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 <
r.
C or an
Further, all
algebras and vector spaces are finite dimensional and all associative algebras have an identity. Let df' be an associative algebras over & of the form of df', and
a is the radical of d
.
1 @ %, where 1 is the identity
. Let N ( d ) be the subset of the group of,units of J' of
the form
Then N ( d ) is a subgroup of the group of units, because
and because
8 is an ideal if nl,nze%, then n l
exists k such that
+ n, + n , n 2 c a . Since % is nilpotent.
there
ak+' = 10) . By the binomial theorem. (1
+ n)-'
= I-n
+ ... + ( - 1)kn k
.
Let 9 c k% be an ideal in a . Let
Then G ( S ) is easily seen to be a normal subgroup of N(df') and N ( d ) / G ( $ ) to N ( d
is isomorphic
/9).
Let the dimension of N(&)
as a &-vector space be m. For I + n
E
~ ( d f ' ) ,and g
E
df',
the mapping
is a representation of N(&)
in GL(m,k).
Further, there is a basis of df' such that
p ( ~ ( & ) ) c U(m,k), where
and p(N(df')) is the set of zeros of a set of linear equations over k . We call N ( d ) the &-algebraic group of N. It is easily verified that N(&) The above has the following generalization.
is a nilpotcnt proup.
Definition:
Let G be a nilpotent group. We call G a k-nilpotent algebraic group if there
exists an isomorphism p:G-+U
(m,k) such that p(G) is a &-algebraic variety in U(m,&). nilpotent if [G,G] is central. If G is a 2-step
A nilpotent group G is called '-step
k-nilpotent algebraic group it is easily verified that G satisfies an exact sequence:
where V, and V2 are k-vector spaces such that V,>[G,G] and V, is in the center of G. With these general definitions out of the way, we can discuss the special cases with which we will be concerned in these notes.
Let V be a k-vector space and let A ( v )
denote the
exterior algebra over V. Then
where &' = Z
i>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,<&. Then
Let
Then, A c Att (V(k)) and B = t
0
A and we have proven that t(Polx (V(k)) = d ( 6 . o ) .
Thus the commutative diagram
enables us to identify d ( 4 , p . a ) and PolX(V(&)). Let us now consider the four groups that appear in the above diagram. g 2 ( V ( Q ) ) and N2qh+l(Ql are rational algebraic groups, and therefore we may consider the group of real points of these algebraic groups, which we will denote by g 2 ( v ( 4 ) and NZqh+l(R), respectively.
In Section 1, we saw that g 2 ( V ( & ) ) and N2q+l(k) a m the k points of rational
algebraic groups; or, equivalently, are defined over O Hence by reducing the field to Q we may form @algebraic groups which are isomorphic to g 2 ( V ( k ) ) and N2q+l(k) and form the group of real points of these Qalgebraic NLa+l(k)mrespectively. Let xi : k--,
groups, which we will denote by (Ir,(V(k)),
BP
i-I.
and
....h be distinct isomorphisms of & into R
25
We claim that
whcrc N 2 , + , ( l x i ) is isomorphic to N2q+l(m, i = l , ....,k. The above assertion may be seen as follows:
I f we reduce the field to
a we obtain
where the right side above is a matrix of h+h matrices. GL(h.R) be such that
where
Lel
Now, as in Section 1, let A
E
whe.? :he matrix on the right side is a (q+2) x ( q x 2 ) matrix of h x h matrices. Then
It is now easily seen that we may rearrange rows and columns of the above matrix t o obtain
This proves our assertidn and explains that the notation N 2 q + l ( l l ~ i stands ) for the 2q+l IlCHeisenberg with k imbedded in Bg by the isomorphism Now t:N2q+ ( k )
--,
xi,
N2qh+ (Q uniquely extends t o a homomorphism
(Bxi)
Further, T restricted t o the center of N 2 q + l ( m ~ iis) an isomorphism for
Nlqh+,(IQ. each i. If N2qh+1(IQis oriented, we may use T t o orient each N 2 q + l ( t ~ iby) requiring that
--,
the induced isomorphism on each center is orientation preserving.
We will then say that
WZq+(@xi) is coherently oriented. We are now going to find all complex structures J on V(B) such that for P c d ( & , p , o ) , (J,P) is a Riemann pair and p(k) J = Jp(k), k c
k.
Assume J satisfies the two conditions above. Since J is an automorphism of S 2 ( v R ) ) that commutes with p(k), it follows that J determines an automorphism of g 2 ( v ( k ) ) = By hypothesis. J preserves the real closure of the kernel of P = t the kernel of s 2 ( V ( k ) ) P
--,
N2q+I(&)B(and s o
J
0
r
0
p
.
Hence, J preserves
induces an automorphism
Jl
of
N2q+l(&)Bt Because k operates on each N 2 q + l ( t ~ imodulo ) its center as xi(&) (i.e.. a s a diagonal matrix) and since X, , .i=l,...,h , are all distinct, it follows that J 1 must leave each If J is a positive definite CR structure relative t o N 2 q h + l ( m , it
N 2 q + l ( @ ~ i invariant. )
follows that J I I N 2 q + l ( P ~ i )all. i, is a positive definite CR structure. Hence, we have that if 1 satisfies our two conditions, then J may be written as
definite CR structure for N2q+l(B,x,). i = l , ...,h.
h
n J, where
each J, is a positive
Now let
) determines a J i be a complex structure automorphism of N 2 q + 1 ( l ~ i that positive definite CR structure on N Z q + , ( l x i ) i-1. ...,h. Let J 1 = J,. Then J 1 is an
n
h
automorphism of ll N 2 q + l ( L ~ , that ) acts trivially o n the center of this group. Hence, T
J,
is an automorphism of N,,.q+,(lE) that is a positive definite complex structure. Clearly, there exists an automorphism J of g 2 ( V ( U ) that commutes with p ( k ) and lifts T
0
J,.
Thus. if 3 is the set of positive definite CR structures in N2q+l(R), we have that the set of all complex structures J on V ( R ) such that for P a d ( k , p , o ) , (J.P) is a Riemann h
pair and p(k)J
Jp(k), k
; .
E
k is of the form n 3
5.
The Involution Problem for Division Algebras Of the First Kind (Part I)
Let 9 be a division algebra over O that has a positive involution of the first kind. Then by the algebraic theory
9
may be described a s follows: Let k be the center of 9.
Then k is a totally real number field and we may assume [ k . a = h. If
k,then 8
9
does not equal
is a quaternion division algebra K = K(a,b) over k defined as a 4-dimensional
k-vector space with k basis 1, i, j, k satisfying
where a.b
E
k.
Let L be any subfield of
5 and consider K as a left L-vector space. The right
regular representation of 5 over L is given by
as L-vector spaces, where r,-(61)62 = 6261, 61.62
E
KLet
N,-(6) = det rL(S) . in particular, we define the norm of
6, N(6). 6
maximal commutative subfield k(i), i2 = a N(6) = x2 -ay2 -bz2
+
E
E
5, by N(6) = NL(6) where
k. Explicitly, if 6 = x
+ yi + zj + tk
abt2. The algebraic theory tells us that because
algebra N(6) # 0, unless 6 = 0. If we identify k with k
.
L E
is the
K, then
1 c 5, we have that N : K X
k X is a homomorphism, where 5' is the multiplicative subgroup of
--
P: is a division
BL We next note that
N(6) = 6 4 6 ) . 6 # 0, implies that a(6) = 6-' N(6). Since N(6) is central in P, we have
Explicitly, if 6 = x
+ iy + jz + kt, then o(6) = x
- iy - jz - k t .
We will use r to denote the right regular representation of 5 over k . Similarly, we
introduce the left regular representation Hom(6LD
1:K-hy 1(61)62=
62. a l , a2
t
K Clearly, r ( a l ) 1(a2) = 1(a2) r(61)
It is well known that H o m ( P D = r ( D
@
61,62~5
1(1().
Define T r S y m X ( D c ~ o m ( 5 . d )by
One verifies that
a(61)62 + ~ ( 6 ~ )=6 N(al ~
+ 62) - N(6%) - N(6,)
and
hence that
T(bl .a2) = x ( ~ ( 6 ~ + ) 60~( 6 ~ ) 6)d. ~ The following two formulas lie at the heart of all we will d o in the rest of this section.
We will now verify these formulas. Since T(661.662) = N(6)T(61,62) we have l ' ( 6 ) ~l(6) = N(6)T = TN(6) and formula (2) is verified. Next, notice that T(o(aI). ~ ( 6 ~ = ) ) ~ ( T ( 6 ~ . 6 ~=) )T(61.82) Hence,
which easily implies 1.
-
if k is a totally real field and
that k
x l .....xhare distinct > 0 , i = l . ...,h.
A is totally positive if x,(K)
E
isomorphisms of & into 81 we say We now distinguish two subsets of
quaternions. The totally positiw quaternions. 5+,are those lK(a.b) where -a and -b arc totally positive. The totally indefinite quaternions, 5-, are those 9 a . b ) for which a,-b arc totally positive. For 5?, o
is the only positive involution; but for BC, a is not a positive involution.
However, there exists u c
K such that
is a positive involution. The complete set of positive involutions of IC can bc described as follows. If r0 is a fixed positive involution of
K,then every other positive involution of
IIL-
can be written as
where w
E
BC is such that
p) The right regular representation of w has positive eigenvalues.
In this section we will solve the following problem: &(iK,r,~) of A
E
Given ( K T ) determine the set
AltX(lQ such that
We have to treat the two cases 5+and BC separately and, since the solution for l@ is most easily motivated, we will begin with (*,a). Consider equation 1. It shows that T
E
symX(E?) satisfies 3.
But if B c Bil(l@)
satisfies 3 and L commutes with r(6), 6 c l@, i.e., if L = P(6). 6 c 5?, then BL also satisfies 3 because
Hence, if B satisfies 3, then BP(m satisfies 3. Now let =
- 6 if
and only if 6 c
$:
= Ai
+ Aj + kk . Clearly, o(8)
a.It is a consequence of equation 2, that T
P(6)
E
AItX(BD for 6
c
4,. To see this. note that ( T P(6))
P (6)T
=
= T P(a(8)) =
- T P(6).
Thus equations 1 and 2 imply that d ( K + , r , o ) > T P(5;)). It is now our purpose to prove that d(6P.r.o)
= T P(E$).
The crucial idea in this proof is the idea of using the action of r ( e ) on ~il(@,lK?). This is suggested by the observation that
Hence
the
space
TP($;)
r1(6)X = N(G)X.This
where X
E
has
the
property
that
if
X~TP($;),
then
s u g t t ~ t rthnt
W, if and only if r1(6)X = N(6)X. This is indeed the case because if X
6
W,
r ( 6 ) ' ~r(6) = Xr(a(6)6) = X r(o(S))r(6)
The converse is obvious. Thus our task becomes to find W I . (Notice, that W I is P I ( 5 + ) invariant because P,r, = r i p I . ) We will now describe the action of r , ( W on Alt(lQ. where K i s a general quaternion dtvrsion algehra over k.
We have already observed that B(o is the set of 6 such that 0(6) = -6. since o(o(6)p 6) = -a(6)p 6
ar&,6cBL
wc havc n(6)B(o6 c 4 , . This implies that r 1 ( 6 ) ( Tr($,))cT r ( 4 , ) .
Now
We have already verified that T [ ( a ) c A l t ( 9 . To verify that T r(6,,)
c
a,then
c A l t ( 9 , let p
and
Hence, T r ( a ) c A l t ( O . Let W3 = T r(B(o). Then dim Wl = dim W3 = 3 and W3 is an r l ( O invariant irreducible subspace of Alt(g). Since dim A l t 5 = 6, and W I fl W3 = 0, we have
and we con conclude that d ( 5 + , r , o ) = T f(&+)
= W,
Because we will need it later and it falls into place naturally here, we will now describe the action of r l ( O on Syrn(g). Clearly, k T c sym(&) and
Let a r {i,j,k). We claim that equations 1 and 2 imply
This is because, for k,) r
q, (T rtko)P(a)) = f
.
= -f
(0)
r*(ko)~
( a ) T r(ko)
= T [(a) r(ko)
Now let W(a) = T !(a) r ( q ) . It is straightforward t o verify that r l ( D W(a) = W(a) and that W(n) is a 3-dimensional r l ( m invariant, irreducible subspace of S y m ( q . Since dim S y m ( 9 = 10, we have easily that
We now turn to the study of T I c S y m ( r ) that satisfy equations
where
7
is a positive involution of 57. It follows from our previous discussion that ~ ( 6 = )
u-' a(8)u where a(u)= -u. We claim that if T is a solution of equations 1 and 2 and T c S y m ( F ) . then
,
satisfies equations I and 2 ' and T I
Sym(Bt). We will now verify this assertion.
Noting that
we have that T I r Sym(K-). We will now verify that T I satisfies equation 1, that T I satisfies equation 2, can be verified in a similar manner.
and ji(S1)r(S2) = r(S2)
P(SI), and so T r l r 0 ( 6 ) T l = r(u)-I r(a(6)) r(u) = r(u
-I
a(6)u) = r ( ~ ( 6 ) )
and equation 1 has been verified. We will now see that A = T I ~ ( 6 b ' ) E d ( ~ , r , r )where ,
We next show that if B
d ( 5 - , r , r ) , then B = kA, &
E
r(6)B. we have BA-' commutes with r(S) all 6 -B and so (P(S)A)* = -P(S)A.
E
4'.
o 6". First
Since A-' r(6)A = B''
ST. Hence, BA" = P(6). S e K . But B* =
But (P(s)A)* = A * P ' ( ~ )=
P(a(8)) = P(6) or o(6) = S and S
E
= 6,'
T
- A P ' ( ~ )= -P(o(6))A.
d . d ( P r , r ) = TIP(&-)
Hence,
.
