

with the following properties :

I I

( i i ) For any to U

v

E

= $(v : - ) from G

~ a p h e r i c a land

is

zB, ( i i i ) For any

$ = $(v)

5, the function

= ~l~,)(z)(X+ iv)$,,

al'%

constants

c

E

~ ( 9 ~and) any

= c(al,a2,

(2 c

aE

9).

~ ( 5 ~ 1there , are

a) > 0 and r = r(a1,a2, a) 2 0 such

that :al;x;a2)1

for a l l

X E

d

SO,

$ be a function of type II(X). given any psgrp P = MAN

constant t e r m

+ o ( x ) ) ~+ (~

G, v c 5.

Here we use the usual interpretation of Let

1 ~ Ir1

5c

We put

a

as a d i f f e r e n t i a l operator on

For fixed

it makes sense ( c f .

v c 5,

$,,

&

A(G :7)

$5) t o speak of the

5.

In studying the behaviour of of

$5 but taking care t h a t

gV

the idea i s t o use the perturbation theory

all estimates are uniform i n

because the estimates in ( 3 ) ( i i i ) above asserting that actually uniform i n

v.

This i s possible

f(v)

E

b ( :~ 7) are

v.

Let us write

(5)

F ( x ) = {VE 5

I f we f i x

v

1

X

+

iv

i s regular].

5, the equations (13) of $5 show t h a t

E

Bp(v) s a t i s f i e s on

MA the d i f f e r e n t i a l equations

If

v

E

it i s not d i f f i c u l t t o deduce from t h i s t h a t

5'(X),

written as a sum of eigenfunctions for

on MA.

$ (v) can be P

~ p r i o r one i would

expect t h i s sum t o be w e r the complex Weyl group; however, the assumption that

@(v) is tempered implies t h a t only the r e a l Weyl group comes in.

formulate t h i s very basic r e s u l t l e t us introduce sane notation. the Lie algebra of linear injections

Ii' a = of

We write m(hl a)

A.

s of

by we write

a

into

h

m(a) = lo(%)

1

s = ~d(k) a

for same k e K.

it is a f i n i t e subgroup

m($II+);

GL(~). Proposition C.

(i) (ti)

'TM)

$(v)(m)

I$

Let

,

& , s ( ~ )E &(MA,

s

E

v

E

5'(~). Then

3 unique functions

tu(b1 a ) = lo with the following properties :

= Zs,,

s(v) = P

Plp

s(v)(m)

(m

E

( i S ) ( h + iv)$ys(v)

MA)

([

E

8(y)).

my/!? (we remnk that k

a be

for the (possibly empty) s e t of

such t h a t for

Let

To

E

K defines

$ s;

and

is

are defined respectively as

they are independent of the choice of

4

and

k, and

I"

where

$ 3 4).

&le.

Let

$(v:x) = cp(v: x ) ,

a =

% = a.

( G = KA$TO i s an Iwasawa decomposition) and

t h e elementary spherical function.

Take P = P o = M 0A $0,

the minimal psgrp; then

where t h e

is t h e

c( .)

c-function.

HL

We also remark t h e following innnediate consequence of ( i i ) :

(7) suggests t h a t when v

The example $,,(v)

II(X).

t h e function

$ of type II(X)

A

regulated if t h e following is t r u e . m($la),

a ' ( & ) tends t o a boundary point,

To avoid t h i s inconvenience we introduce the concept of

may blow up.

regulated elements of type

S E

E

($,,)'

is said t o be

P = MAN

Given any p s g q

and any

which i s well defined on 5'(h) X (MA)'

by

extends (uniquely) t o a Function of type

$ of type II(X)

(on

II(X)

on 5 X (MA)s.

5 x G) is s a i d t o be of type

I1 (x). reg

A regulated

We have t h e

f ollowing Proposition D.

any pserp P =

ww

(i) Let

and any

( A ) on 5 X G. Then, for reg ($,s)s is of type 11 (A) on reg

$ be of type I1 s

m($

E

a),

5 x (MA)'. (ii)

If

(iii)

Let

$ is of type I1

(x) reg

9

be of type II(X)

on

5 X G,

on 5 X G.

Let

and P = MAN

as above,

where

=

system.

Then

is t h e product of coroots of

I&,o

in a p o s i t i v e

(gc,bc)

i s of type

I1 ( A ) on 3 X G. reg I f we take a psgrp P = MAN f o r which a)

Jr

~(bl

is empty, then

$&v) = 0.

By t h e t r a n s i t i v i t y of t h e constant t e r m s t h i s implies t h a t i f

dim A = d h

h,

$(v)lM

then aU further constant terms of Note t h a t i n t h i s case

i s a cusp form.

$(v)lM

e( P( V

a r e zero so t h a t regarded a s a

: .),

is in L~(M:U) (recall- t h a t U

function on M with values i n U,

Hilbert space. ) As a s p e c i a l case of t h i s we may take

is a

P = MAN where

a =

h.

The following proposition shows t h a t t h e constant terms r e l a t i v e t o such P already contain much of t h e information. Proposition E. a l l psgrps

P = MAN

(i) Fix

(ii)

For

v

E

Let

$ be of type II(X).

with

5.

a =

If

qR.

$(v)

P(h)

denote t h e s e t of

We then have t h e following. = 0

f o r all P

$ t o be of type I1

t h a t t h e following be valid:

Let

(x) reg f o r any P E

E

~ ( h ) , then

g(v) = 0.

it is necessary and s u f f i c i e n t

~ ( b and )

itqy

s

E

m (%I

$1,

if

(If

(v) is t h e r e s t r i c t i o n t o M of $ ( v ) (v E 5'(~)), then (v)ll p,s p,s p,s 2 (norm i n L (M:U )) should be locally bounded on 5, i.e , should be bounded

f

.

on every subset of t h e form I

n S'(X)

where

L is a ccanpact subset of

3. Using these properties of eigenfUnctions of type II(X)

5.

and lIreg(k)

i n conjunction with the perturbation theory of $5 (developed with uniformity in

V ) one can prove t h e f i r s t and second wave packet theorems which a r e

analogues of t h e corresponding theorems f o r spherical functions. Theorem 1. Fix a function ol E

6 3 ) (= Schwartz space of

$ of type I1r e g(x) on 5 X G. For 5)

let

Then

#a is well defined

is a continuous map of

and belongs t o

@( G :T) . And

C(G :T ).

(35) into

For t h e second wave packet theorem, we f i x a p s g p P = MAN,

as above, a function of type

I1

(x) on 5 X G.

reg

For any

a! E

#

being,

@(5) we

form t h e "truncated wave packet"

It follows e s s e n t i a l l y *om Proposition D t h a t

We extend

$,a

Theorem 2.

Then, f o r a J l m

Here

(X

E

@(MA : ~

$,a

(Y)

a)

t o a function on

Let

E

6

G by s e t t i n g

be a Haar measure on

f.

Define

MA,

pp and Hp

have t h e i r usual meanings.

and H ( b a n ) = l o g a P

Corollary.

~ 1 .

&?

=

o

Thus

( k K,~ m r M, a c A, n

dess

o

kc

for some k

+(x) E

N).

E

K.

= $tr(ad

x ) ~

I.

For the theory discussed in t h i s lecture, see Haxish Chandxa [ 4

7.

The Eisenstein i n t e r n a l and t h e

c-functions

1. We s h a l l now apply t h e theory of constant terms and wave packets by

choosing for a

t h e so-xlled

8-stable CSA of

corresponding t o

and l e t notation be as i n $6. We write

g

so t h a t

lj

A s before we denote by ?

LI = L

n K,

~ ( k the )

We put

TM

kt

i n a f i n i t e dimensional Hilbert

where, as usual,

=

h.

LR = exp

K

be

L f o r t h e CSG

f i n i t e s e t of p s m s P = %N.

be a unitary double representation of

space U.

B

To define it, l e t

Eisenstein Integral.

having t h e b usual meanings we clef ine, f o r any

=K g

E

n M.

With

cm(M:rM),

V

pp

E Zc

and H

P * (=( h ) c ) ,

t h e Eisenstein i n t e g r a l

for

x

E

G;

here,

g

and we write u?(k) as t o what P or E ( ~ : v : x ) or

is extended t o t h e whole of

for g

u'r2(k)

when u

E

U, k E K.

U = C,

E(P : g :V :x )

When there is no doubt

is, we abbreviate t h e notation E(P : g : v :x ) t o

E(v:x).

The formula (1) i s analogous t o ( 1 ) of $3. psgrp,

G by

?

In fact, i f

i s t h e t r i v i a l double representation of

is just

cp(V

:x).

is t h e minimal and

g = 1,

So there i s a strong andogy of t h e

Eisenstein i n t e g r a l with elementary spherical functions. suppose

K,

P

i s a d i s c r e t e subgroup of

G

such t h a t

On t h e other hand,

GF has

f i n i t e volume

and t h a t U

is a bimodule for

K x

r

the right; then, averaging over

which (for suitable

g, I-)

as an Eisenstein s e r i e s .

r,

with

K acting on the l e f t and

r

on

K w i l l give the sum

instead of

i s what i s known in the theory of automorphic forms

It i s t h i s analogy t h a t prompted Harish Chandra t o

r e f e r t o (1) as t h e Eisenstein integral.

Indeed, the theory of the Eisenstein

i n t e g a l i s illuminated t o a remarkable extent by the two analogies mentioned just now. The Eisenstein i n t e g r a l i s well defined on m

g

E

C (M :'rM); it is holomorphic i n

v

E

5

it i s i n C-(G :7 ) .

C

where

%(z)(iv)

v

-

$m)

(recall gm),

f o r fixed x

for fixed

G;

E

and is the value of

gml)as

v i a t h e interpretation of elements of

an eigenfunction of

Sc

G f o r all

A simple calculation gives, for each

is an element of

values in a m )

E

zc X

z

E

p (z)

8,

at

P

iv

polynomials on 5 with

8(m) @ u ( ( % ) ~ ) ) . In particular, if

E ( :~V : - ) i s an eigenfunction for

8.

g

is

More

precisely we have t h e following r e s u l t t h a t s e t s the stage for applying the theory of %5 and 6. Proposition A.

an eigenfunction f o r

X

Fix a regular dm)

E

(-1) 1/2 gI+ and l e t

al,a2

E

a

II(X)

@(M :

on 5 X G.

u ( ~ ~ ) ,E ~ ( 3 ~ 1we, can find constants

c > 0 such t h a t f o r all v

E

3)be

such t h a t

Then E(P : g : : - ) i s an eigenfunction of type for each

g

E

zc,

x

E

G,

c

Moreover,

> 0, r 2 0,

Eigenfunctions i n t h e Schwartz space of

M

are of course matrix

coefficients of the d i s c r e t e s e r i e s of representations of use other types of matrix coefficients of

M

M.

One can also

in t h e Eisenstein integral; as

long as they are tempered, t h e Eisenstein i n t e g r a l w i l l s a t i s f y t h e weak The s p e c i a l case considered above is hawever t h e important one

inequality.

and is decisive for our purposes. A

It is well known t h a t f a r a given c l a s s

2.

many d i s c r e t e classes Varadarejan [ 1I).

A

W

Gd

E

such t h a t

K there are only f i n i t e l y

E

[a :b] > 0 ( c f . Harish Chandra [ 7

Applying t h i s r e s u l t t o

space spanned by t h e eigenfunctions f o r

b

M

instead of i n C(M:

8(m)

G we see t h a t t h e

i s f i n i t e dimen-

From Harish Chandra's theory of t h e d i s c r e t e s e r i e s one knows t h a t

sional.

these a r e a l s o a l l the a(m)-f i n i t e functions i n C(M: T ~ ) , t h a t t h e eigenhomomorphisms a r e defined by regular the space

'C(M : -rM) of

Varadarajan [

11,

X c (-1)

l/2

qI,+

and t h a t t h i s i s a l s o

-rM-spherical cusp forms ( c f . Harish Chandra [

PBrt 11,

5s 15,

3 1,

16).

Put

Then of

V

dim(v)
0

if

P

chosen s o t h a t the

i s between

For any f i n i t e

P1 h

F C K,

and

P2

;fc and t h e r e

i s i n s u i t a b l e domains of

J's

$ P21 PI)

are

have t h e product property

i n t h e sense t h a t

d(P1,P2)

= d(P1,P)

+ d(P,P2).

we define

The main point i s t h a t the i n t e g r a l representation of t h e

j-functions is

e s s e n t i a l l y t h e same a s t h e i n t e g r a l r e p r e s e n t a t i o n s of t h e i n Theorem D of $7, f o r t h e isomorphism JI,

we have, f o r a l l

c(A)

is a constant

>

V = VF

T

6

a s above.

For i n s t a n c e , using

~ n d ( ~and ~ )s u i t a b l e

v,

p( 1: v)JrT = c ( A ) J ' ~ ~

Cpl

where

and

U = UF

c-functions given

0

and

Moreover,

and

( s e e $11, Harish Chandra [ 5

I). c-functions i s fundamental.

This l i n k between i n t e r t w i n i n g o p e r a t o r s and

It allows on t h e one hand t o a n a l y t i c a l l y continue t h e i n t e r t w i n i n g o p e r a t o r s s i n c e , a s we mentioned i n $7, t h e p e r t u r b a t i o n t h e o r y a l r e a d y g i v e s a n a l y t i c c o n t i n u a t i o n of t h e

c-functions.

On t h e o t h e r hand, t h e

product p r o p e r t i e s which can now be t r a n s f e r r e d t o t h e t o the

p-functions.

j-functions have

c-functions, and hence

This circumstance i s t h e source of t h e product represen-

t a t i o n of t h e P l a n c h e r e l measure. Before doing t h e e x p l i c i t computations it i s necessary t o d e r i v e t h e f u n c t i o n a l equations f o r t h e E i s e n s t e i n i n t e g r a l s .

Harish Chandra does t h i s

v i a what he c a l l s t h e Maass-Selberg r e l a t i o n s ( o r i g i n a l l y obtained i n t h e context of t h e t h e o r y o f E i s e n s t e i n s e r i e s ) . Let

f

E

IA(G: 7 ) and f i x

v

6

These r e l a t i o n s a r e a s follows.

3' ; suppose t h a t

f

has t h e following two

p r o p e r t i e s ( t h a t a r e c e r t a i n l y possessed by t h e E i s e n s t e i n i n t e g r a l s c o r r e s -

ponding t o (a)

v): if

PI = M'A'N'

constant term

fp,

cusp forms of

M'

p, s

i n t h e sense t h a t

0

f o r each

a'

E

i s not conjugate t o A , f p , ,,,

i s orthogonal t o a l l

a)

(m

E

M, a

two such

f,

P1,P2 say

t h i s implies t h a t a l l -Q E P(A);

E

P(A),

s1,s2

E

ID.

for suitable

A)

I n p a r t i c u l a r , i f we know t h a t f o r

fly f 2 ,

one knows t h a t

(fl)p,s

f

(actually, f o r

g = f

1

= f2

= (f2)p,s

-

coupled with ( a ) , one e a s i l y g e t s

f2,

ment f a c t o r s

Oc,

for 9??E ( P Y ~ ) ,

we have

g = 0).

equations of the E i s e n s t e i n i n t e g r a l follow innnediately.