We will close the study of this section by examining how d ( K 7 . r . r ) behaves under the action rl(S), S
E
UL-. Let A
E
d ( K . r , ~ )and consider B = r*(S)A
r(6). We will now
prove that B" r * ( d ) ~= r(dd) where dde K.Further, if ~ ' ( d )= d*, then
T'
is a positive
involution of K.For notational convenience in the proof of these assertions, we let r(6) =
M. We begin by observing that
Hence, d' = a(SdS-')S.
It is easily verified that if rt(d) = dd then
T'
is an involution.
>> 0,where >> 0 Trace(A-'r*(a)~r(S))> > 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
<
and
B r , ( q ) = 0, r # t. Then
Since p ( I Q ( q ) c K, and p*(lQ<
c
<,
we have
Clearly, we may identify Bn with H o m ( q , <)and, hence suppressing the indices, we have
and, with respect t o this identification,
Clearly. Alt = OZ. A,,, where A,, = Ah(%) r.5 follows: As above.
= Alt(BD and
An, r # s, may be described as
B,, @ BSr = Horn(& &) @ Horn(&&), Then Ars = {B-B*
I BEB,,)
-
Horn(&&) and p , ( m
I
r # s.
An = r,(lQ.
A,, = W,(r) O W3(r) where W,(r) corresponds to W a , a = 1.3. under identification A,, Further
-
Thus
Alt(5).
A,, = W,(r.s) O W3(r.s) where Wn(r.s) corresponds t o Wn, a = 1.3, where ArS = ~ o m ( i & l f ) . It follows that Alt(V) = W,,(P) O W3(p) where
Clearly W I ( p ) is a p I ( m invariant subspace satisfying
and W3(p) is a direct sum of 3-dim. p l ( W invariant irreducible subspaces. It is easy t o verify that 2
dim W,(p)=- 4 ( p -P) 2 dim W3(p) =
12(p2-P) 2
+
3p
+ 3P
A similar argument gives Sym(V) = w l " ( ~ )O w ~ " ( ~where )
w ~ ' ( ~ )satisfies ~ ~ ( I6 )
~ , ' ( p ) = N(6)l and w ~ ' ( ~ )is the direct sum of 3 dimensional invariant irreducible p l ( m subspaccs. I t will be important for our later discussion t o note that dim wI'(p) =
As usual. we view V W"
A
4(P2-P) +P 2
V as the dual space t o Alt(V). Then V
A
V = w l O w3 where
is the annihilator of Wn, a = 1, 3. Clearly, p1(6)X = N(S)X, X E w l and w3 is the direct
sum of 3-dimensional p l ( Q invariant, irreducible subspaces. We are now in a position to solve the involution problem for the quaternion algebra 5(a.h).
Recall that when
d ( I K + , p . n ) , o,
5 = 5+, a totally positive quaternion division algebra, we have
thc positive involution of 5+,consists of the A c A l t ( V ) such that
Since N(B) = n(6)d. we have
Hence A
E
d ( 5 + , p I o ) if and only if t o A ; where A
E
WI(p). where t is the trace map
of k. Thus
In terms of matrices, we have A
where 6:.IJ = -o(S..) 1
E
d(@,p,o)
if and only if
i,j 5 p
Now let BT be a totally indefinite quaternion division algebra. Fix a positive involution 7
of BT,which, by our previous discussion, is given by 4 6 ) = 6(,-I 0 ( 6 ) 6 ~
~ ( 6 =~ -) a0
E
BT.
Let d ( 7 ) = d ( K T , p , 7 ) . Fix Sl E K-, 61 # 0 consider ~ ~ ( d 6( 7 ~) . ) By our previous discussion, if u = 7(6,)SI, then ~ ( u = ) u, the mapping 6 tion, and
p
,(61) d ( 7 ) = d ( u - T U ) .
We will now describe d ( r ) . Let W~',C = S P ( ~ ~ Then, ). since ~ ( 6 =~ -)
Hence w1';(6,)
a(,.
--
u-lr(6)u is a positive involu-
Sym(V) be as above. Let
SEW^'
and let A
we have A' = -A and
c d ( ~ and ) the converse if easily seen.
We have therefore verified that
W~'~(G,,) = d ( 7 ) . Let w(7) be the annihilator of d ( ~ ) we , have proven that
defines a monomorphism ~ o i ( $ ~ ( ~ ) / w ( rinto ) ) Pol(V(Q) whose image is d ( X - , p , r ) .
We can refine this picture slightly by noting that if N ( e V p , 3 ) = g 2 ( V ) / W 3 ( p ) then
is a commutative diagram for all subspace containing W(T) for all
T.
This is because W3(p) is the smallest p , ( 5 - ) invariant
T.
Again, in terms of matrices, if T(T) = TP(fjO)r(6,,).
where 6.. = -5(dji) IJ
1
< i,j < p.
Then A C & ~ ( ~ - , P , Tif) and only if
7. Existence of Riemann Matrices for Division Algebras of the First Kind In Section 6, we obtained a description of d ( K p . 7 ) where 5 is a quaternion division algebra over a totally real field d with positive involution
T
and rational representation p. For
~ ~ d ~ ( & p , we r ) need . to describe the set R(K,p.A) of all Riemann matrices J such that (J,A) is a Riemann pair and p(6)J = Jp(S), S E K We will begin by showing how this problem can be reduced to finding for each ~ e d ~ ( K p . one 7 ) element of R(14p.A). The first step in this reduction is t o relate our current problem t o the results in Section 4.
By Section 4, since the Riemann matrix J
that we are seeking commutes K, it certainly
commutes with &. and s o we may structure J as foilows: k t
x,. ...,x,,
be the h =
(kg
distinct isomorphisms of & into R and let &(i) = xi(&). Define g ( l i ) = K@d(it8t Then, using the notation of Section 4, we have a representation pi of E f P i ) on g 2 ( V ( W . i ) . Thus each A E ~ ~ ( K ~determines , T ) Ai as in Section 4. Thus our problem may be formulated as follows.
For i = l ....,h, find the set dyi(A) of all complex structures J i on g 2 ( V ( B ) , i ) such
that (Ji,Ai) is a Riemann pair with the property that
then R(Kp,A) =
nh1 yi(A).
Let G i be the group in non-singular matrices commuting with pi and let G i = C i f l Aut N(Ai), where
Aut N(Ai) denotes the automorphism group of the real Heisenberg group
N(Ai). Then y i ( A )
c
Gi. Let J o < j i ( A ) and let S e G i be such that
Since J' = -I, where I is the identity of Gi, we have, (J-IS)'
= -J-'S
JS= -E.
Hence
J-'S = R is a complex structure. Conversely, if R E G , and R~ = -I, the JR = S is certainly
in Gi. Since R~ = or
J-IS
J-IS
J-IS
J-'
J-1s J
Let G:
g-' and let
=
=
=
-E
we have
-s-'
S-I
c G i be the subgroup generated by g e G i such that J ~ - 'g J,, = g or J - ' ~Jo =
be the subgroup of elements such that
~ , , - ' g Jo = g.
There is a natural
mapping of G,=/G; into the S such that J,) S = R is a Riemann matrix. The subset of
G : / G ~for which positivity holds is precisely j i ( A ) . This discussion suggests that Cartan symmetric spaces enter into the description of $,(A).
This is true and this is where the Siege1
moduli enters into the picture. For a full discussion of this see [4]. We will now show how t o produce J,, E g i ( A ) where A E (K
ad,,, S pi,
T)
dX (5p, T) and Ai
r
dX
and i= 1,...,h. We will begin by looking at two special cases of our problem.
We will first produce for A c
d
(e. r, a), and r the right regular representation, the
desired J,,. Let T E Sym(K+) satisfy equations I and 2. Then we know we may write A = TP(60), 6(,
E
K+, a(6(,) = -6".
Clearly T-'A = f(8,) is a linear transformation commuting with r(6).
T o see that P(60) defines an automorphism of N(A), we simply note that
Of course (P(6,,))' may not be -I. (N(6(,1)
Since k is totally real and a is positive, for each i,
80 > 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<)C(n) l lnll where, f o r some constant C.
Consequently K Proof.
lrll +In1
< C[
C(n)
-1
I
.
2 extends t o a bounded convolution o p e r a t o r on L (V).
5
By easy changes of v a r i a b l e s , one f i n d s t h a t
(K. *f,g) = (K
1n
in
,gtf')
= SCdu P(u) SR+ds s
-
where f -(v) = f (v
-1
(v E V).
)
in-1
(g*f') ( ~ ~ , 0 )
It w i l l t h e r e f o r e s u f f i c e t o show t h a t , f o r
any a i n E, ifR+ds s
in-1
( g * f - ) ( ~ s a ) ~4 c ( n ) l f 1 2 ig12
u This i s e q u i v a l e n t t o showing t h a t t h e d i s t r i b u t i o n s T
s
T~ ( f ) = +ds s5-l f ( ~ ~ , a ) 5 R
5
.
, given
by the r u l e
enjoy t h e property t h a t
in
I ( T O ,g*f-)I i.e.
that
I (T
lfl
2
lg12
If l2 flgR2
l C C(q)
, ,
o r , by the converse of HElder's i n e q u a l i t y , t h a t
IT' for a l l f i n
- .
in
cmF(v)
a The d i s t r i b u t i o n T
i f o = (x,y) i n V. mapping ( s , t )
~ £ 1G c ( ~ ) 2
r.
i s supported i n a subgrcup H 2 Ha = C(sx,ty): ( s , t ) E R )
a
of V:
,
Apart from t h e t r i v i a l c a s e s where x = 0 o r y = 0 , the 2 ( s x , t y ) i s an isomorphism of R onto Ha. Roughly speaking,
i t i s s u f f i c i e n t t o prove t h a t
.fn2
c cc.1
.
it!
f E cmC(~') 2 a a I n e f f e c t , t h i s i s because the c o s e t s H v of H f o l i a t e V smoothly, and cona (I v o l u t i o n with T a c t s on each of t h e c o s e t s j u s t a s on H i t s e l f ; the uniform 5 e s t i m a t e s thus obtained f o r what happens on a l l the leaves of the f o l i a t i o n (1.3)
!TO
in
imply e s t i m a t e s f o r what happens on V.
(See the papers of A.P.
Calderdn and
A.
Zygmund [ c ~ z ] , S. Saeki [ ~ a e land C.S. Herz [ ~ e r.)] The proof of i n e q u a l i t y (1.3) i s n o t t r i v i a l .
i n v o l v e s a s l i g h t m o d i f i c a t i o n of a r e s u l t of E.M. which i s based on work of J . G .
1 ~ y g ]on
p. 197.
Perhaps t h e e a s i e s t method S t e i n and S. Wainger [ s w ~ ] ,
van d e r Corput, d e s c r i b e d i n A. Zygmund's book
I n o u r c a s e , t h e p r o o f s s i m p l i f y , and we reproduce t h e s e f o r
completeness.
u
*f i s t h e p r o d u c t of t h e 5 0 t r a n s f o r m s of t h e f a c t o r s , we may r e s o l v e t h e problem by e s t i m a t i n g (T I-. S i n c e t h e F o u r i e r t r a n s f o r m of t h e c o n v o l u t i o n T
in
We s e t 9in(~,q) = l i m and
c-in,
9 (P,P)
R$( 5)705-15
IOd s
05(p,q) = l i ma40+ 2 5 ~,(ap.a q) = a o~(P,Q)
-
Since and
2 exp(i[sp + s q]
s
- a s 2) .
a E R+, (p,q) E R~ 2 (p,q) E R 9
45(-~,-q) = [ + ; ( P , ~ ) I -
i n o u r i n v e s t i g a t i o n s we may suppose t h a t q = 1. F i r s t we c o n s i d e r t h e i n t e g r a l
I t i s now e a s y t o check t h a t
(1.4)
I l i mpinllRe(()XI
< C(l
+ In1
)
.
lima+O+
Next, we c o n s i d e r t h e i n t e g r a l
L e t JI be t h a t f u n c t i o n f o r which
By changing t h e o r d e r of i n t e g r a t i o n , we f i n d t h a t
Wc n o t e t h a t
I0
1 2 2 d s s5-l e x p ( i [ s p + s ]-as )I
f lwdt
and also that
l ) t ( ) l
cjl"dt 11
I
-
-
~ltS-2
-
+
2d -dt up(-at
+
Ca1 h
2
)
It now follows readily that
Combining the estimates (1.4) and (1.5),
1 (TO. 1-1- G C(IQI 111
+
we obtain the desired result, viz -1 I ~ I1 0
It should be noted that the a b w e proof contains some gaps. For instance, the following question has been glossed over:
the distribution'T a
in
is reali-
with Re(5) > 0. While u there is an obvious correspondence between the distributions T as distribu5 u tions on V and as distributions on H when Re(<) > 0, this correspondence might
sed as an analytic continuation of distributions T
c
conceivably break down when analytic continuation takes place.
We leave it to
the reader to ferret out and grapple with such difficulties. Further study of the distributions K requires the use of differential 5 n-1 operators. We shall identify the Lie algebra of V with F x Im(F); we
x
write (X,Y) for the generic element of
1.
It will be convenient to denote by n-1 the subspaces of 1of all elements of the forms (X,O) (X E F ) and
V and V -1 -2 and (0,Y) (Y E Im(F))
respectively. The exponential map is given by the for-
mula
The reason for these notations should become clear in 54. If E is in
1, we write fE and Ef for the functions on V given by the rule: fE(x,y) = [dldt f((x,y)exp(tE))] Ef (x,~) = [ dldt f (exp(t~) (x,y))l
= .O =
,
i.e. we identify 1 with a space of distributions supported at e, the identity of V.
If E
then we write Xf in place of Ef, and so on. n-1 Let (X.) and (Y ) be orthogonal bases of F and Im(F) over R. J k for (x,y) in V, we have, with the obvious notations, (1.6)
=
(X,O),
(X.f)(x,y) J
Clearly,
= [ d / d t f(tX
f
j
= a/ax. f(x,y)
J = a/ax. f(x,y) J
and analogously,
X,
y
-
E
*
-
2Im(X.x ) ) ] = J * * ~ e ( z I m ( X . x)yk ) a/ayk f (x,Y)
*J
k=l 2zkslq
+
*
Re(xX. Y ) J k
f(x,y)
,
(1.7)
fX.(x,y) 3 * * = alax. f(x,y) 2zkZ1 Re(xX. Y a/ayk f ( x , y ) , J J k and f u r t h e r £Y = Y f = a / a y f k k k 0 We now d e f i n e d i s t r i b u t i o n s A and n on V a s follows: f o r f i n C (v),
-
.
C
(1.8)
A*~=-Z n*f =
P ~ . 2 f j=l 9 J2 Yk Ck=l
*
-
We have a l s o t h a t
The important f a c t s about t h e so-called sub-Laplacian A a r e t h a t A i s a hmogeneous d i s t r i b u t i o n of degree -r-2,
and t h a t A i s h y p o e l l i p t i c .
have t h e following p r o p o s i t i o n , due t o G.B. PROPOSITION 1.6.
I n f a c t , we
Folland [ F o l ] and A. Kaplan [Kap]
Let k be t h e f u n c t i o n N
2-r
.
Then, i n t h e
d i s t r i b u t i o n a l sense,
k*A = A*k = c6
,
where c i s a nonzero c o n s t a n t , and 6 ' i s t h e Dirac measure a t (0,O). Proof.
We s h a l l l a t e r prove a g e n e r a l i s a t i o n of t h i s r e s u l t ( s e e
P r o p o s i t i o n 1.9 and Remark 1.10). P r o p o s i t i o n 1.6 can be proved by b r u t a l c a l c u l a t i o n s , but we p r e f e r a longer but more e l e g a n t approach.
The scope of t h e next two lemmata i s t o
simplify l a t e r calculations.
LEMMA 1.7.
Proof.
The following e q u a l i t y holds:
Since Y
*
i s imaginary, Re(zYk ) = -Re(Y z) k n- 1 i s t h e i n n e r product of c and d i n F , whence
k
.
*
Also, Re(cd )
.
We now r e c a l l t h a t M a c t s on V (formula (1.1)): m 1 -1 R (x,y) = (v xu, v yv)
-
where m = m(u,v), v E F, denote by ' M
( x , ~ )E V,
Ivl = 1, and u i s an isometric (n-l)x(n-1)
matrix. We
t h e subgroup of M of those elements f o r which u i s diagonal.
A f u n c t i o n f on V i s c a l l e d p o l y r a d i a l i f
f
o = ~f
~
mEM'
.
This means t h a t f o r any (x,y) i n V,
-.. S X ~ - ~ , Y=) ~ ( E ~ x ~ , E ~ xn-1 ~ , .n-l"ny) .
f(x1,x2, whenever
-,E
E E
j j
EC*1)
l(j
ET
1
-
X
if F - R
1
E
n
= 1
if F = C
EU l < j < n i f F = Q , j where T and U a r e t h e m u l t i p l i c a t i v e groups of t h e elements of C and Q of abE
s o l u t e value 1.
A distribution T i s called polyradial i f
A s t h e nomenclature would suggest, i t i s p o s s i b l e t o approximate polyradial
d i s t r i b u t i o n s by p o l y r a d i a l f u n c t i o n s . The next lemma u s e s an i d e a which t h e author l e a r n t from A. KorSnyi. LEMMA 1.8.
The space of p o l y r a d i a l f u n c t i o n s forms a
commutative algebra under convolution. Proof.
We show f i r s t t h a t t h e space is closed under convolution.
I f f and g a r e p o l y r a d i a l , then a change of v a r i a b l e s shows t h a t , f o r any u i n V and any m i n M',
m because R i s a n automorphism o f V
To show t h a t t h e a l g e b r a i s commutative, we c o n s i d e r two c a s e s . I f F # C m -1 t h e n , g i v e n v i n V , t h e r e e x i s t s m i n M such t h a t R v = v Consequently, a -1 p o l y r a d i a l f u n c t i o n f i s e q u a l t o i t s ref t e c t i o n f v (f(v) = f (v ) ) . Again,
.
i f f and g a r e p o l y r a d i a l , we may change v a r i a b l e s t o o b t a i n t h a t , f o r any u i n
v. (f
*
*
8 ) (u) = ( f
g)(u-l)
-1 g (V ) -1 = j V d v gv (uv) f' ( v ) = J dv f (u-lv)
v
.
=g*f(u)
-
F i n a l l y , i f F = C, t h e n t h e map v !+ automorphism ( t h i s i s f a l s e i f F = Q). that
fO(v) = f(v).
v
For f on V, l e t f
*
*
g) = ( f
g)"" = ( f O
F u r t h e r , i f f and g a r e p o l y r a d i a l , t h e n f' (f
*
(x,;))
(x,y)
be t h e f u n c t i o n s u c h
*
*
g) = (f*
*
.
go)"
= f o and 'g
gV)O = (g
*i
= g o , whence
f)VO = g
*
f
,
f i s also polyradial.
A f u n c t i o n f on V i s c a l l e d c y l i n d r i c a l ( o r b i - r a d i a l )
m i n M.
is a n
By s i m i l a r c a l c u l a t i o n s , we f i n d t h a t (f
since g
(i.e.
I f F = R, t h e n f ( x ) depends o n l y o n 1x1 ( i . e .
m i f foR = f f o r a l l
i s r a d i a l i n t h e usual
s e n s e ) , w h i l e i f F = C, t h e n f ( x , y ) depends o n l y o n 1x1 and y, and i f F = Q, t h e n f ( x , y ) depends o n l y o n 1x1 and l y l . i f T ( ~ o R =~ T ) (f) f o r a l l f i n C
m
A d i s t r i b u t i o n T is c a l l e d c y l i n d r i c
(V) and a l l m i n M.
Obviously, a c y l i n d r i c a l
d i s t r i b u t i o n may be approximated by c y l i n d r i c a l f u n c t i o n s . We now g e n e r a l i s e P r o p o s i t i o n 1.6, a s promised. The f o l l o w i n g e q u a l i t y i s v a l i d : 2 <-r-4 ~ ' - ~ ) ( x , y )= ( r 5)(< 2) 1x1 ~ ( x , y )
PROPOSITION 1.9. (A
Proof.
-
*
-
The proof p r o c e e d s i n s e v e r a l s t a g e s .
s u f f i c e s to establish the r e s u l t f o r t h e e v a l u a t i o n of A
*
N'-~
<
such t h a t Re(<)
F i r s t , we n o t e t h a t i t
>
r + 2, i n which c a s e
i s by o r d i n a r y d i f f e r e n t i a t i o n .
The r e s u l t w i l l
t h e n e x t e n d t o g e n e r a l 5 by meromorphic c o n t i n u a t i o n . The second s t e p of t h e proof i s t o f i n d a f o r m u l a f o r A , v a l i d f o r polyr a d i a l functions; the t h i r d stage is t o s p e c i a l i s e t o the r a d i a l case.
W e now
p a s s t o t h e second s t e p . From t h e formulae (1.6) and ( 1 . 8 ) , we have t h a t A
*
f(x,y)
* *
2
-r j =1 1 a/ax + 2q=lq Re(xX. yk a/ayk1 3 = - z j-1 ' a 2 / a X j 2 ~ ( x , Y ) j * * 2 P - 4 ~ I zkZlq Re(xX. yk a / a y kI f ( x . y ) j=l A * 2 Ek=l Re(xX. Y ) a / a x . a y k f ( x , y ) -4Cj=l k J =
* *
since
*
J*
a / a x . Re(xX. Y ) = Re(X.X. Y ) = 0 J J k 3 3 k Lemma 1.7 i m p l i e s t h a t t h e second term i s e q u a l t o
and s o we have t h e f o l l o w i n g e x p r e s s i o n f o r A
(formulae (1.10)
f:
*
~
a
-
.
and (1.11) d i f f e r o n l y i n t h e s i g n of @ f )
r a d i a l , we have from Lemma f
E
1;s and
,
.
2 2 =l a y k~ ~ ~( x , Y ) @ ~ ( x , Y ), * * 2 where @ f ( x , y ) = 4C i: R ~ ( X Xyk ) a / a x . a y k f ( x , y ) j=1 k=l j J Analogously, from formulae (1.7) and ( 1 . 9 ) , i t f o l l o w s t h a t -41x1
2
*
f(x.y)
.
Now, i f f i s poly-
a s t a n d a r d l i m i t i n g argument t h a t A
A , f o r A i s obviously c y l i n d r i c a l and a f o r t i o r i p o l y r a d i a l .
*
f =
We deduce
from formulae (1.10) and (1.11) t h a t @f = 0, and (1.12)
A
*
f(x,y)
I f f i s c y l i n d r i c a l , then a f u r t h e r s i m p l i f i c a t i o n i s p o s s i b l e : we may 2 2 a2/ax and E (I a l a y k The key pass to polar coordinates t o evaluate T j =I j k=l n f a c t . i s f a m i l i a r : f o r a r a d i a l f u n c t i o n g on R , n 2 E a /ax 2 .g = [ a 2 l a p 2 + ( ( n - i ) / ~ ~ a / ga ~ ~, j=1 j where p i s t h e r a d i u s .
.
The r e s t of t h e proof i s now s t r a i g h t f o r w a r d c a l c u l a t i o n , and i s o m i t t e d . The r e a d e r may l i k e t o check t h e d e t a i l s .
REMARK 1.10. 1.9.
Let
E
We e x p l a i n how P r o p o s i t i o n 1.6 f o l l o w s from P r o p o s i t i o n
be a small p o s i t i v e q u a n t i t y , and l e t Q be d e f i n e d by t h e formula
Clearly whence
A
* A
2 n ( x , y ) = 1x1 N ( x , ~ ) - ~ N2+~-r E-r = ~ ( r 2 E) QN
-
*
N~-'
-
,
= ( ~ - ~ ) R ~ s ( Q N ' -=~ ;0) L
where the notation should not need explanation. on the same idea a s Lemma 1 . 1 shows that
and the Proposition 1 . 6 f o l l o w s .
,
A simple c a l c u l a t i o n , based
2.
ANALYSIS ON HEISENBERG GROUPS. I1
of t h e 5 I n p a r t i c u l a r , we showed t h a t
We have p r e v i o u s l y c o n s i d e r e d c e r t a i n homogeneous d i s t r i b u t i o n s K form RN'-~,
where R i s homogeneous of d e g r e e 0.
f o r almost a l l v a l u e s of 5 , K if 5 =
i n (n
E R\(O))
then K
5
h a s a d i s t r i b u t i o n a l i n t e r p r e t a t i o n , and t h a t 2 convolves L (V) i n t o i t s e l f (Th'm 1.5). In this
5 s e c t i o n , we c o n s i d e r p o s s i b l e I,'-boundedness
more g e n e r a l p o t e n t i a l - t h e o r e t i c
spaces.
and boundedness of K on 0' 5 Our main t o o l s a r e t h e a n a l y s i s of of K
t h e p r e v i o u s s e c t i o n , and t h e complex method of i n t e r p o l a t i o n , developed by A.P.
Calderon [ C a l l and E.M.
S t e i n [Ste].
L a t e r , we c o n s i d e r some p r o p e r t i e s
of t h e p o t e n t i a l - t h e o r e t i c s p a c e s w i t h which we d e a l . F i r s t of a l l we c o n s i d e r t h e k e r n e l K
0'
LEMMA 2.1.
Let K
5
be of t h e form
N5-',
1 where $2 E L (Z).
Define t h e mean v a l u e MV(R) of R by t h e formula MV(SL)=
I
~
n~ ( U ~ )
. is a 0 h a s a p o l e a t 0 , and t h e
Then t h e r e i s t h e f o l l o w i n g dichotomy: i f MV(R) = 0 , t h e n K d i s t r i b u t i o n , and i f MV(R)
+ 0,
then K
5 r e s i d u e t h e r e i s c6, f o r some nonzero c o n s t a n t c . Proof.
We omit t h e p r o o f , b u t r e f e r t o Lemma 1.1 and Remark 1.7 f o r
the necessary techniques. Lennna 2.1 s u g g e s t s t h a t , i f MV(C2) = 0 , t h e n c o n v o l u t i o n w i t h K might be 0 2 bounded on L (V). I n f a c t t h i s i s s o , b u t o n l y i f some a d d i t i o n a l r e s t r i c t i o n be imposed on R . t o A.W.
S e v e r a l p r o o f s of t h i s f a c t a r e a v a i l a b l e .
Knapp and E.M.
S t e i n [KnS] ( s e e p.
The f i r s t i s due
494; t h e h y p o t h e s i s of t h e i r theorem
may be r e l a x e d s u b s t a n t i a l l y u s i n g t h e same method of p r o o f , b u t n o t t o R i n C(C).
We s h a l l shor.tly p r e s e n t a second p r o o f , which u s e s complex i n t e r -
p o l a t i o n ; t h e philosophy of t h i s second proof i s t h a t i f a n o p e r a t o r e x i s t s i n a weak s e n s e (K ) and i f t h e ambient o p e r a t o r s (K w i t h 5 # 0 ) a r e bounded i n 0 5' some s t r o n g e r s e n s e , t h e n s o i s t h e o r i g i n a l o p e r a t o r . A t h i r d p r o o f , o f f e r e d i-r -i-r l a t e r , r e l i e s on t h e f a c t t h a t N + N = c6 f o r some nonzero c . A fourth
i -i proof, s i m i l a r t o the t h i r d , uses the f a c t t h a t A * A = 6 , and r e l i e s on i -i -2i-r 2i-r d e s c r i p t i o n of A and A a s o p e r a t o r s of t h e form QN and R N , which
-
d e s c r i p t i o n f o l l o w s from work of G.B. F o l l a n d [ F o ~ ]( s e e p. 184). We now c o n s i d e r t h e d i s t r i b u t i o n s K
5
f o r c e r t a i n v a l u e s of 5 w i t h nonzero
real parts. LEMMA 2.2.