E

E

OC(M: T ~ ) . Then

E

for arbitrary

($

the

A'

fpb4= Cshm fpYs(m)eisv(log

(b) f

-

is

i s a psgrp where A '

g

f

For, with t h e a d j u s t -

the function

must be

f

Q,s

= 0.

The d e t a i l e d information regarding t h e

p

and

c-functions and t h e

E i s e n s t e i n i n t e g r a l s allows us t o simplify t h e (second wave packet theorem) Theorem E of $7 ( s e e Harish Chandra [ 5 1, Theorem 20.1). pl,p2

(q

E

where

E

We put, with

P(A)

V[o]).

c 7 0

Then

for

The f u n c t i o n a l

V[W]) has properties ( a ) , ( b ) described above, and i n addition

so that

= 0

Q

$a E @(G:

i s a constant.

T)

If

and one has, f o r

h(

p2

$a

(v)

m

E

M, a

E

A,

denotes t h e Fourier transform,

= 0,

-1 SEW

Formula ( 5 ) is v e r y c l o s e t o a n i n v e r s i o n formula.

L e t us p u t

Define t h e o p e r a t o r

no: U'U

%u=./,T(k)uT(k-')~

(UEU)-

Then m m ( 5 ) one g e t s

: v ) = &(.a'

tu(w)

f o r some

rbr(l) z

~ S V )

s

E ID

: sv) =

d(w1-l where

w' = s w

unless

0 A

$,(w*

(6)

is t h e stabilizer of

in

w

scro(w>

m.

Formula (6) e s s e n t i a l l y completes t h e F o u r i e r transform t h e o r y i n one d i r e c t i o n ; it is necessary t o extend it t o once t h e growth p r o p e r t i e s of possession.

The f a c t t h a t

a)

p(w:v)

a) E

C(5) of course b u t t h i s i s e a s y

described i n Theorem D a r e i n our

i s completely a r b i t r a r y i n (6) l e a d s t o t h e uniqueness

p a r t o f t h e P l a n c h e r e l theorem s t a t e d i n $1.

To complete t h e transform theory i n t h e

i n v e r s e d i r e c t i o n , we s t a r t with f c @(G: T ) and d e f i n e f^(w : V ) ( a s before) a s

his

sum i s o n l y over a f i n i t e s u b s e t of

constant

C(A)> 0

independent of

f,

A

Ma).

Then we f i n d t h a t f o r a

for a l l

Jr e V.

So, i f we consider g =

A

c ( A ) ~ ~

where t h e summation i s over a complete s e t of representatives and

f

g = f

A,

then

g

have t h e same Fourier transform and a not t o o d i f f i c u l t argument yields ( t h i s i s t h e argument t h a t was f i r s t encountered i n t h e s p h e r i c a l case).

Evaluating the r e l a t i o n

at

1 we g e t the Plancherel formula.

A s remarked e a r l i e r , one s t i l l needs Theorem D g i v i n g , t h e growth and

holomorphy properties of case when

v.

Using t h e product formula t h i s comes down t o the

dim(^) = 1. As described i n 81 t h i s is s p l i t i n t o two cases,

according a s whether

rk(G) > rk(K)

or

rk(G) = rk(K).

The e x p l i c i t

Plancherel formula t h a t one obtains here i s by using methods e n t i r e l y d i f f e r e n t from what we have been discussing so f a r .

It i s based on t h e Harish Chandra

linit formula f o r o r b i t a l i n t e g r a l s on

( cf. Varadarajan [ 11 , Part 11,

G

Theorem 13), and i s analogous t o t h e case when by Harish Chandra [12].

Acknowledgement.

r k ( ~ / K )= 1, t r e a t e d e a r l i e r

I cannot go i n t o it here.

I wish t o acknowledge the support of NSF Grant

MCS 79-03184 during t h e preparation of t h i s work.

I am a l s o g r a t e f u l t o

J u l i e Honig f o r her typing and cooperation i n t h e preparation o f t h e s e notes.

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116

( 1966), 2-111. [81 Harmonic Analysis on semisimple Lie groups, Bull. AMS

(1970),

529-551.

191 The Flancherel formula f o r complex semisimple Lie groups, Trans. AMS 1101

(1954), 485-528.

Plancherel formula f o r the Acad. S c i . USA

3 (1952),

2 x 2

r e a l unimodular group, R o c . Nat.

337-342.

[ll] Some r e s u l t s on d i f f e r e n t i a l equations and t h e i r applications, Proc. Nat. Acad. Sci. USA [12]

(1959), 1763-1764.

Two theorems on semisimple Lie groups, Ann. Math.

(1966), 74-128.

A . W . Knapp

[I1

Commutativity of Intertwining operators 11, ~ f l .

82 (1976),

271-273B. Kostant [l]

On t h e existence and i r r e d u c i b i l i t y of c e r t a i n s e r i e s of represent a t i o n s , Lie groups and t h e i r representations, Edited by I. M. Gel'fand, Halsted Press, 1975.

S. Lang [l]

"sL~(R)", Addison-Wesley, Reading, Mass., 1975.

K. R. Parthasarathy, R. Ranga Rao, and V. S. Varadarajan [l]

Representations of complex semisimple Lie groups and Lie algebras, Ann. Math.

& (1967),

383-429.

I. E. Segal

[I]

An extension of Plancherel's formula t o separable unimodular groups, Ann. Math.

(1950), 272-292.

P. C. Trombi [l]

Asymptotic expansions of matrix c o e f f i c i e n t s : case, Jour. of Functional Analysis

( 15 p c 2),

83-105.

[2]

Harmonic a n a l y s i s of

[3]

I n v a r i a n t harmonic a n a l y s i s on s p l i t rank one groups with applications,

cP(G : F)

2 (1978),

t h e r e a l rank one

(preprint).

(p r e p r i n t ) .

P. C. Trombi and V. S. Varadarajan [l]

Spherical transforms on semisimple Lie groups, Ann. Math

& (1971),

246-303. V. S . Varadarajan

[I] Hanoonic Analysis on r e a l reductive groups, Lecture Notes i n Mathematics 4576, Springer Verlag, 1977. [2]

Lie groups, Lie Algebras, and t h e i r representations, Prentice Hall, 1974

[3]

-

I n f i n i t e s i m a l theory of representations of semisimple Lie groups, Lectures given a t t h e Nato Advanced Study I n s t i t u t e a t Liege, Belgium on Representations of Lie groups and Harmonic Analysis, 1977.

N. R. Wallach [l]

Cyclic vectors and i r r e d u c i b i l i t y f o r p r i n c i p a l s e r i e s of represent a t i o n s , Trans. AMS

158 ( 1971)~ 107-112.

G. Warner

[lj Harmonic Analysis on semisimple L i e groups, I, 11. S p r i n g e r Verlag,

1972. D. P. Zhelebenko [l]

The a n a l y s i s o f i r r e d u c i b i l i t y i n t h e c l a s s of elementary represent a t i o n s o f a complex semisimple L i e group, Math-USSR I z v e s t r a

2 (1968),

105-128.

CEK TRO INTERYAZIONALE MATEMATICO ESTIVO (c.I.M.E.

ERGODIC THEORY,

GROUP

REPRESENTATIONS,

AND R I G I D I T Y *

ROBERT J. ZIMMER U n i v e r s i t y of C h i c a g o

* P a r t i a l l ) - s u p p o r t e d by a S l o a n F o u n d a t i o n F e l l o w s h i p and NSF G r a n t MCS 79-05036

These notes represent a m i l d l y expanded v e r s i o n o f l e c t u r e s d e l i v e r e d a t t h e C.I.M.E.

sumner session on harmonic a n a l y s i s and group representations i n

Cortona, I t a l y , June-July 1980.

The author would l i k e t o express h i s thanks

and a p p r e c i a t i o n t o t h e organizers o f t h e conference, Michael Cowling, Sandro ~ i g 2 - ~ a l a m a n c a and , FBssimo P i c a r d e l l o , f o r i n v i t i n g him t o d e l i v e r these l e c t u r e s and f o r t h e i r most warm and generous h o s p i t a l i t y d u r i n g h i s stay i n Italy.

We would a l s o l i k e t o thank t h e o t h e r p a r t i c i p a n t s o f t h e conference

f o r t h e i r i n t e r e s t i n these l e c t u r e s .

F i n a l l y , we would l i k e t o thank Terese

S. Zimmer f o r (among innumerable o t h e r t h i n g s t h a t we need n o t go i n t o here) helping w i t h t h e t r a n s l a t i o n o f [ 2 9 ] .

Contents

............................................... E r g o d i c i t y Theorems ........................................ Cocycles .................................................... Generalized D i s c r e t e Spectrum ............................... Amenability ............................................... Rasic Notions

Rigidity:

4 10 15 19

25

The Mostow- Margul i s Theorem

and a G e n e r a l i z a t i o n t o Ergodic Actions

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

34

Complements t o t h e R i g i d i t y Theorem f o r Actions: F o l i a t i o n s by Symmetric Spaces and Kazhdan' s

................................................ Margulis' F i n i t e n e s s Theorem ................................ Margulis' A r i t h m e t i c i t y Theorem ............................. References .................................................. Property (T)

41 47 49

58

I.

Basic Notions I n these l e c t u r e s we discuss some t o p i c s concerning t h e r e l a t i o n s h i p of

ergodic theory, r e p r e s e n t a t i o n theory, and t h e s t r u c t u r e o f L i e groups and t h e i r d i s c r e t e subgroups. I n studying t h e r e p r e s e n t a t i o n t h e o r y o f groups, t h e assumption o f compactness on t h e group e s s e n t i a l l y allows one t o reduce t o a f i n i t e dimensional s i t u a t i o n , i n which case one o f t e n can o b t a i n complete information.

For non-compact groups, o f course, no such r e d u c t i o n i s p o s s i b l e and

t h e s i t u a t i o n i s much more complex. a somewhat s i m i l a r s i t u a t i o n arises.

When studying general a c t i o n s o f groups, I n t h e compact case every o r b i t w i l l be

closed, t h e space o f o r b i t s w i l l have a reasonable s t r u c t u r e ,

and one can

o f t e n f i n d n i c e ( w i t h respect t o t h e a c t i o n ) neighborhoods o f o r b i t s .

A large

amount o f i n f o r m a t i o n about actions o f f i n i t e and compact groups has been obtained by t o p o l o g i c a l methods.

However, once again, i f t h e compactness

assumption on t h e group i s dropped, one faces many a d d i t i o n a l problems.

In

p a r t i c u l a r , one can have o r b i t s which are dense ( f o r example, t h e i r r a t i o n a l f l o w on t h e t o r u s ) and t h e o r b i t space may be so badly behaved as t o have no continuous f u n c t i o n s b u t constants.

Furthermore, moving from a p o i n t t o a

nearby p o i n t may produce an o r b i t which doesn't f o l l o w c l o s e l y t o t h e o r i g i n a l

If one wishes t o deal w i t h a c t i o n s i n the non-compact case, t h i s

orbit.

phenomenon o f complicated o r b i t s t r u c t u r e must be faced. e.g.,

For many actions,

d i f f e n t i a b l e a c t i o n s on manifolds, t h e r e are n a t u r a l measures t h a t

behave w e l l w i t h respect t o t h e action.

A s i g n i f i c a n t p a r t o f ergodic theory

i s t h e study o f group a c t i o n s on measure spaces.

I n p a r t i c u l a r , ergodic

t h e o r y aims t o understand t h e phenomenon o f bad o r b i t s t r u c t u r e i n t h e presence o f a measure. Throughout these l e c t u r e s , G w i l l be a l o c a l l y compact, second countable group.

S

x

G

Let +

S

(S,u)

be a standard measure space, and assume we have an a c t i o n

which i s a Rorel f u n c t i o n .

Then

u

(which i s always assumed t o be

a-finite)

i s invariant i f

quasi-invariant i f

=

v(A)

f o r a l l A c S and

v ( A ~ )= 0 i f and only i f

v(A) = 0 o r

-

u(S

g € 6 , and

v ( A ) = 0.

The action i s c a l l e d ergodic i f A

Definition 1.1: implies

11(Ag)

C

i s G-invariant

S

A) = 0.

Clearly any t r a n s i t i v e action i s ergodic, o r , more generally, any t r a n s i t i v e on t h e complement of a null

e s s e n t i a l l y t r a n s i t i v e action ( i .e., set).

We can then w r i t e

S

=

GIG0 where GO c G is as closed subgroup.

An

ergodic act ion t h a t i s not essenti a1 1y t r a n s i t i v e will be c a l l e d properly ergodi c

.

Example 1.2.

Let

S

=

Iz

E

CI Izl

=

1)

and

T : S + S be T(z) = e l a z

If A c S is invariant,

a1211 i s i r r a t i o n a l . Then T generates a Z-action. let

xA(z) =

function. aneina

=

1 anzn

where

be t h e ~ ' - ~ o u r i e rexpansion of i t s c h a r a c t e r i s t i c

Then by invariance xA(z) = XA(eiaz) = an and so an = 0

r

f o r n # 0. This implies

Thus

a neinazn.

xA i s constant, so t h e

action is properly ergodic.

Remark:

I f S i s a (second countable) topological space and

11

i s p o s i t i v e on

open s e t s , then proper ergodicity imp1 i e s almost every o r b i t i s a dense null set.

This i s one sense i n which proper e r g o d i c i t y i s a r e f l e c t i o n of compli-

cated o r b i t s .

Another i s t h e following.

Propositon 1.3 [127.

Let G a c t continuously on S where S i s metrizable

by a complete separable metric.

Then t h e following a r e equivalent:

t h e action i s "smooth" i f they hold.) i)

Every G-orbit is l o c a l l y closed

ii)

SIG i s To i n t h e quotient topology

(We say

iii)

The q u o t i e n t R o r e l s t r u c t u r e on S/G i s c o u n t a b l y separated and

!I.e.,

generated.

t h e r e i s a countable family

{Ail

separating

p o i n t s and g e n e r a t i n g t h e R o r e l s t r u c t u r e . ) Every q u a s i - i n v a r i a n t e r g o d i c measure i s s u p p o r t e d on an o r b i t .

iv) Proof. let Then

(i)

p:S + S/G v = P,(~)

(i)

t h e p r o j e c t i o n , and

-

i s supported on a p o i n t , so (iv)

a r e elementary.

u

To see ( i i i )

(iv),

an e r g o d i c p r o b a b i l i t y measure on S.

i s a measure on S/G w i t h t h e p r o p e r t y t h a t f o r any Bocel s e t

v(R) = 0 o r 1.