L e t n be a p o s i t i v e i n t e g e r . Suppose t h a t 2n K = QN~", where n E C ( C ) , and t h a t 5 = 2n + i n (r) E R\{o}). 5 Then nbn
*
K
where
*
5
t12 < c ( n , ~ ) nfu,
1n1
c(n,s)<~(n,n) (lnl-l+ Proof.
n It s u f f i c e s t o check t h a t Q
*
.
,
i s a d i s t r i b u t i o n of the 5 t a k i n g c a r e o f t h e growth of t h e c o n s t a n t s a l l
form c o n s i d e r e d i n Theorem 1.4, t h e while.
f E cmC(v) 2n+l
K
We omit t h e d e t a i l s .
1 i s a meromorphic L (Z)-valued 5 f u n c t i o n , d e f i n e d i n t h e h a l f - s p a c e H, where H = E C:
LEMMA 2.3.
-
Suppose t h a t 5 k R
S e t K e q u a l t o Q N ~ (- i n~ t h e s e n s e of Remark 1 . 2 ) , 5 5 and l e t S be t h e s e t o f p o l e s of t h e d i s t r i b u t i o n - v a l u e d f u n c t i o n 2-r 5 - K . LetkbeN 5 I f Re(<) E ( 0 , r - 2 ) , and i f 5 & S, t h e n t h e i n t e g r a l d e f i n i n g Re(5)
2).
.
*
converges ( a l m o s t everywhere), and i s a 5 locally integrable function 'K o f t h e f o l l o w i n g form: '~-r+2 K' = a- N 9 5 5 1 where Q'? is a L (E)-valued meromorphic f u n c t i o n . the convolution k
R N5-'
" I
The L ( Z ) - v a l u e d and
cm (V)'-valued
f u n c t i o n s Q'
e x t e n d meromorphically t o H, and a l l p o l e s o f K-
5
and f i n a l l y c(s)
in
C(C)
+
I]
5 1
where, i f 5 = 5 + i ~ , C(5)
(
C( l n l
' +
lnl-l)
and K'
l i e i n S.
Distributionally,
lln-clll c
5
.
5
We s h a l l n o t show t h a t t h e i n t e g r a l d e f i n i n g t h e c o n v o l u t i o n
Proof. k
*
converges, b u t t h i s i s s t r a i g h t f o r w a r d .
$2 N ' - ~
5
i s of t h e s t a t e d type i f Re(5) E (0, r-2),
It i s a l s o c l e a r t h a t K
f o r r e a s o n s of homogeneity.
5
We
s h a l l c o n s i d e r t h e q u e s t i o n of t h e meromorphic c o n t i n u a t i o n . We t a k e (I i n set
cW ( R ) which i s e q u a l t o 1 i n a s m a l l neighbourhood of 0 , and
4 equal t o $ION.
Now
The f i r s t p i e c e on t h e r i g h t hand s i d e i s t h e c o n v o l u t i o n of k with a compactly supported d i s t r i b u t i o n , and s o i s a d i s t r i b u t i o n . OD
(0,0), t h i s piece is C
o f f t h e s u p p o r t of
OD
Since k i s C
+. Provided
second p i e c e "is" a l o c a l l y i n t e g r a b l e f u n c t i o n .
away from
t h a t Re(6)
2, t h e
Both p i e c e s c o n t i n u e meromor-
h a s p o l e s . The sum K- i s l o c a l l y i n t e 5 5 g r a b l e o f f s u p p ( @ ) , which may be made a r b i t r a r i l y s m a l l , whence K' is in 5 1 L loc(v\I (0'0) I ) . p h i c a l l y , w i t h p o l e s o n l y where K
By c o n s i d e r a t i o n s of homogeneity, i t must be t r u e t h a t , a s d i s t r i b u t i o n s 5-r+2 =R'N 5 5 1 f o r a n a p p r o p r i a t e L (Z)-valued f u n c t i o n R--. K-
morphic C
OD
C
and R' 5 5 2, and hence they a g r e e everywhere.
(V)'-valued f u n c t i o n s ' K
These a g r e e i f 0
< Re(5) < r -
We now c o n s i d e r t h e two mero-r+2 N , d e f i n e d a s i n Remark 1.2. It i s c l e a r
.
= k * K 5 5 We conclude by n o t i n g t h a t $K
that ' K
N(x,y)
inside the s e t ((x,y):
<
i s a d i s t r i b u t i o n w i t h small s u p p o r t , s a y 5 1 / 2 1 , and of o r d e r a t most 1-E, 1 + 1, where 1 . I
denotes the i n t e g r a l p a r t function.
I f N(x,y) = 1, k
mated by t h e d e r i v a t i v e s of k of o r d e r a t most 112
< N(x,y) < 3/21.
The e s t i m a t e f o r
forward e s t i m a t e s f o r k
*
(1
-
#
5 1
[-c]
*
I$K (x,y) can be e s t i 5 + 1 i n t h e anulus {(x,y) :
u s e s t h i s f a c t and some s t r a i g h t -
@)Kc.
We may now improve somewhat Lemma 2.1. PROPOSITION 2.4.
where C(ft)
2 I f R E C (Z)
, and
i f MV(fl) = 0 , t h e n
depends on t h e d e r i v a t i v e s of fl of o r d e r a t most 2.
Proof. We know t h a t A i s a p o s i t i v e unbounded o p e r a t o r w i t h d e n s e 2 domain i n L ( V ) , w i t h no n u l l space. The s p e c t r a l c a l c u l u s p e r m i t s u s t o de2 f i n e A5I2, a s a n unbounded o p e r a t o r on L (V) w i t h d e n s e domain. The o p e r a t o r s 2 A ~ where ~ s E ~ R, ~e x t e n,d t o isometrics of L (V) G.B. F o l l a n d [Fo2] h a s s t u -
.
died these operators. We c o n s i d e r t h e f a m i l y of d i s t r i b u t i o n s 5 -t Kg where 5' 2 2 4 ) ( 5 + 1 ) QN'-' ~ 9 =' exp(5 ) ( 5 , 5 which i s a n a l y t i c i n t h e s t r i p ( 5 E C: R e ( < ) E [-3, 3 1 1 , because any p o l e s of
-
nd-'
a t -1 o r -2 a r e a n n u l l e d by t h e f a c t o r ( c 2
@
h a s no p o l e a t 0. On one hand, i f 5 j u g g l i n g of e s t i m a t e s , we o b t a i n t h a t
5
= 2,
( t h e c o n s t a n t C(n) i s now independent of n ) .
-
4)(5 +
11, and by h y p o t h e s i s
t h e n from Lemma 2.2 and some
Consequently
and hence
On t h e o t h e r hand, i f 5 = -2,
* fl12 F by Lemma 2.3 and Theorem 1.2. Hk
Kg
c(R') Ilfl12
f E cmC(v)
Analogously,
K9
T i t c h m a r s h [ T i t ] , S e c t i o n 5.65) i m p l i e s t h a t , i f i n e q u a l i t y holds.
,
i t follows t h a t
* f , g ) l C(Q) 11fi2 11gl12 5 A p p l i c a t i o n of t h e Phragmen-Lindeliif " t h r e e l i n e s theorem" ( s e e , e . g . , l(ACI2
*
*
then
5
= 0,
E.C.
then an analogous
C o n s i d e r i n g t h e c a s e where 5 = 0 , we o b t a i n t h e i n e q u a l i t y :
I ( K * ~f , ~ g)l
< c(n)
H~II,
f,g E
c
m
c ( ~ >,
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,
1< q
*pq =
< -, w h i l e I
The s p a c e L " ~ ( v )
+
= m({s E
.
I n o t h e r words, f * i s non-
1 E R+.
Hunt [Hun] ( s e e a l s o
and make t h e f o l l o w i n g d e f i n i t i o n s :
rsl/~
1 ( q ~ p ~)
if 1< p
R+: f 8 ( s ) > X I )
We f o l l o w R.A.
S t e i n and G. Weiss [StW] pp. 188-205)
if 1< p
on R
and f u r t h e r
m ( S ) d e n o t i n g t h e measure of t h e s e t S.
E.M.
*
~ ~ d s l s~ * ( s ) I ~ I ' / ~
< -,
q =
m,
.
s E R+I
= sup{ [s1lp f * ( s ) ] :
i s t h e s p a c e o f t h o s e f u n c t i o n s f o n V f o r which llf
l*
I f p = q , t h e n LPSq(V) i s t h e u s u a l Lebesgue s p a c e LP (V), w h i l e i f q =
Pq
-,
< m. then
t h e s p a c e i s o f t e n c a l l e d "weak LP(V)". The n e x t lemma i s due t o Hunt [Hun] ( s e e p. 273). LEMMA 2.11.
Suppose t h a t t ~ ( 1 m) , and t h a t k E L
t,-
(V).
Then t h e r e i s a c o n s t a n t C ( p , q ) such t h a t
where 1 / p + l / t = l / q + 1, p r o v i d e d t h a t 1 Proof.
Omitted
m.
.
We s h a l l u s e a c o r o l l a r y of t h i s lemma, which w i l l be proved a f t e r t h e n e c c e s s a r y n o t a t i o n h a s been i n t r o d u c e d . We work w i t h H i l b e r t s p a c e s H5 : t h e s e a r e t h e c o m p l e t i o n s of C t h e norm
m
(V) w i t h
I. i 5 : llfll
5
=
It is c l e a r t h a t H
and H-5
*
=
IIA
-514
f,f)
*fn2
112
.
a r e d u a l when equipped w i t h t h e u s u a l p a i r i n g :
COROLLARY 2.12.
I f f E C (V), t h e n f E H
5
<
whenever 0
5
< r,
and f u r t h e r
.
where l / p = 112 + 5 / 2 r Proof.
I f F, = 0 , t h i s r e s u l t i s t r i v i a l .
t h a t 5 i s n o t a n even i n t e g e r .
We p i c k 5 v e r y c l o s e t o r , s o
F o r f i n C (V), P r o p o s i t i o n 2.9 i m p l i e s t h a t 512-r 4 fl12 c E ( 5 ) I N *fl12
.
Since N
512-
l i e s i n L ~ ' = ( V ) , where l / t = 112 + 1
Lemma 2.11 i m p l i e s t h e d e s i r e d r e s u l t .
-
l / p and l / p = 112 + f , / 2 r ,
An a p p l i c a r i o n of complex i n t e r -
p o l a t i o n now p r o v e s t h e c o r o l l a r y f o r g e n e r a l 6 .
Lemma 4.9 c o u l d a l s o be proved by a p p l y i n g r e s u l t s of
REMARK 2.13. G.B.
F o l l a n d [Fo2] ( P r o p o s i t i o n 3.17 o n p . 1 8 4 ) . An e a s y c o r o l l a r y o f t h e p r e c e d i n g c o r o l l a r y i s proved by d u a l i t y .
<
I f -r
COROLLARY 2.14.
f,
< 0,
t h e n 'H
i s contained i n L " ~ ( v ) ,
where l / p = 112 + 5 / 2 r , and
Proof.
Omitted
.
We conclude t h i s s e c t i o n by s p e c i a l i s i n g a n o t h e r r e s u l t of Hunt [Hun] ( s e e p. 271) t o f i t o u r n e e d s .
We g e n e r a l i s e a r e s u l t o f R.S.
Strichartz
[ S t r ] and N. Lohou6 [Loh], b u t t h e proof we o f f e r i s somewhat s i m p l e r t h a n theirs. LEMMA 2.15. l/q = l/t + l/p.
Suppose t h a t 1 < p, q , t
. d L e t M (V) be t h e s p a c e of a l l m i n
PROPOSITION 2.15. m
and t h a t
I f f E L P ' ~and m E L ~ ' ~t h ,e n mf E L~~~ and
Proof. h i t t e d
C (V\{(O,O)])
< m,
which a r e homogeneous of. d e g r e e d , where d E C
<
< 5 1 < r , and 50 = 2Re(d), 0 d t h e n p o i n t w i s e m u l t i p l i c a t i o n by m i n M (V) d e f i n e s a bounded 5~ f,l o p e r a t o r from H to H and Re(d) G 0 .
I f -r
f,
.
Proof.
We c o n t e n t o u r s e l v e s w i t h a b r i e f s k e t c h of t h e p r o o f , and
r e f e r t h e r e a d e r t o M. Cowling and A.M.
Mantero [CoM] f o r f u r t h e r d e t a i l s . The
proof i n v o l v e s f i v e e a s y p i e c e s .
< 0 < ,C 1, t h e n t h e r e s u l t i s a n i n m e d i a t e consequence of 0 C o r o l l a r i e s 2.12 and 2.14 and Lemma 2.15. First, i f 5
...,
Next we t a c k l e t h e c a s e where 5 = 5 = -2. L e t (X X ) be a b a s i s 0 1 1' P -2 of V To show t h a t m f i s i n H i f f i s , i t s u f f i c e s t o prove t h a t X.(mf) i s J 0 in H S i n c e X.(mf) = (X.m)f + m ( X . f ) , and s i n c e X.f l i e s i n H , t h i s can be 3 3 J J done by t h e f i r s t argument.
. -A .
T h i r d , by complex i n t e r p o l a t i o n and d u a l i t y , t h e c a s e s where 5 = 0 C0 E [-2, 2 ] a r e t r e a t e d . The f o u r t h s t e p i s t o c o n s i d e r t h e c a s e where 6
0
done by c o n s i d e r i n g D(m£),
=
El
< 0.
and
T h i s c a n be
where D i s a r i g h t - i n v a r i a n t d i f f e r e n t i a l o p e r a t o r
of o r d e r k and 2k + 5 E [O, 2 ) . 0 F i n a l l y , d u a l i t y and complex i n t e r p o l a t i o n a r e a p p l i e d t o f i n i s h o f f t h e proof.
3.
REPRESENTATIONS OF HEISENBERG GROUPS.
We s h a l l now d e v e l o p t h e r e p r e s e n t a t i o n t h e o r y of V , u s i n g t h e K i r i l l o v method ( s e e A.A.
K i r i l l o v [ K i l ] o r [KiZ], o r L. Pukanszky [Puk])
.
I t would be
p o s s i b l e t o do t h i n g s more d i r e c t l y , b u t t h e chosen p a t h may have d i d a c t i c advantages. The group V i s i n f a c t a group of m a t r i c e s .
By
1 we
denote the Lie alge-
b r a of a l l m a t r i c e s o f t h e form:
( t h e ( n + l ) x ( n + l ) m a t r i x i s d i v i d e d o n t o b l o c k s of s i d e 1 o r (n-1): b l o c k is ( n - l ) x ( n - 1 ) ) .
the central
We a b b r e v i a t e t h i s m a t r i x t o x(X,Y) o r j u s t (X,Y).
The group V i s e x a c t l y t h e s e t of a l l e x p o n e n t i a l s of t h e s e m a t r i c e s : we w r i t e V(x,y) o r j u s t (x, y ) t o i n d i c a t e e x p ( r ( x , y ) )
,
i .e.
T h i s f o r m u l a t i o n g i v e s r i s e t o t h e m u l t i p l i c a t i o n w i t h which we have been dealing. M a t t e r s a r e a r r a n g e d such t h a t exp i s a diffeomorphism of map f t-+
foexp i s a n isomorphism of C
m
C
(V) o n t o C
m C
y
o n t o V;
the
(V), and a l s o of S(V) o n t o
-
S ( 1 ) ( t h e s e a r e t h e Schwartz s p a c e s of f u n c t i o n s which, t o g e t h e r w i t h a l l t h e i r d e r i v a t i v e s , v a n i s h a t i n f i n i t y f a s t e r t h a n any polynomial g r o w s ) . F u r t h e r , t h e Haar measure of V c o r r e s p o n d s t o Lebesgue measure on
1, i . e .
jVdxdy f ( x , y ) = jVdXdY foexp(X,Y)
-
The a d j o i n t r e p r e s e n t a t i o n of V on Ad(x, y) (X,Y) i s t h a t e l e m e n t of
1 is
described a s follows:
1 which
c o r r e s p o n d s t o V(x,y)y(X,Y)V(x,y)
(x,
41m(x~*)
-1
,
i .c. ~ d ( x , y () x , Y ) = The ( r e n 1 ) d u a l
*!
of
the f o l l o w i n g manner.
Y
-
.
i s j u s t Horn (V R ) ; we i d e n t i f y t h i s w i t h F R -' By ( a , X ) we d e n o t e t h e l i n e a r f u n c t i o n a l :
n-1
xIm(F) i n
((a,
A ) I (X,
Y)) = Re(aX
The c s a d j o i n t r e p r e s e n t a t i o n Cd of V on
Y
1!
*
I
+ XY )
(X,Y) E
i s t h e c o n t r a g r e d i e n t of t h e a d j o i n t
representation: (Cd(x,y)(cr,X)l (X,Y)) = ( ( a , X ) l ~ d ( x , y ) - ~ ( ~ , ~ ,) ) whence, by s t r a i g h t f o r w a r d c a l c u l a t i o n , Cd(x,y)(a,X)
= (a
-
.
4Xx,X)
There a r e two d i s t i n c t c a s e s of t h i s a c t i o n : X = 0 , and X # 0 .
I n the
former c a s e , then t h e V-orbit i s a p o i n t , and o t h e r w i s e i t i s a plane 0 Cd(V) (a.0) = C (a,O)
A:
1
0, = cd (v) (a, A ) = (pn-l ,A)
(if A
+ O) .
The s t a r t i n g p o i n t of K i r i l l o v ' s t h e o r y is t h a t we can d e f i n e d i s t r i butions T
(a,h)
on V by t h e formula
T ( f ) = JVdxdy f-exp(X,Y) ( a , X) where e ( r ) d e n o t e s e x p ( i r ) . We o b s e r v e t h a t
-
e ( ( ( a , A ) l (X,Y) ))
,
(f)=(foexp)^(a,X) , (a,h) d e n o t e s t h e Euclidean F o u r i e r transform of a f u n c t i o n g on a Euclidean T
where
space, i n t h i s c a s e phisms of V i n t o
X.
When X = 0 , t h e s e a r e c h a r a c t e r s of V,
(i.e.
T), b u t otherwise t h e y a r e not: some "V-invariance"
hommori s needed.
The necessary V-invariance
can be o b t a i n e d by i n t e g r a t i n g over b e o r b i t 0 A' n-1 The a p p r o p r i a t e measure i s o b t a i n e d by t r a n s f e r r i n g Lebesgue measure on F ( i . e . V / k e r ( ~ d ) ) . We d e f i n e t h e d i s t r i b v ~ t i o nT TA(f) =
I n-ldx F
X
on V by t h e formula
T (f) Cd(x,O) (0,X)
Then T (£1 i s t h e i n t e g r a l of (foexp)- o v e r 0 A AWe s h a l l now s t u d y T and show t h a t i t i s a " c e n t r a l " d i s t r i b u t i o n ; t h i s A' r e s u l t could be obtained by applying Lemma 3 . 2 (below) which d e s c r i b e s T X explicitly. Suppose t h a t f be a f u n c t i o n on V,
t h a t (x,y) E V, and t h a t f ' be d e f i n e d
by t h e formula:
Then we s a y t h a t f i s c e n t r a l i f f = f ' ; we say t h a t a d i s t r i b u t i o n T i s cent r a l i f T ( f ) = T ( f l ) ( f o r a l l c h o i c e s of (x,y) i n V). and T
(a.0)
The d i s t r i b u t i o n s T
a r e c e n t r a l : we s h a l l check t h i s o n l y f o r t h e former.
X
LEt4N.A 3.1.
The d i s t r i b u t i o n s T d e f i n e d above a r e c e n t r a l . h
Proof.
F i r s t we e x p r e s s T i n terms of t h e F o u r i e r t r a n s f o r m of f , X and then we compare t h e F o u r i e r transforms of f and f ' , d e f i n e d a s above. We
x .-
have t h e formula f o r T (3.1)
=
IFn-ldx
= jFn-1
T ~ d ( x , (~0), ~ () 0
dx (foexp)^(Cd(x,O)(O,X))
.
Furthermore,
(f 'oexp)^(a,X) IVdxdy fl(cxp(X,Y)) e ( ( ( a , X ) I (X,Y) )) f ( ( x . y ) e x p ( ~ , ~ ) ( x , y ) - ~e) ( ( ( a , ~ ) (x.Y))) l
= =
=
I;dKdY
=
*I-
f ~ e x p ( A d ( x . y ) ( x , Y ) ) e ( ((.,All
(X,Y) ))
~ O U P ( ( X , Y ) )e ( ( ( a , ~ I) M ( X , Y ) - ~ ( X , Y)) I foexp((X,Y)) e((Cd(x,y) (a,X)l (X,Y) ))
.
T h i s l a s t formula is combined w i t h t h e preceding e q u a l i t y (3.1) t o conclude: Th(f '1 dx' ( f ' oexp)^(Cd(xl ,O) (0,X))
=
a
= I
IFn-1
dx' (f~exp)^(Cd(x,y)Cd(x',O)(O,X))
IFn-1dx' Ip-1dx'
= IFn-1 dx'
( f oexp)^(Cd(x + x ' , y
-
21m(xx1*)) ( 0 , ~ ) )
( f o e x p ) ^ ( ~ d ( x+ xl,O)(O,X)) (f oexp)^(Cd(xl ,O) (OJ))
I n f a c t , i t i s p o s s i b l e t o d e s c r i b e T more e x p l i c i t l y . h
LEMMA 3.2.
The d i s t r i b u t i o n T, i s g i v e n by t h e following formula: A
-
T A U ) = ( 1 ~ 1 2 l h l )j~I m ( F ) d ~ f ( ( 0 , y ) ) e ( ~ e ( ~ y . 1 ) n-1 f o r a l l f i n C (V), where p = dim (F ). Proof.
We omit t h i s p r o o f , which u s e s t h e i d e a s a l r e a d y v e n t i l a t e d
together with standard Fourier a n a l y t i c techniques. The n e x t s t e p of t h e K i r i l l o v t h e o r y i s t o a s s o c i a t e a r e p r e s e n t a t i o n t o
g.
The philosophy behind t h i s i s t h a t s i n c e the o r b i t s g i v e each o r b i t 0 i n h r i s e t o c e n t r a l d i s t r i b u t i o n s , and s i n c e r e p r e s e n t a t i o n s give r i s e t o c e n t r a l d i s t r i b u t i o n s (by t a k i n g the c h a r a c t e r ) , t h e r e might be some connection between the o r b i t s and the r e p r e s e n t a t i o n s .
We s h a l l d e s c r i b e t h e general procedure
very b r i e f l y . To t h e element 13 of
1, we 5!
(3.2)
a s s o c i a t e a b i l i n e a r form B Be(W,Z) = e([W,Zl)
(where [ w , z ] i s t h e Lie product of W and Z, i . e . A " p o l a r i s a t i o n " of
1i s
1) such
that
subspace of
the f u n c t i o n a l
€I
of
H of 1, t h e r e
defining
1
t h e i r commutator a s m a t r i c e s ) . a maximal [
, 1 -closed
.
B e ( g , g = I01 Associated t o t h e subalgebra
1, by
W,Z E
(i.e.
a maximal subalgebra
on
e
i s a subgroup H of V; a s s o c i a t e d t o
x , t h e r e i s a c h a r a c t e r x e of f
H (i.e.
xe
i s a homomorphism
of H i n t o T, where T i s the group of complex numbers of modulus one under mu1t i p l i c a t i o n ) , given by the formula x,(exp(W))
= e(0(W))
One "induces" t h i s c h a r a c t e r t o a r e p r e s e n t a t i o n of V.
.
WE!
One of t h e main r e s u l t s
of K i r i l l o v ' s theory i s t h a t we o b t a i n ( e s s e n t i a l l y ) t h e same r e p r e s e n t a t i o n a s
we v a r y t h e p o l a r i s a t i o n . L e t u s now apply t h i s general method t o the case i n hand.
We take X i n
I ~ ( F ) \ { o ) , and c o n s i d e r t h e b i l i n e a r form B ( c f . (3.2)) given by the r u l e : A BA((X,Y),(XV,Y')) = Re(A[ - 2 h ( M V * ) ] * ) = 2Re(XM1*)
.
We now meet one of t h e u n s a t i s f a c t o r y a s p e c t s of t h e theory, v i z , t h e r e i s no canonical choice of p o l a r i s a t i o n . d i s c u s s i o n would be f u t i l e .
(3.3)
We suppose t h a t F # R, f o r otherwise t h i s
I f F = C, t h e n i t i s p o s s i b l e t o take n-1 H=iJ +V CI1+V 3 -2 *
g
a s follows
while i f F = Q, we t a k e
-H =
(aRn-l + bR
n-1
)
+ x 2 c t 1+I&
,
where a and b a r e two perpendicular v e c t o r s i n the ( r e a l ) space F which a l s o s a t i s f y a A-dependent condition.
The p o i n t i s t h a t a and b must vary with A ,
but t h e r e i s no p a r t i c u l a r l y good way t o choose them.
1 has
Once
We o b t a i n
been chosen, we employ t h e i n d u c t i o n procedure.
f i r s t the character
xX
of H, and t h e n c o n s i d e r t h e s p a c e of (measurable) func-
t i o n s on V which t r a n s f o r m by
x
X
on t h e l e f t , i . e .
such t h a t
f (hv) = xX(h) f ( v )
h E H , v E V
,
and which s a t i s f y t h e i n t e g r a b i l i t y c o n d i t i o n t h a t If! be f i n i t e , where 2 1/2 Of! = (lH,vdc l f ( c ) l 1 We s h a l l i l l u s t r a t e m a t t e r s f o r t h e c a s e where F = C. I f F = C, more e x p l i c i t p a r a m e t r i s a t i o n i s p o s s i b l e . We w r i t e A = Lv, n-1 , and a l s o y = i w , w i t h w i n R. w i t h v i n R, and x = u + i v , w i t h u , v i n R We t a k e
fl
t o be t h e s u b a l g e b r a i n d i c a t e d i n t h e formula (3.3) above and func-
t i o n s f such t h a t
.
f ( ( ~ v ' , ~ w ' ) ( u + i v , ~ w )=) e ( v w l ) f ( ( u + i v , i w ) ) T h i s t r a n s f o r m a t i o n c o n d i t i o n may be r e p h r a s e d a s f o l l o w s : f ( ( u + ~ v i,w ) ) = e(v[w + ~ v u * ] ) ~ ( ( u , o ) ) and f i s determined by i t s r e s t r i c t i o n t o Rn-'x{O),
,
which i s a " s e c t i o n " f o r
We u s e t h e norm:
H\V.
Ilfl = [lRn-ldr
lf(r.O)I
2 1/2
1
Now we l e t V a c t on t h e s p a c e of such f u n c t i o n s , on t h e r i g h t , t h u s : (aX(v)f)(vl) = f(vlv)
v,vl E V
.