B C S/G,

(iii)

S i n c e S/G i s c o u n t a b l y s e p a r a t e d and generated, p

i s s u p p o r t e d on an o r b i t .

v

The i m p l i c a t i o n

( i ) i s d i f f i c u l t (and we w i l l n o t be u s i n g i t ) .

We w i l l be making c o n s t a n t use o f t h e i m p l i c a t i o n ( i )

(iv).

For

example : C o r o l l a r y 1.4.

F v e r y e r g o d i c a c t i o n o f a compact group i s e s s e n t i a l l y

transitive.

I f t h e a c t i o n i s on a m e t r i c space, t h i s f o l l o w s immediately.

However, a

theorem o f V a r a d a r a j a n [451 i m p l i e s t h a t any a c t i o n can be so r e a l i z e d . C o r o l l a r y 1.5.

Every e r g o d i c a l g e b r a i c a c t i o n o f a r e a l ( o r p - a d i c )

a l g e b r a i c group (more p r e c i s e l y , t h e r e a l o r p - a d i c p o i n t s ) on an a1 g e b r a i c variety i s essentially transitive. T h i s f o l l o w s f r o m t h e theorem o f Bore1 and R o r e l - S e r r e t h a t o r b i t s a r e l o c a l l y c l o s e d L3-l [ 6 ! . W h i l e t h e decompositon o f a general a c t i o n i n t o o r b i t s may n o t he s a t i s f a c t o r y t h e r e i s always a good decompositon i n t o e r g o d i c components. P r o p o s i t i o n 1.5. measure space

( E ,v)

Let

,a

( 5 , ~ ) be a G-space.

Then t h e r e i s a s t a n d a r d

c o n u l l 6 - i n v a r i a n t s e t YO C S, and a 6 - i n v a r i a n t

Bore1 map

q:S

+

E

with

cp,(u) = v

such t h a t , w r i t i n g

where

u i s supported on v - ' ( ~ ) , we have Y ergodic under G f o r almost a l l y.

u Y

p

=

$ uy

~v(Y)

i s q u a s i - i n v a r i a n t and

(E,v) i s c a l l e d t h e space o f ergodic components o f t h e a c t i o n (and i s e s s e n t i a l l y uniquely determined by t h e above conditions.) We now discuss some n o t i o n s o f "isomorphism".

D e f i n i t i o n 1.7

Let

(S,u),

(S'

j u g a t e i f modulo n u l l s e t s t h e r e i s

i)

(p

(p

be ergodic G-spaces. :S

+

C a l l them con-

S' w i t h

a b i j e c t i v e Rorel isomorphism.

- p'

ii)

,p8 )

(i.e.,

c~,(v)

same n u l l s e t s ) .

i i i ) (p(sg) = (p(s)g.

If A defining

E

Aut(6)

and

s o g = s

S

i s a G-space,

we have a new G-action on

S by

A(g).

D e f i n i t i o n 1.8.

C a l l two a c t i o n s automorphically conjugate i f they

become conjugate when m o d i f i e d by some automorphism.

An a p r i o r i much weaker n o t i o n i s simply t o ask f o r t h e o r b i t p i c t u r e s t o Here, we can compare a c t i o n s o f d i f f e r e n t groups.

be t h e same.

D e f i n i t i o n 1.9.

Suppose

(S,p)

i s a G-space,

(S',p8)

a 6'-space.

t h e a c t i o n s o r b i t e q u i v a l e n t i f (modulo n u l l s e t s ) t h e r e e x i s t s with

i) ii) iii)

v a b i j e c t i v e Rorel isomorphism. v*(P) (p

-

M'.

(G-orbit) = GI-orbit.

v :S

+ S'

Call

If v:(X,v)

+

i s a measure c l a s s p r e s e r v i n g G-map o f G-spaces we

(Y,v)

c a l l X an e x t e n s i o n o f Y o r Y a f a c t o r o f X. have

v(Y

-

v ( X ) ) = !!.

If H

C

Observe t h a t we a u t o m a t i c a l l y

G i s a subgroup, and X i s an e r g o d i c 6-space,

we can r e s t r i c t t o o b t a i n an a c t i o n o f H, which o f course no l o n g e r need be ergodic.

I n t h e o t h e r d i r e c t i o n , we can induce.

ergodic H-space and Y C S

i s a closed subgroup.

associated 5-space as f o l l o w s . and l e t

X = (S

x

Namely, suppose S i s an

Let

H

a c t on

Then G acts on

G)/'il.

t h i s a c t i o n comnutes w i t h t h e H-action.

S

x

S

Then we o b t a i n a n a t u r a l l y x

G

G by

by (s,g)h = (sh,gh) (s,g)G = (s,G-lg),

and

Hence t h e r e i s an induced a c t i o n o f G

on X which w i l l be e r g o d i c w i t h i t s n a t u r a l measure class.

D e f i n i t i o n 1.10.

i s c a l l e d t h e ergodic G-space induced from t h e G-

X

action, and we denote i t by

F o r example,

(S

x

[0,1!)/-

indG(7).

i f !-I = Z, G = R, then X can be i d e n t i f i e d w i t h

where

-

i d e n t i f i e s (s,l)

w i t h (Ts,O).

IJnder t h e induced R-

a c t i o n a p o i n t simply flows up along. t h e l i n e i t i s i n w i t h u n i t speed. Given an e r g o d i c S-space X, an a c t i o n o f a subgroup.

The f o l l o w i n g i s h e l p f u l i n t h i s regard.

P r o p o s i t i o n 1.11 C52-J. closed subgroup, t h e n X = f a c t o r o f X,

i-e.,

i t i s u s e f u l t o know when i t i s induced from

I f X i s an ergodic 5-space and H C G C ind;(S)

f o r some H-space S i f and o n l y i f G/H i s a

t h e r e i s a measure c l a s s preserving G-map

I f X i s a S-space,

is a

X + G/H.

i s t h e r e a unique (up t o conjugacy) smallest closed

subgroup from which- i t i s induced?

The answer i n general i s no, b u t we have

the following P r o p o s i t i o n 1 - 1 2 [547.

Yuppose G i s ( t h e r e a l p o i n t s , o r k-points,

p-adic f i e l d ) of an a l g e b r a i c group and X an ergodic 6-space.

k a

Then t h e r e i s a

unique conjugacy class of algebraic subgroups such that algebraic (and some S),

X = i n d z ( ~ ) for H

if and only if H contains a member of t h i s conjugacy

class. nefinition 1.13 [541. of the action.

If H i s in t h i s class, call H the algebraic hull

If t h i s i s all of G , call the action Zariski dense.

If X = GIG", then the algebraic hull i s just the usual algebraic hull of the group Go.

Proof of 1.12.

There e x i s t minimal such groups from the descending chain

condition on algebraic subgroups. algebraic groups. We have VJ =

( q l , q2):X

+

measure on G/H1

v,(v)

x

G/H1

x

G/H2.

qi:X G/Y2.

+

G/Hi.

Then

C G are two such minimal

Let v , ( ~ ) i s an ergodic quasi-invariant

R u t the G-action on t h i s product i s algebraic, so

i s supported on an orbit.

G / ( ~ ~ HC, ~ g2~2g;1). ~ ; ~

Suppose H1,H2

R u t as a G-space, an o r b i t i s

By d n i m a l i t y assumptions, q1 and Hp are conjugate.

Theorem 1.14 (Bore1 Density Theorem [43).

I f G i s a connected semisimple

real algebraic group with no compact f a c t o r s , and S an ergodic 6-space with f i n i t e invariant measure, then S i s Zariski dense in G. As an example of an ergodic action of such a group, we point out the following example.

(One can show there are uncountably many inequivalent

actions of such groups [with f i n i t e invariant measure'.)

Example 1.15.

Let SL(n,Z) act on R ~ / zby~ automorphisms.

This i s ergodic.

The induced SL ( n ,R) act ion wi 11 be properly ergodi c , essentially f r e e ( i .e. almost a l l s t a b i l i z e r s t r i v i a l ) , and have f i n i t e invariant measure.

2.

E r g o d i c i t y Theorems A n a t u r a l c l a s s o f actions t h a t a r i s e s i n a v a r i e t y o f s i t u a t i o n s are

a c t i o n s on homogeneous spaces.

Thus, i f H1,H2 C G

are subgroups w i t h Hz

closed, H1 a c t s on G/H2 and t h e question a r i s e s as t o when t h i s i s ergodic. This i s a s p e c i a l case o f t h e f o l l o w i n g question. space and H C G ergodic?

i s a subgroup.

Suppose S i s an ergodic G-

When w i l l t h e r e s t r i c t i o n t o H s t i l l be

I n t h e s p e c i a l case i n which S has a f i n i t e i n v a r i a n t measure,

r e s u l t s about u n i t a r y r e p r e s e n t a t i o n s can be d i r e c t l y appl ied. (ugf)(s) = f(sg],

Vamely, l e t

where f r ~ ~ ( 5 ) This . defines a unitary representation o f G

on L ~ ( s )and G i s ergodic on S (assuming f i n i t e i n v a r i a n t measure) i f and only Thus, t o s e t t l e

i n v a r i a n t vectors i n L ~ ( s ) @ C.

i f t h e r e are no "on-zero

t h e question about e r g o d i c i t y o f r e s t r i c t i o n s i n t h i s case, we have a representation U

g

o f G w i t h no i n v a r i a n t vectors and we ask whether o r n o t

i n v a r i a n t vectors.

UIH has

L e t ' s consider some o f t h e c l a s s i c a l examples, when G i s

t r a n s i t i v e on 5 .

Example 2 .l.Suppose 6 i s compact, S. = G. and o n l y if ti i s dense.

Then H

C

G

This i n c l u d e s Example 1.2.

Now l e t Y be a simply connected n i l p o t e n t L i e group,

'(i a ;)I

r i s d i s c r e t e and N/r

(i-e.,

N=; Then

N/r

+

x,y,z

torus. [N,N!/[N,Y1

r

and

e x h i b i t s t h e 3-manifold

I n general

n r.

V/r

+

V/TN,Vlr

r C N a lattice

has f i n i t e i n v a r i a n t measure).

~it)and

cN,Y1 = i A r N I X = y = 01, N/!N,Nlr

i s ergodi c on S i f

= Nz,

V/[N,Nlr N/r

For example,

t h e subgroup w i t h x,y,z i s a torus.

1.

The map

as a c i r c l e bundle over t h e

w i l l he a bundle over t h e t o r u s w i t h f i b e r

Theorem 2.2 i s ergodic on

(L. Green '17).

H c Y i s ergodic on

N/r

i f and o n l y i f i t

N/[N,Nlr.

As t h e l a t t e r i s a t o r u s , e r g o d i c i t y can be determined as i n Example 2.1. The p r o o f o f t h i s depends on w r i t i n g down t h e r e p r e s e n t a t i o n s o f N which appear i n L [ ( N / ~ ) and examining them w i t h respect t o r e s t r i c t i o n t o subgroups.

See

C11 f o r d e t a i l s .

Results f o r 1-parameter subgroups a c t i n g on compact homogeneous spaces o f s o l v a b l e L i e groups have been obtained by Auslander r 2 1 and R r e z i n and Moore

L71. r C G i s a l a t t i c e i n G and H C 6 i s t h e group o f

I f G = SL(2,R),

p o s i t i v e diagonal matrices, then

G/r

i s i n a n a t u r a l way t h e u n i t tangent

bundle o f t h e f i n i t e volume n e g a t i v e l y curved m a n i f o l d

D/r

S O ( ~ , R ) \ G i s t h e Poincare disk, and H i s t h e geodesic flow. classical [211.

where D = Thus, a

r e s u l t o f Hedlund and Hopf says t h a t Y i s e r g o d i c on

G/r

f207

Moore g e n e r a l i z e d t h i s t o a l l o w G t o be a very general semisimple

C.C.

L i e group, and H t o be an a r b i t r a r y subgroup. Theorem 2.3.

(C.C.

Moore [32!).

Let

G

=

nPJi

where Gi i s a non-compact

connected simple L i e group w i t h f i n i t e c e n t e r and l e t ducible l a t t i c e .

Then H C I; i s ergodic on

G/r

r C G be an i r r e -

i f and o n l y i f ;ii i s n o t

compact. T h i s theorem was proved by showing t h e f o l l o w i n g general r e s u l t about a r b i t r a r y r e p r e s e n t a t i o n s ( n o t n e c e s s a r i l y one appearing i n

L'(G/~)).

Let G

be a non-compact connected simple L i e group w i t h f i n i t e c e n t e r and n a u n i t a r y r e p r e s e n t a t i o n o f G w i t h no non-zero i n v a r i a n t vectors. x

*

0, {g

theorem.

E

GI n ( g ) x

= X}

i s compact.

Then f o r any v e c t o r

This r e s u l t easily implies the

A s t r o n g e r r e s u l t about such r e p r e s e n t a t i o n s t h a t we w i l l need has

subsequently come t o l i g h t .

Theorem 2.4.

i s any u n i t a r y r e p r e s e n t a t i o n o f a connected

If n

non-compact simple L i e group w i t h f i n i t e center, then t h e m a t r i x c o e f f i c i e n t s f (g) =

+

as

O

g

-

+

,

assuming t h e r e are no n ( G ) - i n v a r i ant

vectors.

A n i c e p r o o f o f t h i s appears i n a paper o f Howe and b o r e [221 although t h e basic i d e a i s present i n t h e work of Sherman [43].

(See a l s o [491.)

i d e a o f t h e p r o o f i s t o l e t G = KAK be a Cartan decomposition. compact, i t s u f f i c e s t o see

+

0

as

a +

-.

so t h a t A i s t h e p o s i t i v e diagonals.

G = SL(2,R),

triangular

f(a)

2

x

2

Since K i s

Consider t h e example L e t P be t h e upper

m a t r i c e s i n G w i t h p o s i t i v e diagonal e n t r i e s .

r e p r e s e n t a t i o n t h e o r y o f ? i s we1 1 known.

The

The

There are 1-dimensional repre-

s e n t a t i o n s which f a c t o r through [P ,P] and 2 i n f i n i t e dimensional representat i o n s induced from [P,P].

For t h e l a t t e r , i t i s c l e a r t h a t t h e r e s t r i c t i o n o f

a r e p r e s e n t a t i o n t o A i s j u s t t h e r e g u l a r r e p r e s e n t a t i o n o f A f o r which i t i s c l e a r t h a t m a t r i x c o e f i c i e n t s vanish a t

Thus i t s u f f i c e s t o see t h a t

m.

r ) P has a s p e c t r a l decomposition which assigns measure 0 t o t h e 1-dimensional r e p r e s e n t a t i o n s .