With t h e n o t a t i o n (x,y) = ( u + Lv, i w ) , we f i n d t h a t ( f o r t h e complex c a s e )
I f we w r i t e F ( r ) i n s t e a d o f f ( ( r , O ) ) , we have t h a t [ s (U
A
Thus
+ i v , i w ) ~ (]r )
= exp(iv[w
+ 4rv* + 2vu*]) F ( r + u)
.
( ( u , 0 ) ) a c t s by t r a n s l a t i o n s , n X ( ( i v , 0 ) ) a c t s by m u l t i p l i c a t i o n by X c h a r a c t e r s , and n ( ( 0 , i w ) ) by s c a l a r s . T
X
L e t u s c o n s i d e r some F o u r i e r t r a n s f o r m s .
I t i s n a t u r a l t o d e f i n e a (X) A
by t h e r u l e t h a t n (X)f(r) = [ d / d t aA((tX, O ) ) f ( r ) l t = A
and s o on.
I f X l,...,Xn-l
a r e the v e c t o r s (1,0,
,
...,01, ..., ( 0 , 0 , ...,1 )
i n -V1
'
then (x.)f(r) = (a/ar.)f(r) J J and r (iX.)f(r) = 4ivr f ( r ) 2 A - J 2 j 2 2 2 I t f o l l o w s t h a t -{a (X ) + a (iX.) 1 i s the o p e r a t o r -3 / a r + 16" r A j A- J j j ' t h i s o p e r a t o r becomes t h e Hermite o p e r a t o r (-dL/dtL + t L ) when we e f f e c t the IT
A
.
.
.
1n f a c t change of v a r i a b l e s s = 2 1 ~ 1 r 2 2 2 2 -{a (X ) + a ( i x . ) I f ( s ) = 4 1 ~ 1{-a /as + s I f(s) j A - J jj The Hermite o p e r a t o r on R i s known t o have a d i s c r e t e spectrum, with eigenv a l u e s 1, 3, 5,
..., and
f u n c t i o n s DO, Dl,
D2,
.
1-dimensional eigenspaces g e n e r a t e d by the Hermite
....
I t f o l l o w s t h a t , i f we form f u n c t i o n s of the form
then t h e s e a r e e i g e n f u n c t i o n s f o r a, ( A ) , and n-1" nA(b)[Dm B . . . @ D m J = ~ I A I [ ~ (2m ~ = ~ +l)lDm @ . . . @ D ~ ; j 1 n- 1 1 n-1 the expansion we o b t a i n i n t h i s way i s a complete e i g e n f u n c t i o n expansion f o r
on R"
&,
aA(A). The e i g e n v a l u e s a r e 41XJ(n
- 1),
41XI(n + 1 ) , 41XI(n + 3 ) , and s o on.
I t may be worth p o i n t i n g o u t t h a t t h e a c t i o n of M on V g i v e s r i s e t o arc a c t i o n on each of t h e e i g e n s p a c e s , which permutes t h e o r d e r of t h e D and t a k e s m, weighted sums of t h e s e . J I t i s a p p r o p r i a t e t o view a (A) a s a diagonal o p e r a t o r of t h e form A
where the blocks correspond t o t h e d i f f e r e n t eigenspaces.
2 Thus r (A ) c o r r e s A
ponds t o
.... The same h o l d s when F =
.... ....
Q: p i s dim ( F ~ - ' ) , a s b e f o r e .
R 2 Analogously, a (n) corresponds to I A l I , where I i s t h e i d e n t i t y o p e r a t o r . A
The r e a s o n f o r t h i s a n a l y s i s i s the following lemna, which i s proved by t h e methods of 5 1 .
LEMMA 3.3.
The following equality holds: 5-r-4 ( A ~+ B(c)n) N5-' = C(5) N 2 where B(5) = -4(5-2) and C(5) = (5-r) (c-2) (5-p-2) (5-4)
.
Proof. Omitted. On the Fourier transform side, we find that there are recurrence relations which connect the Fourier transforms of the functions ."'N
These Fourier
transforms are also diagonal, and are made up of blocks of the same size as the th blocks which make up the Fourier transform of A. We obtain, for the k block, the equality C(5) (k = 1, 2, 3,
zA(N
- -Ik
= 41112 [(p + 4k
...). It is clear that, if N
definite, then N'-~
- 4) 2 - (5 - 2) 2I
5-r-4
nA(~c-r)k
is positive (or negative)
is also positive (or negative) definite if and only if 2 2 (p + 4k 4) - (5 - 2) > 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
<5
0
p + 29, while if F = Q, then this same condition holds only if 2q.
This will correspond to the fact that the groups Sp(n,l)
(F = Q) have "property T", while the groups SU(n, 1) (F = C) do not.
4.
SOME SIMPLE GROUPS
n+l We s h a l l c o n s i d e r the v e c t o r space F over F, where s c a l a r s a c t on t h e l e f t , with b a s i s ( e l , e 2 ,
..., e n )
over F, and t h e s e s q u i l i n e a r form q given by
t h e formula
1j = l
zjcj
-
-
.
z,c€Fn+l n+1 which preserve We denote by O(q) t h e group of l i n e a r t r a n s f o r m a t i o n s of F q ( z , 5) =
zoSO
t h i s form ( t h e s e transformations a c t on t h e r i g h t ) . a n a l y s i s on O(q).
We s h a l l consider harmonic
In what f o l l o w s , we must assume t h a t F i s e i t h e r C o r Q,
s i n c e i n t h e r e a l c a s e t h e r e a r e some ( s i l l y ) complications due t o t h e nonconnectness of O(q) (which has f o u r components); t h e r e s u l t s s t a t e d below a r e sometimes t r u e f o r t h e connected component of O(q) only and sometimes t r u e f o r t h e whole group only.
We leave t o t h e r e a d e r t h e t a s k of deciding f o r which
of t h e s e groups t h e theorems hold.
To t a c k l e harmonic a n a l y s i s on O(q), we
s h a l l need harmonic a n a l y s i s on some o t h e r groups. I n o r d e r t o d e s c r i b e some of t h e o t h e r groups t h a t we s h a l l meet, we s h a l l n+l l e t f 0 = (en eo) / J 2 , f n = ( e + e ) / J 2 , and consider a second b a s i s of F n 0 j and f = e i f 1 < j < n-1. Coordinates i n the new system w i l l be w r i t t e n z j
..
-
.
j
Then t h e following formulae hold: j (z = (2.1 U
J
j (2.1 = ( 2 ) U 3
z E pn+l
,
where the m a t r i x U being made up of blocks of s i d e 1 o r n-1, being (n-l)x(n-1);
t h e number c i s 1/J2.
represented a s follows:
I n t h e new system, t h e form q i s
r.
n-1 j - j zO;n z c j=1 One of t h e groups of i n t e r e s t i s t h e following: s(z9 5) =
and t h e c e n t r a l block
K=O(q)nO(I.l)
+
+
zn;O
,
n+l where 0(1 . I ) i s t h e compact group of a l l l i n e a r t r a n s f o r m a t i o n s of F which preserve t h e l e n g t h of v e c t o r s , d e f i n e d i n the u s u a l way.
The group K i s made
up of e l e m e n t s k ( v , w), which, i n t h e f i r s t c o o r d i n a t e system, a r e d e s c r i b e d by m a t r i c e s o f t h e f o l l o w i n g form: kv,
1
=
;1 I:
where v and w a r e 1x1 and nxn norm-preserving m a t r i c e s r e s p e c t i v e l y . e l e m e n t of K i s t h e e x p o n e n t i a l of a n e l e m e n t of i t s L i e a l g e b r a
5,
Every which i s
t h e s e t of a l l m a t r i c e s o f t h e same form, b u t which s a t i s f y t h e c o n d i t i o n s t h a t (The a d j o i n t w* of w i s t h e c o n j u g a t e t r a n s p o s e . )
vw = -v and w* = -w.
O t h e r groups which w i l l a t t r a c t o u r a t t e n t i o n a r e b e s t viewed i n t h e o t h e r c o o r d i n a t e system: A
V = {v(x, y): x E
F
-
{ a ( t ) : t E. R ) , where
n- 1
,y
E I m ( F ) ) , where v ( x , y ) i s t h e e l e m e n t V(x, y) o f 13;
N = { n ( x , y ) : x E F ~ - ' , y E I m ( F ) ) , where
M = {m(v, u ) : m(v, u ) E K ) , where
I t t u r n s o u t t h a t v s a t i s f i e s t h e c o n d i t i o n Ivl = 1, w h i l e u i s a n i s o m e t r i c (n-l)x(n-1)
matrix.
The group V i s e x a c t l y t h e group which we have d i s c u s s e d
f o r the f i r s t three sections.
I t i s e a s y t o check t h a t M c e n t r a l i s e s A ( i n f a c t M i s t h e c e n t r a l i s e r of
A i n K), and both M and A n o r m a l i s e V and N. MAV a r e a l l subgroups of G.
Thus MA, MN, AN, MAN, MV, AV, and
The a c t i o n of M and A on V of I 1 i s c o n j u g a t i o n .
I t w i l l be h e l p f u l t o s t u d y c e r t a i n n a t u r a l g e o m e t r i c a c t i o n s of t h e group
S i n c e G p r e s e r v e s t h e form q , i t p r e s e r v e s t h e f o l l o w i n g cone r: r = {z E Fn + l : q ( z , z ) < 0 ) n The u n i t d i s c D i n F (D = { z E F ~ :l z l < 1 ) ) can be i d e n t i f i e d w i t h t h e pro-
G.
.
j e c t i v e space F
where
* F
X
\r
a s follows: z cr ( 1 , z)
= F\{O).
-
{(w, WZ): w E F * )
Then, s i n c e t h e G-action comnutes w i t h t h e F - a c t i o n ,
we have
a G-action on D by f r a c t i o n a l l i n e a r t r a n s f o r m a t i o n s : 1 zog = ( a + zc) ( b + zd)
-
where, i n t h e f i r s t c o o r d i n a t e system,
We f i n d i m e d i a t e l y t h a t
a being a 1x1 m a t r i x and d being nxn. oog = 0
i f and o n l y i f g E K, and t h a t K a c t s on D by r o t a t i o n s . Hence 0 i s t h e unique The a c t i o n o f A on D may be c a l c u l a t e d a s follows: i n the
K-fixed p o i n t o f D.
f i r s t c o o r d i n a t e - system,
where c ( t ) and s ( t ) a r e t h e h y p e r b o l i c c o s i n e and s i n e of t r e s p e c t i v e l y . Theref o r e Ooa(t) = t a n h ( t ) e
n We thus o b t a i n a " p o l a r decomposition" of D: D = OoAK and hence t h e "Cartan decomposition'' of G: G = KAK
.
We now c o n s i d e r t h e same G-action i n t h e second system of c o o r d i n a t e s . s i n c e (2;)-
=
wi (w, z E F) ,
lj=l
I f q (z, z)
<
*
n-1 Iz j 1 ~ ( Z 2) Y = 0 n 0 0 , then Re(z ) < 0 and z # 0.
z
+ Z R (zOzn) ~
z E Fn+l
.
Thus t h e p r o j e c t i v e cone
may be i d e n t i f i e d w i t h the "Siegal domain" S: S = Cz E F ~ h : (z)
< 01 ,
where h i s t h e h e i g h t f u n c t i o n g i v e n by t h e formula
-
The i d e n t i f i c a t i o n proceeds by t h e two correspondences:
z
c--,
('1,
Z)
*1
{(-W, WZ): w E F
zEF"
.
Before we s t u d y the a c t i o n of G on S, we s t a t e f o r m a l l y t h e r e l a t i o n between D and S.
LEMMA 4.1.
The spaces D and S a r e isomorphic.
l e t 41 and J, be the maps given by t h e formulae:
In p a r t i c u l a r ,
-1 2
z j -1 $ ( ~ ) ~ = ( l - z ~( l) + z n )
-
= (1
2
'
2 .
and $ i s i t s i n v e r s e .
l),
(1 G j < n
-
I),
as u 1-1,
Then $ i s a b i j e c t i o n of D o n t o S and of a D o n t o t w i n e s t h e G-actions,
-
<j
,
2 zj
$ ( z ) , = ( z n + 1j-l
< n
(1
which i n t e r -
Moreover, $(O, 0 ) = ( 0 ,
11,
@ ( - I , 0 ) = (0, O), and -2 2 I ( 1 - lzl ) n 1zn+ 11-~4h(z)
h($(z)) = 1 1 - 2 1$(z)1 = 1
whilst Proof.
-
then f o r any z i n (-1,
2,
1;P-l
z E S
.
F i r s t , i f x E EP-l,and
and w i n F,
w) v ( x , y) = (-1,
( 2 , W ) O V ( X y) , = (Z
whence
,
Obvious from t h e c o n s i d e r a t i o n s above.
We now c o n s i d e r t h e a c t i o n o f G on S, and o n 3 s . y E Im(F),
Z E D
z
- x,
-
x, w
w
-
-
zx* + (XX*
zx* + (xx*
-
-
y)/2)
y)/2)
,
.
We n o t e t h a t , s i n c e Re(y) = 0 , h ( ( z , W ) O V ( Xy, ) ) = Re(w
-
zxf + (xx* -y)/2) 2 = Re(w zx*) + (112) 1x1 2 = Re(w) - (1/2)121
V a c t s on S and o n
as,
S i n c e , i f h E R,
preserving the height.
( 0 , h ) o v ( x , y) = (-x,
(4.4)
2
,
= h ( ( z , w))
i.e.,
- (1/2)1z - xl 2 - (1/2)l z - X I
h + (xx*
-
y)/2)
,
t h e n a n a r b i t r a r y e l e m e n t o f S of h e i g h t h i s t h e image of ( 0 , h ) by a n a p p r o p r i a t e element of V, and f u r t h e r o n l y t h e i d e n t i t y element e o f V f i x e s (0, h ) , i . e .
V a c t s simply t r a n s i t i v e l y o n t h e l e a v e s of t h e f o l i a t i o n induced
by h. Next, we c o n s i d e r t h e a c t i o n of M. (-1,
We have t h a t
z , w) m(v, u ) = (-v,
zu, wv) -1 1 = v (-l,v zu, v wv)
-
,
whence i n S, (4.5)
(-1,z
, w)
m(v, u) = ( v
-1
zu, v
-1
wv)
.
I n p a r t i c u l a r , M s t a b i l i s e s t h e e l e m e n t s ( 0 , h) ( h To p r o c e e d , we c o n s i d e r t h e a c t i o n of A .
> 0)
of S and ( 0 , 0 ) i n
It i s c l e a r t h a t
as.
whence i n S t 2t ( z , w ) o a ( t ) = ( e z , e w) 2t (0, l ) o a ( t ) = (0, e )
(4.6) In particular,
.
.
We conclude o u r s t u d y of t h e a c t i o n of G on S by c o n s i d e r i n g one s p e c i a l element of K .
8H
L e t w be t h a t e l e m e n t of G g i v e n by t h e m a t r i c e s
-a
and
i n t h e f i r s t and second c o o r d i n a t e s y s t e m s r e s p e c t i v e l y ( t h e f i r s t m a t r i x h a s an nxn i d e n t i t y s u b m a t r i x , w h i l e t h e second c o n t a i n s an (n-l)x(n-1)
submatrix
It i s c l e a r t h a t w a c t s on D by m u l t i p l i c a t i o n by -1 W e n o t i c e a l s o t h a t w = w , and t h a t
i n the central position). -1.
w a ( t ) w = a(-t)
.
t € R
I t i s e a s y t o check t h a t w commutes w i t h e a c h m i n M, and t h a t
.
w V(X, y) w = n(x, y) We f i n d t h a t , i n t h e second c o o r d i n a t e system, (-1,
z , x)
W
= ( x , 2, -1) =
-x (-1,
-x
-1
2, X
whence t h e a c t i o n of w on S i s g i v e n by t h e formula -1 -1 (4.7) (2, x ) o w = (-x 2, X ) The same f o r m u l a a p p l i e s f o r ( z , x) i n
LEMMA 4.2.
-1
,
)
.
We now prove a lemma.
as.
Suppose t h a t v ( x , y) be i n V , and t h a t v ( x , y) # e .
Then, f o r t h e a c t i o n of G on aS a l r e a d y d e s c r i b e d , we have t h e formula (0, O)ov(x, y ) o w = ( 0 , O)ov(x * -1 xt = -(XX - y) 2x t -2 y =-Jxx*-yl 4y
where
proof.
t
,
The formulae above ( ( 4 . 4 ) and ( 4 . 7 ) ) imply t h a t
( 0 , O)OV(X,y).w
=
(-~(xx*
* * ((XX -
= ((xx
while
,y
.
and
and t h e n
t
t
( 0 , O)OV(X, y
t
=
-1
-1 ~ ( x x *- y) ) -1 y)-12x, (XX* y) 2) , -1 -1 2 y ) 2x, ( ((xx* - y) 2x1
-
y)
(-x),
-
-
t
y )/2)
= ((xxf = ((XX*
-
*
-1
2x, Ixx -1 y) 2x, (XX* y)
-
yl
- y)
-2 -1
2(xx 2)
*
+
y))
,
a s required. I t f o l l o w s from o u r s t u d y of t h e a c t i o n s of G o n D and on S t h a t G = KAV
(because (0.1) i s t h e K-fixed
p o i n t i n S, and t h e r e i s a n obvious b i j e c t i o n of
AV o n t o S , g i v e n by t h e formula a v k ( 0 , l ) o a v ) ,
G = KAN ( o b t a i n e d from t h e p r e v i o u s decomposition by c o n j u g a t i n g by w), and G = NAK ( o b t a i n e d from t h e p r e c e d i n g decomposition by t a k i n g i n v e r s e s ) .
Finally,
G = ANK ( s i n c e AN = NA); t h i s i s what we s h a l l c a l l t h e Iwasawa decomposition of G. F u r t h e r , c o n s i d e r a t i o n of t h e a c t i o n of G on
as
l e a d s t o t h e s o - c a l l e d Bruhat
decomposition: G = MANV
.
u MANw
T h i s i s a d i s j o i n t u n i o n , and MANw i s of a lower dimension t h a n MANV.
For t h e
s a k e o f c o m p l e t e n e s s , we o f f e r a b r i e f d i s c u s s i o n of t h i s d e c o m p o s i t i o n . LEMElA 4 . 3 .
Evcry c l e m c n t g of G h a s e i t h e r a unique d e c o m p o s i t i o n
o i Lhc form g = m a w o r a unique d e c o m p o s i t i o n of t h e form g = manw. Proof. malise N, ((4.1),
We r e c a l l t h a t M c e n t r a l i s e s A , and t h a t M and A b o t h nor-
s o t h a t MAN i s a group.
(4.2),
Moreover, i t i s c l e a r from t h e d e f i n i t i o n s
(4.3)) t h a t M
~
A
=
M
~
N
=
N {~e ) A
,=
s o t h a t e a c h element of MAN h a s a unique e x p r e s s i o n of t h e form man. We now c l a i m t h a t i f g E G , and (0, 0 ) o g = ( 0 , 0 ) ( w i t h t h e a c t i o n of G on S and aS a l r e a d y d e s c r i b e d ) , t h e n g E MAN.
To show t h i s , we w r i t e g ( i n t h e
s t ~ c o n dc o o r d i n a t e system) a s t h e f 01 lowing m a t r i x :
S i n c e (1, 0 , 0 ) g = ( a , b, c ) , i t must be t h a t ( 0 , 0 ) o g = ( 0 , 0 ) i f and o n l y i f
Moreover, i f g i s of t h i s form, t h e n
b = O and c = 0 .
n-1 X E P ,YE?' n- 1 Since G p r e s e r v e s t h e form q, t h e f o l l o w i n g e q u a l i t y h o l d s f o r any x i n F (y, x , 0)og = (ya + xd, x e , x f )
.
and y i n F: 2~e(y6) +
1x1
2
= 2Re((ya + xd)(xf)-)
+
(xet
2
.
By l e t t i n g y v a r y , we conclude t h a t x f = 0; by l e t t i n g x vary, we deduce t h a t 2 n-1 f = 0 and t h a t 1x1 = l x e l f o r a l l x i n F It now f o l l o w s t h a t g l i e s i n
.
MAN, a s claimed,
I f , on t h e o t h e r hand, g l i e s i n MAN t h e n g is of t h e form j u s t d e s c r i b e d . I t is c l e a r t h a t (0, 0)og = (0, 0 ) . t r o p y group of (0, 0 ) i n
We have t h u s - p r o v e d t h a t MAN i s t h e i s o -
as.
Now suppose t h a t g E G, and c o n s i d e r ( 0 , O)og. l i t i e s : e i t h e r (0, 0 ) o g E aS o r (0, 0 ) o g =
-.
There a r e two p o s s i b i -
We s h a l l c o n s i d e r o n l y t h e f i r s t
of t h e s e p o s s i b i l i t i e s , s i n c e t h e second i s t r e a t e d i n a s i m i l a r manner.
There
e x i s t s a unique v i n V such t h a t (0, 0)og = (0, Olov, and f o r t h i s v , (0, 0)Og.V -l = (0, 0 ) i.e.,
gv
-1
E MAN.
t h a t g = manv.
By now e x p r e s s i n g
,
-1 u n i q u e l y gv
i n t h e form man, i t f o l l o w s
I n t h e o t h e r c a s e , we f i n d a n a l o g o u s l y t h a t g = manw: i t i s
enough t o remark t h a t (0, 0 ) ow =
13
a.
More g e n e r a l i n f o r m a t i o n about t h e geometric s i d e of t h i n g s can be found i n S . Helgason's comprehensive work
el]) and t h e r e f e r e n c e s t h e r e c i t e d .
We now need a n a n a l y t i c consequence of t h e a l g e b r a i c and geometric cons i d e r a t i o n s , v i z , a n e x p r e s s i o n f o r t h e i n v a r i a n t measure on G.
i s t o f i n d t h e commutator subgroup of G , H say.
The f i r s t s t e p
It i s e a s y t o s e e t h a t t h e
commutator subgroup of AN i s N and t h a t , w i t h the. a c t i o n of G on the d i s c D, OoN i s a connected s u b s e t o f D which h a s a l i m i t p o i n t on t h e boundary a D .
f o l l o w s t h a t t h e c l o s e d subgroup KH of G h a s t h e p r o p e r t y t h a t OoKH = D.
It Ve
conclude t h a t KH = G , and t h a t H i s co-compact i n C . We may now a r g u e t h a t G must be unimodular, i . e .
on G i s a u t o m a t i c a l l y a r i g h t liailr measure.
t h a t a l e f t Maar measure
For t h e k e r n e l of the modular
a homomorphism of G i n t o R
junction,
a f o r t i o r i co-compact;
+
, contains
t h e commutator subgroup, and i s
t h e modular f u n c t i o n must be t r i v i a l , and s o G i s i n d e e d
unimodular. S i n c e G = ANK, i t must be p o s s i b l e t o e x p r e s s t h e Haar measure o n G by means of a formula of t h e f o l l o w i n g type:
,
JGdg f ( g f = lAda JNdn JKdk $ ( a , n, k ) f (ank) where dn, d a , and dk a r e t h e Haar measures on N, A , and K.
T h i s formula im-
plies that jGdg f k k ' )
= lAda JNdn JKdk $ ( a , n, k ) f ( a n k k ' ) = jAda jNdn JKdk $ ( a , n , kk'
(by a change of v a r i a b l e s o n K ) , $ ( a , n, k l
-1
) f (ank)
and s o + ( a , n, e )
.
An a n a l o g o u s zrgument shows t h a t $ d o e s n o t depend o n t h e f i r s t v a r i a b l e ( a ) . Finally, since fGdg f (n'g) where n'
= a-'na,
= lAda l N d n
/K dk
$ ( a , n, k ) f ( a n - n k )
,
and s i n c e n' E N , t h e same r e a s o n i n g p e r m i t s u s t o c o n c l u d e
that
S i n c e we may n o r m a l i s e Haar measure a s we l i k e , we choose 4 t o be 1. lGdg f (g) = jAda l N d n JKdk f (ank)
Thus
.
I t i s p o s s i b l e t o e s t a b l i s h by e x a c t l y t h e same method t h a t
(4.8) where C
C.
lGdg f ( g ) = CG
'! M dm /Ad a jNdn V dv f (manv)
,
depends o n l y o n G.
We conclude t h i s i n t r ~ d u c t o r ys e c t i o n w i t h some v e r y b r i e f remarks a b o u t the a d j o i n t representation, ture.
t h e c o a d j o i n t r e p r e s e n t a t i o n , and t h e o r b i t s t r u c -
We c o n s i d e r o n l y t h e c a s e where F = R and n = 2, s i n c e i n g e n e r a l ,
t h i n g s g e t r a t h e r messy.
For t h e c a s e i n c o n s i d e r a t i o n , t h e L i e a l g e b r a con-
s i s t s of a l l m a t r i c e s of t h e form X(a, b , c ) , where, i n t h e f i r s t b a s i s ,
The a d j o i n t r e p r e s e n t a t i o n , which a c t s by c o n j u g a t i o n , o b v i o u s l y p r e s e r v e s t h e b i l i n e a r form q'-
q'(X,
Y)
= tr(XY)/Z
- and
since
q-(X(a.
b, c ) , X(a, b, c ) ) = a 2 + b
2
-
c
2
,
t h e r e i s a n o b v i o u s homomorphism o f G i n t o O(q-), whose k e r n e l i s {+I).
The
coadjoint r e p r e s e n t a t i o n i s e s s e n t i a l l y equivalent t o the a d j o i n t represent a t i o n , and t h e o r b i t s t r u c t u r e i s as f o l l o w s : t h e r e a r e two-sheeted hyperb o l o i d s l y i n g i n s i d e t h e cone I-,
t h e r e a r e one-sheeted
h y p e r b o l o i d s wrapping
around t h e cone, t h e r e a r e t h e two h a l v e s o f t h e cone, and t h e r e i s t h e o r i g i n . A t t e m p t s t o copy K i r i l l o v ' s t h e o r y i n t h i s s i t u a t i o n meet w i t h s e v e r a l difficulties.
One of t h e s e i s t h a t n o t a l l o r b i t s i n t h e c o a d j o i n t r e p r e s e n -
t a t i o n o r b i t s p a c e can g i v e r i s e t o r e p r e s e n t a t i o n s : one of t h e s t e p s i n t h e K i r i l l o v method i s t h e " e x p o n e n t i a t i o n "
of a b i l i n e a r form o n a s u b a l g e b r a t o
a c h a r a c t e r o f t h e c o r r e s p o n d i n g subgroup.
e
Now, j u s t a s t h e l i n e a r f u n c t i o n a l
k he g i v e s r i s e t o a c h a r a c t e r o f t h e t o r u s e i e
I-+
eiAe o n l y i f X i s
i n t e g r a l , s o i n o u r c a s e o n l y a d i s c r e t e set o f t h e e l l i p t i c o r b i t s ( i . e . two-sheeted h y p e r b o l o i d s ) o r b i t s g i v e r i s e t o r e p r e s e n t a t i o n s ,
the
the so-called
There i s no such o b s t r u c t i o n t o t o e x t e n d i n g f u n c t i o n a l s
discrete series.
associated t o t h e hyperbolic ( i . e .
one-sheeted
h y p e r b o l o i d ) o r b i t s , and t h e s e
give r i s e t o a continuous s e r i e s of r e p r e s e n t a t i o n s , c a l l e d t h e p r i n c i p a l s e r i e s , and r e a l i s e d by i n d u c t i o n .