Rut i f i t assigned p o s i t i v e measure, [P ,PI

would have t o l e a v e a v e c t o r f i x e d , say v. b i - i n v a r i a n t under [P,P!

= N.

Then ~ ( g )= < r ( g ) v ( v >

G/N can be i d e n t i f i e d w i t h R~

-

o r b i t s on G/N a r e t h e h o r i z o n t a l l i n e s except f o r t h e x-axis,

I01 , and t h e N and s i n g l e

A continuous f u n t i o n on GIN constant on t h e o r b i t s must

p o i n t s on t h e x-axis.

c l e a r l y be constant on t h e x-axis as w e l l . a l l g~ P, and s i n c e

would be

n

This t r a n s l a t e s i n t o cp (g) = 1 f o r

i s unitary, v i s P-invariant.

under P , and s i n c e P has a dense o r b i t on GI?,

P (g) =

Thus

(p

i s bi-invariant

1 f o r a1 1 g G, showing

t h a t v i s G-invariant. We t h u s have good i n f o r m a t i o n about some basic examples f o r t h e question o f e r g o d i c i t y o f a c t i o n s on homogeneous spaces o f f i n i t e i n v a r i a n t measure. For t h e general homogeneous space we make use o f t h e f o l l o w i n g observation.

P r o p o s i t i o n 2.5.

[49?

I f 5 i s an e r g o d i c %space (general quasi-

i n v a r i a n t measure) and H C G i s a closed suhgroup, t h e n H i s ergodic on S i f and o n l y if G acts e r g o d i c a l l y on the product A c G/H

To see t h i s , suppose Ax= { s

E

51 (x,s)

S

S.

x

i s G-invariant.

For each x c G/H,

let

Ry quasi-invariance one e a s i l y sees t h a t A and a l l Ax

A].

E

x

G/H

i s an H-

a r e simultaneously e i t h e r n u l l , o f n u l l complement, o r n e i t h e r .

i n v a r i a n t set, and c l e a r l y any H - i n v a r i a n t s e t B C S i s o f t h e form B = !A ,[ f o r some G - i n v a r i a n t A. C o r o l l a r y 2.6. ergodic on

[321

If

r, H C G a r e closed subgroups, t h e n H i s

G/r i f and o n l y i f

r i s ergodic on $/H.

T h i s enables us t o use i n f o r m a t i o n about e r g o d i c i t y o f r e s t r i c t i o n s on spaces f o r which t h e r e i s a f i n i t e i n v a r i a n t measure t o o b r t a i n r e s u l t s i n t h e case no such measure e x i s t s . C o r o l l a r y 2.7.

(Moore)

t r a n s i t i v e G-space,

then

G =nGi,

r as i n Theorem 2.3.

If S i s a

r i s ergodic on S i f and o n l y i f t h e s t a b i l i z e r s i n

G o f p o i n t s i n S are n o t compact.

F_u.ample 2.8.

(Moore)

SL(n,Z)

i s ergodic on R ~ , n

2.

This follows since

i s e s s e n t i a l l y t r a n s i t i v e on R~ and t h e s t a b i l i z e r s i n t h e o r b i t o f

SL(n,R)

f u l l measure are n o t compact.

Example 2.9.

Consider t h e a c t i o n o f SL(2,R)

SL(2 ,R)/S0(2 ,R)

.

T h i s a c t i o n extends t o t h e boundary c i r c l e ,

can be i d e n t i f i e d w i t h SL(Z,R)/?,

G.

If

r

C SL(2,R)

on t h e Poincare d i s k

where P i s t h e upper t r i a n g u l a r m a t r i c e s i n

i s a t o r s i o n free l a t t i c e , then

r

acts i n a properly

discontinuous f a s h i o n on t h e disk, and t h e q u o t i e n t space surface o f f i n i t e volume.

and t h e boundary

n/r

i s a Riemann

On t h e o t h e r hand, s i n c e P i s n o t compact, t h e

action o f -

r

on t h e boundary will be properly ergodic.

More generally, i f G

i s any semisimple Lie group and P C G i s a minimal parabolic subgroup, then G/P i s t h e unique compact G-orbit i n t h e boundary of a natural compactif i c a t i o n of t h e symmetric space X = GI!?, K C G maximal compact.

r

i s ergodic on G/P.

Yere again,

r on homogeneous spaces

Thus these ergodic actions of

of G a r i s e very n a t u r a l l y in a geometric s e t t i n g , and t h e study of t h e s e ergodi c actions i s extremely useful in understanding

r.

Since t h i s i s such an important example, l e t us point out t h a t f o r G / P compact (e.g. P a parabolic) t h a t ergodicity of i n a much l e s s s o p h i s t i c a t e d fashion. vector in

2

L (G/r) Q C

r on G / P can be demonstrated

Vamely i f t h e r e i s

a ?-invariant

then t h e r e i s a compact G-orbit in t h e Hilbert

4s i s well known, t h i s implies t h a t t h e r e e x i s t f i n i t e dimensional

space.

subrepresentations, which f o r G , i t i s a l s o well known, must be t h e identity.

This i s impossible.

Corollary 2.7 d e a l s with t h e r e s t r i c t i o n of t r a n s i t i v e G-actions t o

r.

We now deal with t h e properly ergodic case. Theorem 2.10 c491.

If G

groups with f i n i t e c e n t e r , ergodic S-space, then Proof. let

n

C G an i r r e d u c i b l e l a t t i c e and S

i s a properly

r i s ergodic on S.

AS = {x e

a s e t of p o s i t i v e measure.

Let A C S

x

G/r be invariant.

G/r(

(s,x)

E

A}.

We can suppose

L [ ( G / ~ ) , then

o:S

+

O

O on

i s a quasi-invariant ergodic measure on S .

o,(u)

weakly as

+

5, @ ( s ) = f S i s a

B u t by vanishing of t h e matrix c o e f f i c i e n t s (Theorem 2.4), f o r +

fS

Invariance of A i s e a s i l y seen t o imply t h a t i f we

t h e unitary representation of S on

w-g

For each s,

be t h e unit ball and l e t G a c t on t h e r i g h t i n 6 v i a

R C L2(~/rQ ) C.

Then

Gi connected non-compact simple Lie

be t h e image under orthogonal projection of t h e charac-

fse LZ(G/r) Q C

G-map.

,

Gi

Suppose not.

t e r i s t i c function of

let

r

=

g

+ m.

we R,

This implies G-orbits in R a r e l o c a l l y closed,

i.e.

t h e a c t i o n i s smooth.

so we can suppose

@:S

It f o l l o w s t h a t

GIGo

+

i s supported on an o r b i t ,

where GO i s t h e s t a b i l i z e r o f a p o i n t i n t h i s

G S = indG (SO) where So i s an ergodic GO space. But Go 0 i s compact, so Go i s t r a n s i t i v e on So. T h i s i m p l i e s G i s t r a n s i t i v e on S,

orbit.

This imp1 i e s

which c o n t r a d i c t s our hypotheses. S i m i l a r r e s u l t s can be proven f o r o t h e r groups f o r which t h e r e i s a v a n i s h i n g theorem f o r m a t r i x c o e f f i c i e n t s . Theorem 2.11 r501. e r g o d i c G-space.

L e t G be an e x p o n e n t i a l s o l v a b l e L i e group and S an

[G,G1

Suppose

i s ergodic on S.

r

Then

i s a l s o e r g o d i c on

r c 6.

S f o r every cocompact

The p r o o f uses t h e r e s u l t o f Howe and b o r e [22] t h a t f o r such a group, the matrix coefficients P

= {gln(g)

3. Cocycl es



i s scalar).

Here

+

0

as

g

+

-

in

.

measurable f u n c t i o n s

(g

a n u l l set.

" t w i s t e d " actions.

the l e f t ) ,

E

X

+

6 a c t s on

f ) ( x ) = f(xg).

and f o r f

Y, X

x

by

Y

-

g = (xg,y)

g = (xg, y

and on F(X,Y)

by

a(x,g))

where

ct(x,g)

c

H,

(where f o r convenience we u s u a l l y t a k e H t o be a c t i n g on For these t o d e f i n e actions, w need t h e

(9-f)(x) =a(x,g)f(xg).

a:X x G

(x,y)

I f Y i s a l s o an H-space f o r some group H, we can d e f i n e

Namely, (x,y)

F(X,Y)

be t h e space o f

two f u n c t i o n s b e i n g i d e n t i f i e d i f t h e y agree

f o l l o w i n g compatibility condition: function

where

n i s assumed i r r e d u c i b l e .

I f X i s a G-space and Y a Borel space, l e t F(X,Y)

off

G/P,

+

a(x,gh)

= a(x,g)a(xg,h).

H w i l l be c a l l e d a cocycle.

Such a B o r e l

(The q u e s t i o n as t o whether

t h i s holds everywhere o r almost everywhere i s an important t e c h n i c a l p o i n t which we w i l l not discuss.

See c411.j

When endowed w i t h t h i s a c t i o n we s h a l l

denote

X

x

by

Y

Y

Y.

xu

a H i l b e r t space, H = U ( K ) , t h e u n i t a r y

If Y =

group o f t h e Y i l b e r t space, and t h e measure on X i s i n v a r i a n t , then t h e a-twisted

a c t i o n on F(X,Y)

ua.

representation

2

L (X;K )

restricts t o

X

I f a,B:

x G

t o y i e l d a unitary

H are cocycles t h e r e i s a c e r t a i n

+

r e l a t i o n which immediately imp1 i e s equivalence o f t h e a c t i o n s o r representations. a(x,g)

Namely i f we have a Bore1 map

= q(x) ~ ( x , g ) q(xg)-I,

: X + H

(p

t h i s w i l l be t h e case.

equivalent, o r cohomologous, and w r i t e

such t h a t

We then c a l l

a

and B

- B.

a

To get some f u r t h e r f e e l i n g f o r t h i s notion, consider t h e case

X = GIG0.

If

GIGg

a:

homomorphism GO

+

G

x

Y.

+

H i s a cocycle, then

Namely, l e t

(x,g)

E

GIGn

x

C,

y: GIGO y(xg)

( x ) g ( x g ) c Go

(x,g)

+

y(x)g y(xg)-l

xGO

+

a(x,g)

a r i s e s from a cocycle

H

g

i s a cocycle

a i n this

Then f o r

are equal when p r o j e c t e d t o GIGo.

We can suppose

yields the identity

homomorphism,

+

be a Rorel section.

G

and y ( x )

Thus

[el

defines a

Equivalent cocycl es y i e l d conjugate homomorphi sms.

Furthermore, every homomorphism Go way.

a(Ce1 x GO

y([el)

GIGo

G

x

= e,

* GO which when r e s t r i c t e d t o

Thus i f

G o + Go.

= n(y(x)gy(xg)-l)

and t h e n

n: G o + H ' i s a

i s t h e r e q u i r e d cocycle.

Thus we

have Theorem 3.1.

a

classes o f cocycles Go

+

+

a ( [eJ

GIGo

x

G

x +

5, H

d e f i n e s a b i j e c t i o n between equivalence and conjugacy classes o f homomorphisms

H.

We remark t h a t i f H = U(K ) , and n: Go

we have an associated cocycle representation

ua

a: G I G o

o f G on L?(G/G,;

x

G

K).

+

H,

Y

i s a u n i t a r y representation, and then an associated

O f course

f o r t h i s approach t o induced representations. We now consider some o t h e r exam~les.

+

ua

= i n d G (n). Gr)

See [45!

Fxample 3.2 a(x,g) Go

+

a) I f

h:G

i s a homomorp+ism, X a G-space, t h e n

H

+

i s a cocycle.

= h(g)

I f X = GIG0,

Thus i n general we s h a l l sometimes c a l l

which i s simply h/Go.

H,

the r e s t r i c t i o n o f h t o

X

x

t h i s corresponds t o a homomorphism

G

and w r i t e

a = h l X x G.

b ) Suppose X i s an e r g o d i c 6-space w i t h q u a s i - i n v a r i a n t measure

t h e Radon-Nikodym d e r i v a t i v e .

rIJ(x,g) = dp(xg)/dp(x), r :X IJ

x

~ I = J

fdv,

r IJ

G

f

- rv'

>

IJ

-

v,

so

i.e.

I n p a r t i c u l a r , t h e r e i s an e q u i v a l e n t

i n v a r i a n t measure i f an o n l y i f t h e cocycle i s t r i v i a l (i.e.

equivalent t o the i d e n t i t y

a(x,g)

c ) Suppose X i s a 6-space, o r b i t equivalent, w i t h X

x

8 ( x ) a(x,g)

=

1).

X ' a f r e e 6'-space,

8: X + X '

and t h a t t h e a c t i o n s a r e

t h e o r b i t equivalence.

8(x) and ~ ( x g )are i n t h e same 6 ' - o r b i t ,

G,

= ~(xg) for

I f G = G',

a(x,g)

r 6'.

Then a: X

x

6

Then f o r say

+

6'

i s a cocycle.

we have t h e f o l l o w i n g .

P r o p o s i t i o n 3.3 automorphism A

C551 I f

Aut(G) t o

X

a x

6,

i s e q u i v a l e n t t o t h e r e s t r i c t i o n o f an t h e n X and Xi a r e a u t o m o r p h i c a l l y con-

I f t h i s automorphism i s inner, t h e n X and Xi a r e conjugate.

jugate.

Proof. el(xg)

Let

Therefore t h e cohomology c l a s s we o b t a i n does n o t depend upon t h e

a-finite

P

If

0, t h e n d ~ ~ ( x g ) / d p ( x =) f(x)-l(dv(xg)/dv(x))f(xg)-l,

measure, o n l y t h e measure class.

(x,g)

IJ.

The c h a i n r u l e imp1 i e s

R+ i s a cocycle, c a l l e d t h e Radon-Nikodym cocycle.

+

a

If a

= ol(x)A(g),

e2(x) = el(x)h.

) = ( x ) ~ ( g ) ( x g ) ~ ,t h e n

el(x)

so we have automorphic conjugacy.

= e(x) a(x)

satisfies

I f A(g) = hgh-l,

let

T h i s i s t h e n a 6-map.

There are many o t h e r n a t u r a l l y a r i s i n g s i t u a t i o n s i n which cocycles appear, b u t we s h a l l n o t have t i m e t o d i s c u s s them here.

Instead, we t u r n t o

an important i n v a r i a n t attached t o a cocycle, namely t h e Mackey range. a:S S

x

x

G H

+

H

where !i i s a l s o l o c a l l y compact.

Let

Form t h e t w i s t e d G-action

where we view H as a c t i n g on i t s e l f by r i g h t t r a n s l a t i o n s .

Y also

acts on

S

by

H

x

t h e G-action. and a = i( S

(s,h)

ho = (s,hglh),

i:G

Note t h a t i f G,

x

+

and t h i s H a c t i o n commutes w i t h

i s an embedding o f G i n t o a l a r g e r group

H

t h i s i s e x a c t l y t h e s i t u a t i o n i n t h e i n d u c i n g procedure.

As i n t h e l a t t e r , we o b t a i n an a c t i o n o f H on t h e space o f G - o r b i t s .

Rut t h i s

space may n o t be a decent measure space, so instead, we l e t X be t h e space o f G-ergodic components o f t h e a c t i o n o f G o f

Then H w i l l a c t ori X as

S xaH.

w e l l , and t h i s w i l l be an ergodic H-action.

If

D e f i n i t i o n 3.4.

a: S

G

x

+

w i l l be c a l l e d t h e h c k e y range o f

Example 3.5

a) I f

and a(s,g)

= i( g ) ,

H

a.

i s a cocycle, t h e associated H space X This i s a cohomology i n v a r i a n t o f a.

i : G + H i s an embedding o f G as a closed subgroup o f H,

i.e.

a = i IS

x

G,

t h e n t h e h c k e y range o f

a

is

.

ind: ( G ~ )

e:X

b) I f

+

X'

i s an o r b i t equivalence,

a:X

x

G

+

G'

t h e associated

cocycle, then t h e Mackey range i s t h e GI-space X ' . c ) I f S = G/Go and a Mackey range o f

a:G/Go

x

corresponds t o a homomorphism n:Go + H, G

+

H

i s t h e H-space

then t h e

H / w .

F i n a l l y , t h e f o l l o w i n g r e l a t e s t h e Mackey range t o t h e cohomology c l a s s

P r o p o s i t i o n 3.6.

-

I f a:S

G

x

+

H,

t h e f o l l o w i n g are e q u i v a l e n t .

G) c Ho , Ho c H

i)

a

ii)

H/Ho i s a f a c t o r o f t h e Mackey range.

$

where

H i i i ) X = i n d (So)

Ho

For a proof, see [47],

[52!.

$(S

x

a closed subgroup.

f o r some So, where X i s t h e Mackey range.

4. Generalized D i s c r e t e Spectrum Suppose

(S,p)

i s an ergodic space w i t h

p

f i n i t e and i n v a r i a n t .

In

t h i s l e c t u r e we t r y t o see what t h e a l g e b r a i c s t r u c t u r e o f t h e r e p r e s e n t a t i o n n

o f G on L'(s)

says about t h e geometric s t r u c t u r e o f t h e action.

D e f i n i t i o n 4 .l. We say t h a t t h e a c t i o n has d i s c r e t e spectrum i f n

is

t h e d i r e c t sum o f f i n i t e dimensional i r r e d u c i b l e subrepresentations.

Example 4.2.

L e t K be a compact group, H a closed subgroup, and cp :G

homomorphism w i t h q(G) dense i n K.

L e t G a c t on K/H by

+

a

K

[ k l - g = [kcp (g)].

Then t h i s a c t i o n has d i s c r e t e spectrum. Theorem 4.3.

(von Neumann-Halmos- Mackey)

.

These are a l l t h e examples.

That i s , i f S i s a G-space w i t h d i s c r e t e spectrum, t h e n t h e r e e x i s t s a compact group K, a closed subgroup H, and a homomorphism cp:G + K w i t h dense range such t h a t S and K/H are conjugate G-spaces. T h i s was o r i g i n a l l y proved by von Neumann and Halmos f o r G = Z o r R, and by FBckey [261 f o r general G. Let s i onal

.

2 L (2) = Let

d

Wi

We sketch Mackey's p r o o f .

where Wi

are n(G)

t h a t K i s a l l 0 compact.

Further,

and f i n i t e dimen-

n(g)Fhr(g)- = M,

n:G

+

R.

Let

K =

m,

so

L e t M be t h e a b e l i a n von Neumann algebra on L*(s)

c o n s i s t i n g o f m u l t i p l i c a t i o n by elements o f

a l l TcK.

invariant

B = nU(Wi ) , t h e product o f t h e associated u n i t a r y groups, which

i s a compact subgroup o f u(L'(s)).

1

-

L-(S) .

Then c l e a r l y

and by passing t o t h e s t r o n g l i m i t , we o b t a i n ~ t 4 T - l = M f o r

From t h i s one can deduce t h a t each o p e r a t o r T i n K

i s induced by

a p o i n t t r a n s f o r m a t i o n o f S, and thus t h e G-action on S extends t o an a c t i o n o f K.

(There i s some d e l i c a t e measure t h e o r y we a r e i g n o r i n g here i f G i s n o t

discrete.)

Since t h e G-action i s already ergodic, so i s t h e K a c t i o n .

K i s compact, K must a c t t r a n s i t i v e l y ,

so we can i d e n t i f y S z K/H.

Since

Theorem 4.3 can be generalized t o extensions.

X

Namely, suppose

The H i l b e r t

an extension o f e r g o d i c G-spaces w i t h f i n i t e i n v a r i a n t measure.

space L 2 ( x ) n o t o n l y has a n a t u r a l r e p r e s e n t a t i o n o f G on i t , but L'(x) also an

Lm(y)-module

i n a n a t u r a l way.

f u n c t i o n on X and m u l t i p l y . )

A l t e r n a t i v e l y , we can express t h i s by saying

where Wi

-

on L2(x) based on Y.

n

We say t h a t X has r e l a t i v e l y d i s c r e t e spectrum over

[471

2 L (X) = Z e q

Y is

is

(Namely, l i f t a f u n c t i o n on Y t o a

t h a t t h e r e i s a n a t u r a l system o f i m p r i m i t i v i t y f o r D e f i n i t i o n 4.4.

is

Y

+

are G - i n v a r i a n t subspaces t h a t are f i n i t e l y

generated as

L-(Y)

Example 4.5.

Suppose Y i s an ergodic G-space w i t h f i n i t e i n v a r i a n t measure,

a:

Y

x

group.

G+ K Then

spectrum. L'(K/H) have

modules.

i s a cocycle where K i s compact, and H c K i s a closed sub-

X = Y

xu

K/H

i s an extension o f Y w i t h r e l a t i v e l y d i s c r e t e

To see t h i s , observe t h a t L 2 ( x ) = L'((Y);

=

kq

where

+

are f i n i t e - d i m e n s i o n a l

L'(K/H)).

Write

and K - i n v a r i a n t .

We then

2 82 L (X) = z L (Y; Z i ) and L2(y; + ) w i l l be G - i n v a r i a n t since G a c t s from

f i b e r t o f i b e r i n X by an element o f K, and Zi 2 L (Y; Zi)

= Lm(y ,zi)

Theorem 4.6.

i s K-invariant.

Clearly

and t h e l a t t e r i s f i n i t e l y generated over [471.

These are a l l t h e examples.

Lm(y).

X

That i s , i f

+

Y

is

an ergodic e x t e n s i o n w i t h r e l a t i v e l y d i s c r e t e spectrum, t h e n t h e r e e x i s t s a compact group K, a closed subgroup H C K, and a cocycle t h a t as extensions o f Y, X

=

Y

x

a:

Y

x

G

+

K,

such

K/H.

Thus Theorem 4,6 t e l l s us how t o recognize extensions o f the form Y x

K/H

extension.

from i n f o r m a t i o n about t h e u n i t a r y r e p r e s e n t a t i o n o f t h e There i s now a l a r g e r c l a s s o f actions whose " s t r u c t u r e " we know.

D e f i n i t i o n 4 -7 [ 4 8 j .

We say t h a t X has general ized d i s c r e t e spectrum i f

X can be b u i l t from a p o i n t v i a t h e operations o f t a k i n g extensions w i t h

r e l a t i v e l y d i s c r e t e spectrum and i n v e r s e l i m i t s . countable o r d i n a l

a

and f o r each

i)

X0 = p o i n t

ii)

Xu+l

+

o(a

a factor

u




0.

Yowever, n o t every d i s t a l a c t i o n

i f N i s a n i l p o t e n t L i e group and

i s a l a t t i c e , t h e n t h e a c t i o n o f N on

shown i n




2.

irre-

Then

r a r e isomorphic, t h e n G and G ' a r e l o c a l l y

isomorphic. ii)

I n t h e c e n t e r f r e e case, any isomorphism r a t i o n a l isomorphism 6

+

r

+

r'

extends t o a

G'.

T h i s was f i r s t proved f o r cocompact l a t t i c e s by Mostow r367 and f o r non-cocompact 1a t t i c e s by Margul is [271.

I n an e x t r a o r d i n a r y and h i g h 1y

o r i g i n a l and i n n o v a t i v e paper, M a r g u l i s t h e n gave an a l t e r n a t e p r o o f i n [281 which subsumed b o t h cases, gave s t r o n g e r r e s u l t s on t h e e x t e n s i o n o f homomorphisms f r o m

r

t o G, and which was p o w e r f u l enough t o p r o v e t h e a r i t h -

meticity of lattices.

I n 155: we showed how N a r g u l i s ' t e c h n i q u e s c o u l d be

incorporated i n t o a proof o f r i g i d i t y f o r ergodic actions. a l s o t r u e w i t h o u t t h e R-rank assumption as l o n g as t o Mostow [36!

and Prasad t391.

D e f i n i t i o n 6.2

r551.

G

#

Theorem 6.1 i s

PSL(2,R).

T h i s i s due

We now d e s c r i b e r i g i d i t y f o r e r g o d i c a c t i o n s .

5rlppose 6 i s a semisimple connected L i e group w i t h

f i n i t e c e n t e r and no compact f a c t o r s .

An e r g o d i c G space S i s c a l l e d

i r r e d u c i b l e i f e v e r y n o n - c e n t r a l normal subgroup o f G i s a l s o e r g o d i c on S.

r i s a lattice,

(Thus i f

G/r

i s i r r e d u c i b l e i f and o n l y i f

r is

irreducible. ) Theorem 6.3

1551 ( R i g i d i t y f o r ergodic a c t i o n s ) .

L e t G, G' be connected

semisimple L i e groups w i t h f i n i t e center and no compact f a c t o r s , S, S' f r e e i r r e d u c i b l e ergodic G, G ' -spaces, r e s p e c t i v e l y , w i t h f i n i t e i n v a r i a n t measure.

L e t R-rank(G) i)

>

2.

Suppose t h e a c t i o n s a r e o r b i t equivalent.

Then

G and G' are l o c a l l y isomorphic, I n t h e c e n t e r f r e e case, we can t a k e 6 = G',

ii)

and then t h e a c t i o n s

on S and S' a r e automorphically conjugate. Thus, t h i s theorem a s s e r t s t h a t one has behavior t h a t i s d i a m e t r i c a l l y opposed t o t h e hehavior o f a c t i o n s o f amenable groups.

Although theorems 6.1

and 6.3 l o o k r a t h e r d i f f e r e n t , l e t us show t h a t t h e y are b o t h d i r e c t consequences o f t h e f o l l o w i n g theorem. Theorem 6.4

[551.

L e t G, G' be connected semisimple L i e groups w i t h

t r i v i a l center and no compact f a c t o r s , and l e t S be an i r r e d u c i b l e e r g o d i c G space w i t h f i n i t e i n v a r i a n t measure. a:S x G a

+

G'

Assume R-rank(G)

> 2.

Let

be a cocycle whose Mackey range i s Z a r i s k i dense i n G'.

i s e q u i v a l e n t t o a cocycle

Then

that i s the restriction of a rational

8

epimorphism n:G + G'. To deduce theorem 6.1 from theorem 6.4, t h e case

5 = G/r

and

morphism

r

theorem 6.4 y i e l d s t h e f o l l o w i n g theorem o f Margulis.

+

G',

Theorem 6.5

r

(Margulis).

x

Z a r i s k i dense i n G'

G -. 6 ' .

.

G

+

6'

a c o c y c l e corresponding t o a homo-

G, G' as i n Theorem 6.4,

an i r r e d u c i b l e l a t t i c e .

C G

n(r)

a:G/r

one observes t h a t when a p p l i e d t o

Then

Suppose n

n:T

+

G'

(R-rank(G)

> 2),

i s a homomorphism w i t h

extends t o a r a t i o n a l epimorphism

To deduce Theorem 6.3 f r o m Theorem 6.4,

Theorem 6.1 t h e n f o l l o w s .

s i m p l y a p p l i e s Theorem 6.4 t o t h e c o c y c l e equivalence.

a:S

x

6

+

G'

one

cominq f r o m an o r b i t

The h y p o t h e s i s o f Theorem 6.4 a r e s a t i s f i e d s i n c e t h e Mackey

range o f

a

i s t h e GI-space S' and t h e Bore1 d e n s i t y theorem i m p l i e s Z a r i s k i

density.

The c o n c l u s i o n o f 6.4 i m p l i e s t h a t o f 6.1 b y P r o p o s i t i o n 3.3.

L e t us g i v e an example o f how t o a p p l y Theorem 6.3 t o some n a t u r a l examples.

[551.

C o r o l l a r y 6.6 w i t h f i n i t e center, ergodic

r, r'

and t h a t t h e

,

L e t 6, G ' be connected s i m p l e non-compact L i e groups '

C

,

G'

l a t t i c e s and suppose S, S' a r e f r e e

spaces w i t h f i n i t e i n v a r i a n t measure.

r a c t i o n on S and r ' - a c t i o n

5uppose R-rank(G)

> 2,

on S ' a r e o r b i t e q u i v a l e n t .

Then

G and G' a r e l o c a l l y isomorphic.

Proof.

Let

G

X = indr(S),

G'

X' = indr,(5').

Then one e a s i l y checks t h a t

t h e h y p o t h e s i s o f Theorem 6.3 i s s a t i s f i e d .

Example 6.7 [551.

R"Z"Y

As we v a r y n,

n ) 2,

t h e n a t u r a l a c t i o n s o f SL(n,Z)

on

automorphisms a r e m u t u a l l y n o n - o r b i t e q u i v a l e n t .

We now t u r n t o some p r o o f s . r a t h e r o n l y Theorem 6.5,

Ue w i l l n o t prove Theorem 6.4 here, h u t

(which t h e r e f o r e g i v e s us a p r o o f o f Theorem 6.1).

The f i r s t p a r t o f t h e p r o o f we p r e s e n t i s d i f f e r e n t f r o m M a r g u l i s ' o r i g i n a l argument.

I n s t e a d , we p r e s e n t an argument which g e n e r a l i z e s n i c e l y when one

a t t e m p t s t o p r o v e Theorem 6.4. o r i g i n a l argument.

I t i s perhaps a l s o more t r a n s p a r e n t t h e n t h e

The remainder of t h e p r o o f w i l l be t h a t o f Margul i s ,

a l t h o u g h we s h a l l t r y t o g i v e some m o t i v a t i o n . see [551.

F o r t h e p r o o f o f Theorem 6.4,

P r o o f of Theorem 6 -5. IJe have a homomorphism n : able of

r

G'

+

and hence G ' / P t becomes a compact m e t r i z -

On t h e o t h e r hand, a s we o b s e r v e d i n Example 5.4, t h e a c t i o n

r-space.

r

L e t P C G , P1 C G' h e minimal p a r a b o l i c s u b g r o u p s .

on G/P i s e r g o d i c and amenable.

5.2, t h e r e i s a m e a s u r a b l e

r-map

(p:G/P

Ry t h e remarks f o l l o w i n g D e f i n i t i o n

t h e s p a c e of probabi 1 i t y measures on G 1 / P ' on M(G'/P1) i s smooth, s o and g e n e r a t e d . M(G'/P1).

Since

cp

M(G'/P1),

+

.

By Theorem 5.7, t h e a c t i o n o f G'

is c o u n t a b l y s e p a r a t e d

M(G1/P' ) = [M(G1/P')l/G'

is a

r-map, (p(xy)

t h e l a t t e r space being

cp(~)n(y),

=

SO

r on G/P, t h e p r o j e c t i o n o f

By e r g o d i c i t y o f

(p(xy) z q ( x )

in

into ~(G'IP')

cp

i s e s s e n t i a l l y c o n s t a n t , i .e. (p(G/P) can be assumed t o l i e i n one G I - o r b i t i n M(G8/P').

Thus, we can view

(p

as a

r-map

5.8, H ' i s an amenable a l g e b r a i c subgroup.

r-map

(p

(p:

G/P

+

G'/H'

, and by Theorem

Ghat we have done i s t o o b t a i n a

where t h e image i s no l o n g e r an i n f i n i t e dimensional s p a c e M(G1/P')

b u t an a l g e b r a i c v a r i e t y G 1 / H '

.

The e x i s t e n c e o f such a m e a s u r a b l e map cp i s The second s t e p i s t h e f o l l o w i n g

t h e f i r s t main s t e p i n t h e p r o o f . fundamental lemma of k r g u l i s . Lemma A .

cp

:G/P

+

G'/H1

i s a r a t i o n a l mapping o f a l g e b r a i c v a r i e t i e s .

L e t u s show why t h i s lemma s u f f i c e s t o p r o v e t h e theorem. c7 : G/P + G ' / H 1

i s a r a t i o n a l mapping such t h a t

R(G/P, G ' / H ' ) be t h e s p a c e o f r a t i o n a l mappings.

Suppose

cp(xy) = q ( x ) n ( y ) Then

G

x

G'

.

Let

a c t s on

R(G/P, G 1 / H ' ) by T(g,gl) The f a c t t h a t

(p

is a

fl(x) = f(xg)-(g')-l

r-map

means cp i s f i x e d u n d e r

i d e n t i f i e d w i t h t h e subgroup o f t h e a1 g e b r a i c h u l l o f claim that dense i n G,

r r

r in G

G x

x G I

G1

.

g i v e n by Then

i s t h e g r a p h o f homomorphism must p r o j e c t o n t o a l l o f 6 .

r , where r i s

{(y ,n(y))].

Let 7 be

(p

is a1 s o f i x e d u n d e r 7.

G

+

G'

.

Since

W e

r i s Zariski

So suppose ( g , h l ) ,

(g,h2)

E

r.

Then

'P(xg) = ' ~ ( x ) h and ~ d x g ) = 'P (x)h3. -

pointwise fixed.

Rut

n(r)

T h e r e f o r e hlh$l

l e a v e s 'P(G/P) i n v a r i a n t , and s i n c e

Z a r i s k i dense i n G ' , v(G/P) must be Z a r i s k i dense i n G ' / Y t

n

l e a v e s a l l G1/H' p o i n t w i s e f i x e d , and s i n c e an amenable normal subgroup), hlhil function

G

+

6'

,

leaves n(r)

is

T h e r e f o r e hlh;l

g ~ ' g - l = ( e l (since it i s g EG' T h e r e f o r e i: i s t h e graph o f a

= e.

w h i c h i s a homomorphism s i n c e

t h e map i s a homomorphism on

.

q(G!P)

r C 6 i s Z a r i s k i dense and

r.

We now r e t u r n t o t h e p r o o f o f t h e lemma.

We must show t h a t a c e r t a i n

measurable mapping between v a r i e t i e s i s a c t u a l l y r a t i o n a l .

There i s one we1 1

known s i t u a t i o n i n w h i c h a measurable map i s known t o have much s t r o n g e r p r o p e r t i e s , namely i f t h e map i s a homomorphism. homomorphism between L i e groups i s

cm,

and s i m i l a r l y , any measurable

R + 9'

homomorphism between r e a l a l g e b r a i c groups r a t i o n a l on a l l u n i p o t e n t subgroups o f 9 .

G/P which i s n o t a group.

F o r example, any measurable

,

w i t h R r e d u c t i v e , w i l l be

Of course o u r map V i s d e f i n e d on

However, up t o a s e t o f measure 0, i t i s a group.

F o r example, c o n s i d e r G = SL(n,W),

P = u p p e r t r i a n g u l a r subgroup.

t h e lower t r i a n g u l a r unipotent matrices.

Then t h e n a t u r a l map

c a r r i e s U o n t o an open subset o f measure 1.

U + G'/H1.

+

G/P

Furthermore, t h i s e s t a b l i s h e s an

isomorphism o f IJ w i t h i t s image as a l g e b r a i c v a r i e t i e s . as a map

G

L e t IJ be

Thus, we can view q

Now, a l t h o u g h we have a d e f i n e d on a group, i t i s n o t a

homomorphism ( a s t h e image i s n o t even a group.)

We do however, have some

s o r t o f a l g e b r a i c r e l a t i o n , namely t h e f a c t t h a t 'P i s a

r-map.

Thus we

m i g h t hope t o be a b l e t o f o r c e t h i s a l g e b r a i c r e l a t i o n t o show t h a t q o n l y depends upon a homomorphism o f

IJ.

T h i s , however, i s n o t p o s s i b l e .

As we

s h a l l see, when we t r y t o f o r c e t h e a l g e b r a , we s h a l l need some c o m m u t a t i v i t y w i t h U f r o m elements o f A, where A i s t h e p o s i t i v e d i a g o n a l s . centralizer of U i n A i s trivial.

I n t h e R-rank 1 case, we can proceed no

f u r t h e r , b u t i n h i g h e r rank a l l i s n o t l o s t . c o n s i d e r t h e c e n t r a l i z e r Ct.

Rut t h e

F o r example,

L e t us f i x an element t

i n SL(S,R), l e t

E

A, and

1 0 0 10)). 0 0 1

Let CY = Ctn U { ( a

Now U z R3, and C?

I

R.

Thus

c

w i l l give us

one d i r e c t i o n i n U , and Ct i s a reductive group t h a t has a c e n t r a l i z e r i n A.

As we s h a l l see, t h i s will be enough t o show t h a t r a t i o n a l l y on

c;.

B u t now i f we vary t

d i r e c t i o n s in t h e same way.

c

cp:U

+

G1/H'

depends

A , we can pick up t h e o t h e r

The following lemma of FBrgulis i s now c l e a r l y

re1 event. Let cp be a measurable function defined on Rn xRk.

Lemma B.

r a t i o n a l i n x f o r almost a l l y

I f cp is

Rk and r a t i o n a l i n y f o r almost a l l x

E

c

R",

then cp i s r a t i o n a l . The above remarks about SL(n, R) extend t o general G . U

Thus, i f we l e t

be t h e unipotent radical of t h e parabolic o p o s i t e t o P, then

U

G/P

+

is

an isomorphism of a l g e b r a i c v a r i e t i e s with i t s image, t h e l a t t e r being open and of f u l l measure. and t

E

t

A,

reductive.

t.

Let A C P be a maximal abelian R-di agonalizable subgroup Let Ct be t h e c e n t r a l i z e r of t i n G.

0.

f

Letting CY

=

Then Ct i s

U n Ct, U can be b u i l t from t h e various C!

Thus, using Lemma B , i t s u f f i c e s now t o prove t h e following.

cp:G/P

+

Gi/H'

as a map cp:G

E

Proof. g t G , define

G'/H1,

For almost a l l g

Lemma C. ( f o r any t

+

E

with cp(pg) = rp(g), f o r p

c

by varying

View P.

G , cp(cg) depends r a t i o n a l l y on c f o r c

c

C$

A). Let Ct w :C 9

= +

C.

We want t o study dependence of (P on C , s o f o r each

G 1 / H ' by

Thus we have a map w:G measurable maps

C

+

G1/H' .

wg(c) +

=

cp(cg).

F(C,G'/H1), Let

T = {tn).

t h e l a t t e r being t h e space of Then w

tg

(c)

=

w(ctg)

=

w!tcg)

~ ( c g )( s i n c e t e P), and so we have a map

w:G/T

-t

( c ) = ~ ~ ( c ) Thus . we can vjew t !.le now use r - i n v a r i a n c e o f q :

F(C, G 1 / H ' ) . 9Y

( c ) = q ( ~ g y )= q ( c ~ ) T ( Y )= wg(c)

w

as

n(y).

Thus,

w and w a r e i n t h e same G I - o r b i t i n F(S,G8/H'), where G' a c t s on gv 4 t h e l a t t e r p o i n t w i s e . We now need a n o t h e r smoothness r e s u l t . Lemma

n.

Every ( ; ' - o r b i t

i n F(X, G 1 / H ' ) i s l o c a l l y c l o s e d , where X i s a

measure space. We observe t h a t i f X i s f i n i t e , t h i s i s immediate f r o m t h e f a c t t h a t F(X, G ' / H 1 ) would t h e n be a v a r i e t y .

M a r g u l i s observed t h a t t h e lemna i s t r u e

F o r a s i m p l e p r o o f , see t h e appendix o f c551.

f o r any measure space Y,.

R e t u r n i n g now t o t h e p r o o f o f Lemna C, we have t h a t j e c t e d t o [F(C,G'/4')1/G1.

w

-

9 =

ws

when p r o -

Ry Lemma D, t h i s l a t t e r space i s c o u n t a b l y gene-

r a t e d and separated, and by Moore's theorem

i s e r g o d i c on G/T.

Therefore,

all

w a r e equal when p r o j e c t e d t o [F(C,G'/H')]/G1, o r equivalently, a l l 9 w l i e i n t h e same G I - o r b i t . 50 f o r a, g e G, we have w = w h(a,g) 9 ag 9 I where h(a,g) e G' and h i s measurable.. F o r any f e F(C, G 1 / H ' ) , l e t Gf be t h e I

.stabilizer, so f o r a s C,

and Nf t h e n o r m a l i z e r o f Gf i n 6 ' . h(a,g)

e

Thus, f o r almost a l l g,

Nu

. 9

Suppose now t h a t a,l

a

h(a,g)

+

Clearly f o r a a2 e C.

s

C, :G

Then

i s a measurable homomorphism

=

ag

GI: , g

We have q(cag) = q ( c g ) a

+

q(cag)

obtain

7.

h(a,g).

Thus f o r a i n any u n i o o t e n t subqroup o f C, Choosing t h i s subgroup t o be c-'c$c,

depends r a t i o n a l l y on a.

a(bg) depends r a t i o n a l l y on b

c$.

E

we

T h i s completes t h e proof.

Complements t o t h e R i g i d i t y Theorem f o r Ergodic Actions: F o l i a t i o n s by Symmetric Spaces, and Kazhdan's P r o p e r t y (T). The r i g i d i t y theorem f o r ergodic a c t i o n s s t a t e d i n s e c t i o n 6 allowed us

t o d i s t i n g u i s h ergodic a c t i o n s o f l a t t i c e s on t h e b a s i s o f o r b i t equivalence i f t h e actions had f i n i t e i n v a r i a n t measure (e.g.

6.7).

c o r o l l a r y 6.6

and example

However, some o f t h e most i n t e r e s t i n g a c t i o n s o f l a t t i c e s , e.g.,

a c t i o n o f SL(n,Z)

on gn-'

f i n i t e i n v a r i a n t measure.

the

o r o t h e r f l a g and Grassman v a r i e t i e s , do n o t have We now i n d i c a t e how t o extend t h e r i g i d i t y theorem

t o enable us t o deal w i t h t h i s s i t u a t i o n .

The main step i s t o f i r s t extend

t h e r i g i d i t y theorem t o a c t i o n s o f general connected groups. L e t H be a connected group.

Every l o c a l l y compact group has a unique

maximal normal amenable subgroup N.

I f H i s connected H/N w i l l be a product

of non-compact connected simple L i e groups w i t h t r i v i a l center.

We s h a l l say

t h a t an ergodic a c t i o n o f H i s i r r e d u c i b l e i f t h e i n v e r s e image i n H o f each o f these simple f a c t o r s o f H/N i s s t i l l ergodic. Theorem 7.1.

C561

L e t H, H' be connected l o c a l l y compact groups, N, N'

t h e maximal normal amenable subgroups.

Suppose R-rank(H/N)

) 2.

L e t S, S'

he f r e e ergodic i r r e d u c i b l e H, HI-spaces w i t h f i n i t e i n v a r i a n t measure, and suppose t h e a c t i o n s are o r b i t equivalent.

Then H/N and H1/N' a r e isomorphic,

and Y i s compact i f and o n l y i f N' i s a l s o compact.

Thus, f o r connected groups, o r b i t e q u i v a l e n c e i m p l i e s isomorphism o f t h e semisimple p a r t s of t h e groups.

The p r o o f o f t h i s r e s u l t i s an e x t e ~ s i o no f

t h e p r o o f o f t h e r i g i d i t y theorem f o r e r g o d i c a c t i o n s o f semisimple groups. To see how t o a p p l y t h i s t o o b t a i n r e s u l t s about a c t i o n s o f l a t t i c e s w i t h o u t i n v a r i a n t measure, observe t h a t t h e o r b i t space o f i d e n t i f i e d w i t h t h e o r b i t space o f H a c t i n g on t h e l a t t e r s i t u a t i o n s i n c e now G/r

r

a c t i n g on G/H can be

G/r.

Theorem 7.1 d e a l s w i t h

has a f i n i t e H - i n v a r i a n t measure, so we

can t r y t o a p p l y t h i s t o t h e a c t i o n o f

r on G/H.

One can t h e n prove t h e

following precise result. Theorem 7.2 c e n t e r , r,

r'-

C567.

L e t G, G ' connected semisimple L i e groups w i t h f i n i t e

irreducible

non-compact subgroups.

lattices.

L e t H C G, H' C G' be almost connected

Assume t h e a c t i o n s o f

e s s e n t i a l l y f r e e and o r b i t e q u i v a l e n t . normal amenable subgroups,

r on G/Y and r ' on G1/H' a r e

L e t N, N'C H, H' be t h e maximal

2.

and suppose R-rank(H/N)

Then H/N and H1/N'

a r e l o c a l l y isomorphic. Example 7.3 [56].

As we v a r y n, n )_ 2, t h e a c t i o n s o f SL(n,Z)

mutually non-orbit equivalent.

on pn-l a r e

T h i s f o l l o w s by s i m p l y o b s e r v i n g t h a t t h e

semisimple p a r t s o f t h e c o r r e s p o n d i n g maximal p a r a b o l i c s i n SL(n,R) isomorphic.

(Actually,

are not

Theorem 7.2 w i l l n o t a p p l y t o compare t h e cases n=2

However, t h e a c t i o n o f SL(2, Z) on P iI s amenable, w h i l e t h e a c t i o n

and n=3. o f SL(3,Z)

on iP2 i s not.)

I n a s i m i l a r f a s h i o n , one can r e a d o f f a l a r g e

number o f r e s u l t s about a c t i o n s o f l a t t i c e s on t h e f l a g and Grassman varieties. A n a t u r a l quest-ion t h a t a r i s e s i n l i g h t o f Theorem 7.1 i s how s e n s i t i v e o r b i t e q u i v a l e n c e i s t o t h e way i n which H i s b u i l t f r o m N and H/V. example, what i s t h e r e l a t i o n o f a c t i o n s of SL(n,R) SL(n,R)

x

For

Rn t o t h a t o f

Q Rn, where t h e l a t t e r s e m i d i r e c t p r o d u c t j u s t r e s u l t s from t h e

n a t u r a l a c t i o n of SL(n,R) on R ~ ?To answer t h i s q u e s t i o n , we r e c a l l Kazhdan's

n o t i o n o f p r o p e r t y ( T ) f o r groups, and t h e n i n d i c a t e how t o d e f i n e t h i s f o r actions. L e t G he a l o c a l l y compact group, and I t h e one dimensional t r i v i a l representation. nn

+

r

vectors

If

rn, a

a r e u n i t a r y r e p r e s e n t a t i o n s o f 6, t h e n r e c a l l

means t h a t f o r any u n i t v e c t o r s

vl

,...,vk

E

there exist u n i t

H

..

n ,vkc Y

such t h a t < n n ( g ) v ? l v n > + < = ( g ) v . l v . > u n i f o r m l y on 'n J 1 J compact s e t s i n G f o r each i,j. Kazhdan [241 d e f i n e d a group t o have p r o p e r t y (T) i f

vy,.

rn +

I implies

I( 1 ,

for n s u f f i c i e n t l y large.

Theorem 7.4 (Kazhdan) 1241, [91.

i)

Semisimple L i e groups w i t h a l l

s i m p l e f a c t o r s h a v i n g R-rank a t l e a s t 2 have p r o p e r t y (T). p r o v e d t h i s assuming R-rank

) 3.

( A c t u a l l y , Kazhdan

) 2 was

That one o n l y need assume R-rank

observed by a number o f persons, e.g.

191.)

Any l a t t i c e subgroup o f a group w i t h p r o p e r t y (T) a l s o has p r o p e r t y (T).

ii)

We s h a l l a l s o need t h e f o l l o w i n g r e s u l t o f Wang. Theorem 7.5 does

SL(n,Z)

@

SL(n,l) 8 R~ has p r o p e r t y (T)

(Wang '467).

zn

, and

hence so

( n ) 3).

We now d e f i n e p r o p e r t y (T) f o r e r g o d i c a c t i o n s . r e s c r i c t a t t e n t i o n t o a c t i o n s o f d i s c r e t e groups.

F o r s i m i p l i c i t y , we

This notion f o r actions

o r i g i n a l l y appeared i n C571. L e t G be a d i s c r e t e group, S an e r g o d i c G-space. a u n i t a r y group v a l u e d c o c y c l e . I

I

fa,V,W

= 1 1 1 1 =

(s,g)

F(S,C), cocycles

1.

Let

f

a,V,W'

Let

an, a,

we say

a

+

.SxG+C

= < a ( s , g ) v ( s g ) lw(s)>.

and we endow F(S,C)

v,w:S

H

Let

a:S

x

G

+

U(H) be

be Bore1 f u n c t i o n s w i t h

begivenby

We c o n s i d e r

f

a,V,W

as a f u n c t i o n 6

w i t h t h e t o p o l o g y o f convergence i n measure. + a

i f given

v

vk:S + Ha, 11v.n= 1, 1

+

For there

exist (i.e.

vl

n

n ,...,vk:S

+

such t h a t

H

an

implies

an ) I

-+ f n a,Vi,V, ,v. J G) f o r a l l i,j.

n

an ,vi

i n measure on S f o r each g 3 e f i n i t i o n 7.6

f

'571.

E

a

>_


2.

a non-amenable q u o t i e n t

Then N C Z(G), t h e center o f 6, and i n p a r t i c u l a r , i s f i n i t e .

I f we f u r t h e r assume t h a t t h e R-rank o f every simple f a c t o r o f G i s a t

l e a s t 2, then

r has p r o p e r t y (T) o f Kazhdan 1241, and hence i f H = r / N

is

an amenable q u o t i e n t , FI must a1 so have p r o p e r t y (T) and hence i s f i n i t e . Thus, we conclude t h e following. L e t G be a connected semisimple L i e group w i t h f i n i t e

C o r o l l a r y 8.2.

center and assume R-rank o f each simple f a c t o r o f G i s a t l e a s t 2. be an i r r e d u c i b l e l a t t i c e .

Let r c G

Then every normal subgroup o f G i s e i t h e r f i n i t e

o r o f f i n i t e index. Y a r g u l i s ' r e s u l t s are i n f a c t s i g n i f i c a n t l y more general, both i n terms o f t a k i n g l a t t i c e s i n products o f a1 gebraic groups d e f i n e d over various l o c a l f i e l d s and i n terms o f rank r e s t r i c t i o n s .

The b a s i c d i f f i c u l t step i n t h e

p r o o f o f Theorem 8.1 i s t h e f o l l o w i n g r e s u l t concerning t h e a c t i o n o f G/P.

L e t P ' be another p a r a b o l i c subgroup c o n t a i n i n g P.

r-map G/P.

+

G/P1,

Theorem 8.3

i.e.

(Margulis r301),

minimal p a r a b o l i c . form

G/P + G/P'

G/P' i s a

Then t h e r e i s a

r -space f a c t o r o f G/P. Let

G, r

PC G a

as i n theorem 8.1,

Then any measurable f a c t o r o f t h e

r-space

G/P i s o f t h e

f o r some p a r a b o l i c P ' 2 p.

I n o t h e r words, every measurable factor.

l' on

r-factor

o f G/P i s a c t u a l l y also a G-

This theorem i s d i f f i c u l t and we w i l l n o t prove i t here.

Instead, we

show how t o deduce theorem 8.1 from it. Let

H = r/N

be a non-amenable q u o t i e n t .

Then t h e r e i s a compact m e t r i c

H-space X so t h a t t h e r e i s no H - i n v a r i a n t measure on X. a compact m e t r i c

r-space.

Since t h e a c t i o n o f

t h e d i s c u s s i o n f o l l o w i n g d e f i n i t i o n 5.2, q:G/P

we l e t

+

Y(X),

u

Ye can a l s o view X as

r on G/P i s amenable, by

t h e r e i s a measurable

r-map

where t h e l a t t e r i s t h e space o f p r o b a b i l i t y meaures on X.

If

be a m a s u r e on G/? i n t h e n a t u r a l measure class, then

M(X),()) P' so t h a t as

is a

r

r-space

-spaces,

no f i x e d p o i n t s i n fJ!X)

f a c t o r o f G/P.

i s conjugate t o G/P'.

(Y(X),q,(u))

under

r,

P'

Thus, t h e r e i s some p a r a b o l i c

+

6,

Since t h e r e are

Rut N acts t r i v i a l l y on M(X) by

d e f i n i t i o n , so N i s t r i v i a l on normal suhgrcup o f G. ohserve t h a t i f

I. (.

G/P' which i m p l i e s

n

g ~ l g - ~ a, proper

i l i v i d i n g G by i t s center, i t c l e a r l y s u f f i c e s t o

fl

i c I

Gi

i s an i r r e d u c i b l e l a t t i c e i n a product o f simple

L i e groups w i t h t r i v i a l c e n t e r , t h a t proper subset.

VC

N = I n

Rut s i n c e ?i i s normalized by

n

i E ,I r and

i s trivial for J C I a

Gi

ll Si,

i t i s normalized by

I-J t h e product o f these groups which i s dense i n G by i r r e d u c i b i l i t y .

The r e s u l t

follows.

9.

Margul i s ' A r i t h m e t i c i t y Theorem.

( T h i s s e c t i o n w i l l r e q u i r e a b i t m r e knowledge about a l g e b r a i c groups than previous sections.

We a l s o c a u t i o n t h e reader t h a t i n t h i s section, by

a l g e b r a i c group, Z a r i s k i closure, etc.,

we s h a l l mean w i t h respect t o t h e

a1 g e b r a i c a l l y closed f i e l d , unless we e x p l i c i t l y declare otherwise i n a given instance.) I n t h i s s e c t i o n we d e s c r i b e t h e p r o o f o f M a r g u l i s ' a r i t h m e t i c i t y theorem f o r l a t t i c e s i n semisimple L i e groups.

The p r o o f o f t h e r i g i d i t y theorem i n

s e c t i o n 6 was based on a r e s u l t a s s e r t i n g t h a t under s u i t a b l e hypotheses, a homomorphism o f

G.

r

i n t o a r e a l a l g e b r a i c group extended t o a homomorphism of

T h i s r e s u l t i s a l s o b a s i c t o t h e p r o o f o f t h e a r i t h m e t i c i t y theorem.

However, we s h a l l a l s o need r e s u l t s concerning homomorphisms of complex groups and a l g e b r a i c groups over l o c a l f i e l d s .

r

into

With some a d d i t i o n a l

comments, t h e p r o o f o f theorem 6.5 can be a p p l i e d t o g i v e us these needed r e s u l t s , so t h a t t h e b u l k o f t h e work o f t h e p r o o f o f a r i t h m e t i c i t y has i n f a c t already been done.

Rut before passing t o these arguments,

t h e statement o f t h e problem.

l e t us r e c a l l

The f i r s t example o f a l a t t i c e i n a L i e group i s t h e i n t e g e r l a t t i c e

Z" C R

l a t t i c e t h e r e i s an automorphism A : R ~ " a r i t h m e t i c a l 1y " d e f i n e d

= L.

Thus, L i s

.

group d e f i n e d o v e r Q, i.e. E

A(Z")

Rn such t h a t

+

To g e t o t h e r examples o f l a t t i c e s , suppose G

G = {a

However i f L i s any

~ . T h i s i s o f c o u r s e n o t t h e o n l y l a t t i c e i n Rn.

C GL(n, C) i s an a l g e b r a i c

t h e r e i s an i d e a l I C Q[ai j, d e t ( a i j)-l]

GL(n,C) ( p ( a ) = 0 f o r a l l p

c

.

I}

such t h a t

As ~ r s u a l , i f B C C i s any. subring,

we l e t GR = {a c GI a.

1

Theorem 9.1

.i

R, f o r a l l i,j and d e t ( a i j ) - l C

c

(Borel-Harish-Chandra)

R}.

I f G i s semisimple, t h e n G Z i s

151.

a l a t t i c e i n GR. F o r example, f o r G = SL(n, C), we have SL(n, Z ) i s a l a t t i c e i n SL(n, R).

The q u e s t i o n t h e a r i t h m e t i c i t y theorem answers i s t o what e x t e n t

t h i s i s a g e n e r a l c o n s t r u c t i o n , i.e.

t o what e x t e n t a r e l a t t i c e s

We now e x h i b i t two ways o f m o d i f y i n g a g i v e n l a t t i c e

a r i t h m e t i c a l l y defined? t o o b t a i n a new l a t t i c e .

r, r '

D e f i n i t i o n 9.2. commensurable i f

[r:r

P r o p o s i t i o n 9.3. commensurable, t h e n

fl

If

r and r 1 a r e c a l l e d

d i s c r e t e groups, t h e n

r'1
2.

r.

L e t G be as i n d e f i n i t i o n 9.5,

and assume

Then any i r r e d u c i b l e l a t t i c e i n G i s a r i t h m e t i c .

As we i n d i c a t e d above, t h e p r o o f i s based on two f u r t h e r r e s u l t s about homomorhi sms o f Theorem 9.7

r

.

( M a r g u l i s C281).

Let

r C G an i r r e d u c i b l e l a t t i c e , G as

above, R-rank(G) ) 2.

i! I f H i s a (complex) simple a l g e b r a i c group, connected and w i t h t r i v i a l center, then any homomorphism n :r + H w i t h sati sfies

i:

n ( ~ ) Z a r i s k i dense i n H e i t h e r

;;li;r compact o r extends t o a r a t i o n a l endomorphism

+ H,

where

i s t h e Zariski c l o s u r e o f G (embedding G i n t h e l i n e a r t r a n s f o r m a t i o n s i n

t h e c o m p l e x i f i e d L i e algebra f o r example).

ii) Any homomorphism

n:r

+

HK where H i s a semisimple a l g e b r a i c group over K,

and K i s a l o c a l t o t a l l y disconnected f i e l d o f c h a r a c t e r i s t i c 0, w i t h ~ ( r ) Z a r i s k i dense, s a t i s f i e s

i s compact.

The p r o o f we present i s i n t h e s p i r i t a f t h e p r o o f

\e gave o f Theorem 6.5 so

as t o be g e n e r a l i z a b l e t o cocycles d e f i n e d on general ergodic G-spaces.

We

expect these g e n e r a l i z e d r e s u l t s t o be o f use i n d e s c r i b i n g " a r i t h m e t i c " f e a t u r e s o f an e r g o d i c a c t i o n , b u t we do not discuss t h i s here. i ) The p r o o f we gave o f Theorem 6.5 can be a p p l i e d i f we can f i n d

Proof. a measurable

n

that

r-map

(p:G/P

+

H/HO where Ho i s an a l g e b r a i c subgroup o f H such

As i n Theorem 6.5,

h~"h-l = {el.

we can l e t P ' C H be a minimal

-

r

p a r a b o l i c subgroup, use a m e n a b i l i t y t o f i n d a

map

:G/P + M(H/P')

(p

Again, as i n 6.5,

prove t h a t each o r b i t i n M(H/P1) under H i s l o c a l l y closed. we can then assume i n M(H/$). braic.

p:G/P

+

H/H1

where HI

and

i s t h e s t a b i l i z e r o f a measure

U n l i k e t h e r e a l case however, t h i s s t a b i l i z e r need not be alge-

For example, t h e group may be compact which i n t h e r e a l case imp1 i e s

t h a t i t i s t h e r e a l p o i n t s o f an a l g e b r a i c group, w h i l e i n t h e complex case, o f course, a compact group w i l l n o t be a l g e b r a i c .

However, we can suppose H

i s r a t i o n a l l y represented on a f i n i t e dimensional complex space i n such a way that

P' i s t h e s t a b i l i z e r o f a p o i n t i n p r o j e c t i v e space.

Let

u

be t h e

I f H1 i s not compact, then u s i n g an

measure on HIP' s t a b i l i z e d by H.I

must be

argument as i n F u r s t e n b e r g ' s lemma, (lemma 5.9) we see t h a t

supported on t h e i n t e r s e c t i o n o f H/P1 w i t h t h e union o f two proper p r o j e c t i v e subspaces.

Choose a proper subspace V so t h a t

u(H/P1 A [V])

minimal dimension among a l l subspaces w i t h t h i s property. p r o p e r t y o f [V]

and H1 - i n v a r i a n c e o f

r) h~,h-l

=

{el,

and V has

must c l e a r l y

Hence, i f we l e t Ho be t h e Z a r i s k i

then VoC H i s a proper a l g e b r a i c subgroup.

and w i t h t r i v i a l c e n t e r

0,

By t h e m i n i m a l i t y

t h e H1 - o r b i t o f [ V ]

p,

be a f i n i t e union o f p r o j e c t i v e subspaces. c l o s u r e o f H,I

>

Since H i s simple

and as we remarked a t t h e beginning

o f t h e proof, t h i s s u f f i c e s . We must now consider t h e case i n which H1 i s compact. r-map q:G/P

+

H/Y1,

so t h a t i f we l e t

v =

(p,(p),

v

We then have a

i s a quasi-invariant

ergodic measure f o r t h e a c t i o n o f graph, c~ x

on H/H1.

( U n l i k e t h e previorrs para-

i s now t h e n a t u r a l measure c l a s s on GI?.)

p

v:G/P

r

x

G/P

H/H1

+

H/H1.

x

Consider t h e

It i s w e l l known t h a t on G/P, t h e P-action i s

e s s e n t i a l l y t r a n s i t i v e , t h e c o n u l l o r b i t having P P i s the opposite p a r a b o l i c t o P. Thus as a G-space,

as s t a b i l i z e r , where G/P

x

G/P

Floore's e r g o d i c i t y theorem ( s e c t i o n 2), r i s t h e r e f o r e ergodic on

r must a l s o be ergodic on (H/H1,v)

compact, t h e H - o r b i t s on

H/Hl

r, and r - o r b i t s

under

t h e H-action on orbit.

H/H1

x

x

H/H1

(H/H1,v).

x

are closed.

Since

implies that

v

v

x

on an H2 o r b i t i n H/H1 where H2 i s a conjugate o f H,I

n(r),

Thus, support ( v )

support ( v )

follows that

n(r)

is

x

i s compact.

n(r)-invariant,

x

Since

By

G/P

x

G/P.

Since H1 i s v)

i s ergodic

smoothness o f

must be supported on an H-

From F u b i n i ' s theorem, one e a s i l y deduces t h a t

compact.

(V

are o f course contained i n H - o r b i t s , H/H1

w i l l be

P n 7 , which i s non-compact.

essentially transitive with stabilizer

It f o l l o w s t h a t

r-map

v

must be supported

and i n p a r t i c u l a r i s

i s q u a s i - i n v a r i a n t under

and s i n c e H1 i s a l s o compact, i t

i s contained i n a compact s e t .

This completes t h e p r o o f

of (i). Let P '

ii)

c

H be a minimal p a r a b o l i c K-subgroup,

so t h a t HK/Pk i s

compact, and P i c o n t a i n s no normal a l g e b r a i c subgroup.

We again wish t o apply

t h e same t y p e o f argument as i n t h e p r o o f o f Theorem 6.5. t o prove t h a t analogue o f Theorem 5.7 over K. t h a t GL(n,K)

The f i r s t s t e p i s

I n f a c t t h e p r o o f i n [521 shows

We can assume t h a t we have a

a c t s smoothly on M(P"'(K)).

f a i t h f u l r a t i o n a l r e p r e s e n t a t i o n o f HK on Kn so t h a t HK/?i i s an o r b i t i n Ry a m e n a b i l i t y o f t h e r - a c t i o n

pn-l(K). r-map

,+,:G/P

+

M(HK/P;( ) C M ( P " ~ ( K ) .

M(pn-l(K)), we can view rp as a map where p

c

M(HK/P,',).

compact i n PGL(n,K), i n PGL(n,K),

and so

:G/P

on G/P, t h e r e i s a Ry smoothness o f t h e GL(n,K)-action +

[u

L e t S be t h e s t a b i l i z e r o f

on

GL(n,K)I n C(HK/P;)

u

i n GL(n,K).

then by t h e argument i n p a r t ( i ) , w i l l also be compact i n H.

If S i s

w i l l be compact

I f not, then u s i n g an

argument as i n Furstenberg's Lemma ( 5 . 9 ) , we can, as i n p a r t ( i ) , assume t h e Zariski c l o s u r e L of S i s a proper a l g e b r a i c subgroup.

Furthermore, we can

c l e a r l y assume from t h e construction of L as i n part ( i ) , t h a t f o r any g c GL(n,K), dim(H n g ~ g - l )< dim H . therefore suffices t o see that have q:G/P

+

By t h e condition of Zari ski density, i t n ( r ) C g ~ K g - l f o r some g r GL(n,K).

GL(n,K)/LK a measurable

r-map.

We

In t h e real case we showed

y

was rational by showing i t could be b u i l t from homomorphisms of unipotent subgroups of G which had t o be r a t i o n a l .

In t h e present s i t u a t i o n , we can

construct t h e same type of homomorphism using t h e argument of Theorem 6.5, but now, since t h e image group i s t o t a l l y disconnected, these maps must be constant.

We t h u s conclude t h a t

constant.

Since r ( r ) leaves (p(G/P) f i x e d , t h i s implies

i n a GL(n',K)

-

(p

:G/P

+

GL(n,K)/LK

i s essentially n(r)

i s contained

conjugate of L K , and t h i s completes t h e proof.

We now turn t o t h e proof of theorem 9.6 i t s e l f . semisimple Lie group t o be G;, group defined over Q .

where now G

We may take t h e

c GL(n, C) represents an algebraic

r C G; i s an i r r e d u c i b l e l a t t i c e . The following

Thus

lemna i s c l a s s i c a l , and follows f o r example from an argument of Selberg [42! (see a l s o [40 ,Prop. 6.61 f o r t h e same argument.)

This argument i s based on

r i n t o G a s an algebraic v a r i e t y , and then

expressing t h e embeddings of

choosing a real a l g e b r a i c point of t h i s v a r i e t y .

However, with theorem 9.7 a t

hand, we present an a l t e r n a t i v e argument due t o Margul i s [291. Lemma 9.8.

There i s a real a l g e b r a i c number f i e l d k and a rational

f a i t h f u l representation of G such t h a t , i d e n t i f y i n g G with i t s image under t h i s representation, Proof.

r C Gk.

The f i r s t s t e p i s t o show t h a t f o r K t h e f i e l d of real algebraic

numbers we have

Tr(Ad(y))

be an automorphism of C. taking

( z i j)

+

( a ( z ij ) )

E

K

for all

Then u

,

y

.

Following Margulis, we l e t

a c t s on matrices with e n t r i e s in C by

and since G i s defined over Q, a induces an

u

automorphism o f G.

(Of course t h i s i s an automorphism o f G o n l y as an

a b s t r a c t group, and w i l l i n general n o t be measurable.) moment t h a t 6 i s simple. ii)

air

9.7.

L e t us assume f o r t h e

o ( r satisfies either i )

Then

extends t o a r a t i o n a l automorphism o f 6.

I n t h e f i r s t case, a l l ejgenvalues o f

i s compact; o r

T h i s f o l l o w s f r o m Theorem

A ~ ( u ( ~ ) )w i l l have a b s o l u t e

value one, and i n t h e second case, these eigenvalues c o i n c i d e w i t h those o f (We remark t h a t i f

Ad(y).

A:G

dA o Ad(A(g)) o ( d ~ ) - ' = Ad(g),

r, {u(Tr(Ad(y))) l o

y

+

G

i s an automorphism we have

so Tr(Ad A(g)) = Tr(Ad g).)

The same can e a s i l y be seen i f G

A u t ( C ) l i s bounded.

i s semisimple by examining t h e composition o f simple f a c t o r s .

olr

Tr(Ad(y))

i s algebraic f o r a l l

r.

, r.

y

r C G with Tr(y) r K f o r a l l

Thus, i d e n t i f y i n g G w i t h Ad(G), we have E

w i t h p r o j e c t i o n on t h e

However, s i n c e Aut(C) i s t r a n s i t i v e on t h e transcendental

numbers, i t f o l l o w s t h a t

y

Hence f o r each

The next step, which i s c l a s s i c a l , i s t o observe t h a t t h i s i m p l i e s

t h a t t h e r e i s a f a i t h f u l r a t i o n a l r e p r e s e n t a t i o n o f G, d e f i n e d over K, such that

r

once again, i d e n t i f y i n g G w i t h i t s image under t h i s represen-

CGK,

tation.

We r e c a l l t h e c o n s t r u c t i o n .

1 i n e a r span o f

r-translates

be 6 - i n v a r i a n t .

o f Tr.

Consider

Tr:G

+

C,

and l e t V = C-

By t h e Bore1 d e n s i t y theorem V w i l l a l s o

Choose a b a s i s o f V o f t h e form yi

Then one can

Tr.

v e r i f y i n a s t r a i g h t f o r w a r d manner t h a t w i t h respect t o t h i s basis, t h e m a t r i x elements o f

y r

r a c t i n g on t h i s space a r e a l l i n K.

Since

generated ( i n t h e p r o p e r t y (T) case t h i s f o l l o w s e a s i l y [9]) a1 gebraic number f i e l d k w i t h

r C Gk

r i s finitely

we can f i n d an

.

We now r e c a l l t h e b a s i c o p e r a t i o n o f r e s t r i c t i o n o f scalars. i s an a l g e b r a i c group d e f i n e d over an a l g e b r a i c number f i e l d k . e x i s t s an a l g e b r a i c group i)

There i s an i n j e c t i v e map a:Gk

p ( c ) = Gk

Q

and

Then t h e r e

d e f i n e d over Q such t h a t +

cQ;

and

i i ) There i s a s u r j e c t i v e r a t i o n a l homomorphism such t h a t

Suppose G

p o a:Gk

+

Gk

p:?:

+

6

i s the i d e n t i t y .

d e f i n e d over k

lde can t a k e

We r e c a l l h e r e two ways o f d e s c r i b i n g t h i s c o n s t r u c t i o n .

6

=

n

u(G)

where

u r u n s t h r o u g h t h e d i s t i n c t embeddings o f k i n C .

0

a:Gk

+

6

i s t h e map

a ( g ) = (ul(g),

-

onto t h e f a c t o r corresponding t o choose an i d e n t i f i c a t i o n and we l e t formulation,

kn

...,ur(g)),

be t h e Z a r i s k i c l o s u r e o f

a(Gk)

a l l o w us t o d e f i n e a l i n e a r map f r o m

Gk.) of

and t h u s a map

+

G.

nr

x

a(g). nr

a:Gk

.

+

GL(nr,Q),

In this

Gk a r e d e s c r i b e d

E

These l i n e a r expressions

Q-matrices t o

n

x

n

(Recall t h a t G i s t h e Zariski closure o f

S i n c e t h i s map i s c l e a r l y a homomorphism on a

l e t [ k : Q l = r, and

i n GL(nr,C)

p a r i s e s from t h e f a c t t h a t t h e e n t r i e s o f g

i s projection

G

+

Then we have a map

by k - l i n e a r c o m b i n a t i o n s o f t h e e n t r i e s o f

k-matrices,

p:6

Alternatively,

u = id.

on'.

and

Then

a(Gk),

), i t i s a l s o a homomorphism on i t s Z a r i s k i - c l o s u r e ,

(being t h e inverse

6.

Completion o f p r o o f o f Theorem 9.6. We l e t k be as i n Lemma 9.8, and we l e t HC

and

F,

p, a

r.

he t h e Z a r i s k i c l o s u r e o f

r

Zariski density o f

i n 6.

as above.

rc

We have

Gk,

We s t i l l have p(H) = G by

Since G i s semisimple, p t r i v i a l on t h e r a d i c a l

of H, and, rep1 acing H by t h e q u o t i e n t o f H by i t s r a d i c a l , we can assume H i s a semisimple group d e f i n e d over Q, and w i t h t r i v i a l center. L e t F be a simple f a c t o r o f

We now c l a i m t h a t ( k e r p)R i s compact. k e r p.

Then as a l g e b r a i c groups defined over R, we can w r i t e H

where F ' i s t h e product of t h e remaining simple f a c t o r s . Z a r i s k i dense i n H, of H onto F.

( q o a)(r)

G

E

Since

F

x

a(r)

x

F'

is

i s Z a r i s k i dense i n F where q i s p r o j e c t i o n

We c l a i m FR must be compact.

I f not, then

( q o a) ( r )

cannot

have compact ( t o p o l ogi c a l ) c l osure since compact r e a l m a t r i x groups a r e r e a l T h i s would i m p l y by Theorem 9.7 t h a t

p o i n t s o f a l g e b r a i c groups.

extended t o a r a t i o n a l ho~nomorphism h:G + F.

I (g,h(g) ,f'l( g c G , f ' ~ F'l a(r),

q o a

Rut t h e n

would be a proper a1 gebraic subgroup c o n t a i n i n g

contradicting Zari ski density o f

a ( r ) i n H.

T h i s v e r i f i e s compactness

of FR, and doing t h e same f o r each f a c t o r , compactness o f ( k e r P ) ~ . Now consider

a:r

+

H4.

For each prime a, t h e image o f

in H Qa This means t h a t t h e powers o f each prime

must be bounded by Theorem 9 . 7 ( i i ) .

3 p ~ e a r i n gi n t h e denominators o f m a t r i x e n t r i e s o f u n i f o r m l y over

y c

r.

Rut

r

a(y)c H

Q

a(r)

a r e bounded

i s f i n i t e l y generated, and hence o n l y

f i n i t e l y many primes w i l l appear a t a l l .

T h i s i s r e a d i l y seen t o imply t h a t

a ( r ) n HZ

and hence, applying p, t h a t

i s o f f i n i t e index i n

a(r),

r n p(HZ) i s o f f i n i t e index i n r . i s a l a t t i c e i n GR. GR,

and since

'p(YZ):

r

Ry Theorem 9.1 and P r o p o s i t i o n 9.4,

( r n p(HZ)) C p(HZ)

n p(HZ)l