A second problem w i t h t h e o r b i t method i s
t h a t t h e c o n s t r u c t i o n of t h e d i s c r e t e s e r i e s i s n o t a s t r a i g h t f o r w a r d u s e of t h e method of i n d u c t i o n , and i s i n g e n e r a l q u i t e d i f f i c u l t . Nevertheless,
t h e o r b i t p i c t u r e o f f e r s some u s e f u l i n t u i t i o n s .
stance, t h e n i l p o t e n t o r b i t s ( t h e half-cones)
For i n -
a r e l i m i t s o f b o t h e l l i p t i c and
h y p e r b o l i c o r b i t s ; t h e l a t t e r must be c u t i n h a l f , a s i t were, t o match t h e two h a l f - c o n e s .
These i n f a c t c o r r e s p o n d t o r e p r e s e n t a t i o n s l i n k e d w i t h b o t h
t h e p r i n c i p a l and t h e d i s c r e t e s e r i e s . We c o n c l u d e t h i s d i s c u s s i o n by n o t i n g t h a t w, t h e s p e c i a l element o f K , maps t h e o r b i t s i n t o t h e m s e l v e s , and t h a t , r e s t r i c t e d t o t h e p l a n e perpend i c u l a r t o t h e a x i s of t h e cone, t a k e s e l e m e n t s t o t h e i r n e g a t i v e s .
5.
THE PRlNCIPAL SERIES AND THE INTERTWINING OPERATORS.
From t h e o r b i t t h e o r y o r f o r o t h e r r e a s o n s , one s u s p e c t s t h a t t h e f o l lowing s e r i e s of r e p r e s e n t a t i o n s w i l l b e of i n t e r e s t . unitary representation
, and a c h a r a c t e r
H
One t a k e s a n i r r e d u c i b l e
of t h e g r o u p M o n a f i n i t e - d i m e n s i o n a l H i l b e r t s p a c e
p
cr o f A , and forms t h e r e p r e s e n t a t i o n p @ a o f MAN g i v e n by
11
t h e formula
.
v @ a : man k ~ ( m ) a ( a ) T h i s i s a r e p r e s e n t a t i o n b e c a u s e N i s normal i n MAN.
We t h e n i n d u c e t h i s rep-
T h i s c o u l d be done by c o n s i d e r i n g H -valued f u n c t i o n s f o n G
r e s e n t a t i o n t o G.
v
which t r a n s f o r m a c c o r d i n g t o t h e r u l e
f (mang)
= p
@3a (man) f ( g )
(where, o b v i o u s l y , m E M, a E A , e t c . ) and t h e n l e t t i n g G a c t on t h e r i g h t on t h i s s p a c e , t h u s o b t a i n i n g a r e p r e s e n t a t i o n n' (a-
v ,a
v,a
:
( g l f ) (g') = f (g'g)
g,gl E G
.
En o r d e r t o have a H i l b e r t s p a c e norm, we s h o u l d i n t e g r a t e o v e r V o r K ( b e c a u s e
f i s e s s e n t i a l l y d e t e r m i n e d by i t s v a l u e s on V o r on K) : we d e f i n e t h e norms
f . I I , ( ~ )and I I . I I , , ( ~ by ) t h e formulae: L
2 1/2
llfll,'"'
= (fKdk I l f ( k ) I )
n f ll,
= (JPV
2 112 I f ( v ) 11 )
We now e n c o u n t e r t h e problem t h a t G d o e s n o t a c t i s o m e t r i c a l l y on t h e s e s p a c e s . f o r instance,
Il n-
v,a
-1 2 112 va) 1 ) -1 21/2 = (JVdv l l f ( a va)A ) 2 112 # (J,dv I l f ( v ) l ) ,
( a ) f ll
= (Jvdv Jlf ( a a
t v a ( s = e ) are n o t measure-t t p r e s e r v i n g ; a t w i s t w i t h r h e Radon-Nikodym d e r i v a t i v e i s needed t o p u t t h i n g s
unless a is t r i v i a l .
The d i l a t i o n s D':
right. We d e f i n e p : A
+
---+
fi
by t h e r u l e
v I-+ a
where, a s u s u a l , ,n-1 r = dim ( P ) + 2dim ( I m ( f ' ) ) R R Then t h e f o l l o w i n g i n t e g r a l formulae h o l d : 1 -2 (5.1) JVdv f ( a v a ) = p ( a ) lVdv f (v) -1 2 (5.2) lNdn f ( a na) = o ( a ) dn f ( n ) N We may now d e f i n e t h e induced r e p r e s e n t a t i o n s i n such a way t h a t t h e y a r e u n i -
.
-
/
.
tary. LEMMA 5 . 1 .
Let
M on t h e s p a c e H
u b e a n i r r e d u c i b l e u n i t a r y r e p r e s e n t a t i o n of
and l e t a : A
v
-+
T b e a c h a r a c t e r of A.
L e t f be
a m e a s u r a b l e H -valued f u n c t i o n s u c h t h a t
u
f (mang)
-
.
u(m) ( p a ) ( a ) f ( g )
Then t h e f o l l o w i n g e q u a l i t y h o l d s : 2 2 /Kdk!f(k)! =CGIVdvflf(v)R
,
where C n i s t h e c o n s t a n t of t h e formula ( 4 . 8 ) , and c o n s e q u e n t l y t h e C1
group G a c t s u n i t a r i l y o n H
u a
t h e H i l b e r t s p a c e of f u n c t i o n s f
('I,
9
f o r which t h e above i n t e g r a l c o n v e r g e s , equipped w i t h t h e norm Proof.
11.1
,.*,
{Vl
2
We t a k e any f u n c t i o n @ i n C (AN) s u c h t h a t C
JANdadn @ ( a n ) = 1 and $ ( a n ) = @(marun-'),
,
and d e f i n e t h e f u n c t i o n $ on G by t h e formula $(ank) = @(an) p(a)-2
Then, by c h a n g i n g v a r i a b l e s , we f i n d t h a t JKd* llf (k)l12 = JANdadn jKdk @ ( a n ) R f ( k ) ll = JANdadn JKdk [$(ank) p ( a )
= C = C
/
G. MAN
G
dmdadn
/ANdadn
We w r i t e v = A'(v)N'(v)K'(v) -1 and l e t n' be A'(v) nA-(v);
V
2 2
1 I (pa) (a)
dv $I (manv) Il f (manv)
JVdv $I(anv) p ( a )
2
Ilf(v)8
11 2
-1Ilf (ank) ll 1 2
L
.
t o e x p r e s s t h e Iwasawa d e c o m p o s i t i o n of v i n V , then
IKdk I I ~ ( I C ) I I ~
-
= CG jANdadn jVdv $J(aA- (v) n-N-(v)K-(v) ) p ( a ) = CG jANdadn JVdv $(aA-(v)n')
p(a)
= CG jANdadn JVdv $ (aA- ( v ) n ) P ( a ) = C
G
AN
dadn JVdv $ ( a n )
= CG JVdv
f ( v ) 11
2
I1 f
2
2
I f (v)l
2
2
I f (v)
2
P (A-(v)
2
I f (v) A
2
2
;
t h e c r i t i c a l s t e p i s t h e change of v a r i a b l e s on N , j u s t i f i e d by t h e f o r m u l a (5.2).
Now 2
IKdk!f(k)!
.
2
=CGJVdvIf(v)l
To show t h a t G a c t s u n i t a r i l y o n H
i s now t r i v i a l . Both K and V a c t u n i u,a 2 2 t a r i l y , s i n c e r i g h t t r a n s l a t i o n s p r e s e r v e t h e L (K)- and L (V)-norms, and K and
V g e n e r a t e t h e whole group G.
I t w i l l be c o n v e n i e n t t o r e p a r a m e t r i s e t h e s e r e p r e s e n t a t i o n s : by a
5
we
d e n o t e t h e e x p o n e n t i a l g i v e n by t h e formula n : a ( t ) t-+ exp(~tl2) , 5 and we w r i t e n ('), t h e H i l b e r t s p a c e of f o r t h e r e p r e s e n t a t i o n of G on H 1195 1195 Lemma 5.1. Of c o u r s e , i n t h e above 5 i s p u r e l y i m a g i n a r y ; i f 5 h a s a r e a l component t h e n t h e c o r r e s p o n d i n g e x p o n e n t i a l i s unbounded ( c f . p ) .
It i s not hard
t o modify t h e proof of Lemma 5 . 1 t o o b t a i n t h e f o l l o w i n g r e s u l t . LEMMA 5.2.
Suppose t h a t 5 i s i n t h e t u b e T, g i v e n by t h e f o r m u l a T = ( 5 E C: Re(5) E [-r,
rl 1 ,
and l e t p be g i v e n by t h e formula l / p = Re(c)/Zr
+
112
.
Then, i f f i s a m e a s u r a b l e f u n c t i o n on G s u c h t h a t (pa ) ( a ) f ( g ) , 5 and i f Cc i s d e f i n e d a s i n t h e formula ( 4 . 8 ) , t h e n f (mang) = u(m)
-
jKdk
Il f (k) llP
and G a c t s i s o m e t r i c a l l y o n H.,
= CG jVdv
I1 f ( v ) llP
,
( V ' p ) , t h e s p a c e of such f u n c t i o n s
P, 4
f f o r which llfll p =
m,
i s f i n i t e , equipped w i t h t h e same norm. P t h e i n t e g r a l i s r e p l a c e d by t h e e s s e n t i a l supremum.
Proof.
If
S u f f i c e i t t o remark t h a t , a s i n t h e proof of Lemma 5 . l w e
use t h a t f a c t t h a t
2 2 l~a(a)l = ~ ( a )
.
i n t h e p r e s e n t s i t u a t i o n we have t h e formula I P ~ ~ (= c~( a)) 2I ;~ t h e f a c t o r p(a12 i s t h e J a c o b i a n which makes t h e argument work.
0
I n s t u d y i n g t h e s e r e p r e s e n t a t i o n s , i t i s e s p e c i a l l y i n t e r e s t i n g t o cons i d e r the "K-finite vectors", i.e., dimensional r e p r e s e n t a t i o n s .
t h o s e f which t r a n s f o r m under K by f i n i t e
For t h i s s t u d y , i t i s c o n v e n i e n t t o u s e t h e so-
c a l l e d compact p i c t u r e of t h e r e p r e s e n t a t i o n s n K.
v, S
: one r e s t r i c t s a t t e n t i o n t o
Thus
[=u, 5
(g) f l(k) = £(kg) = f (A'(kg)N-(kg)K'(kg)
= (pa5) (A-(kg))
I f we c o n s i d e r t h e H -norms,
)
.
f (K'(kg)
then
1.I
For f i x e d g i n G , A-(kg) and K'(kg)
r a n g e o v e r compact s e t s a s k v a r i e s i n K,
and t h e r e p r e s e n t a t i o n always a c t s by bounded o p e r a t o r s on a l l t h e s p a c e s H
u, c
(K'p) f o r any 5 i n C.
on 5.
The r e p r e s e n t a t i o n o b v i o u s l y depends holomorphically
A s i s customary ( a t l e a s t a t t h e p r e s e n t t i m e ) , we s h a l l c o n s i d e r o n l y
t h o s e r e p r e s e n t a t i o n s f o r which t h e K - f i n i t e v e c t o r s form a dense subspace ( i n some s e n s i b l e topology) of t h e r e p r e s e n t a t i o n space. m
We now a s k how t h e K - f i n i t e v e c t o r s o r t h e C - v e c t o r s
(i.e.
m
t h o s e whose
r e s t r i c t i o n t o K i s C ) look i n t h e "noncompact p i c t u r e " , o b t a i n e d when we r e K m s t r i c t o u r a t t e n t i o n t o V. I f f E H or H ( t h e s p a c e s o f K - f i n i t e and u, 5 u 9 c m C v e c t o r s r e s p e c t i v e l y ) , then f (v) = f (A'(v)N-(v)K-(v)) = (pac)(A"(v))
f(K'(v))
,
and s i n c e I l f ( ~ - ( v ) ) l i s bounded, (5.3)
Ilf(v)II G C p a (A'(v))
5
I t i s t h e r e f o r e i n t e r e s t i n g t o know what A'(v)
LEMMA 5.3.
. looks l i k e .
With t h e n o t a t i o n e s t a b l i s h e d above, t h e
f o l l o w i n g formulae a r e v a l i d :
P(A-(v)) N'(v)
= n(x',
= I l +(1x1
2
,
+y)/21-r
y e ) , where x- = - [ I
and K'(v)
2
+
(1x1
2
- y ) / 2 1 I1 + (1x1
2
+ y)/21-1 x
Y- = -Y , -1 1 = N'(v) A-(v) V .
-
P r o o f . To c a l c u l a t e A-(v) and N'(v), t h e r e i s a simple t r i c k . I f -1 -1 -1 -1 -1 v = ank, t h e n v = k n a , and v-'w = k-lw(wn w)a. Thus, w i t h t h e u s u a l C-action o n S, we have t h a t
-
( 0 , 1 ) o v 'w = ( 0 , l ) o k
-1 -1 w(wn-lw)a = ( 0 , l ) o ( w n ")a.
Since V preserves height, while h((x,y)oa(t))=e
2t
h((x,y))
,
-
we c a l c u l a t e t h e number A-(v) by f i n d i n g t h e h e i g h t o f ( 0 , 1 ) o v 'w, easy.
which i s
We f i n d NU(v) s i m i l a r l y . The formula f o r
stant C
G
whence
(A'(v))
we o b t a i n e d p e r m i t s t h e c a l c u l a t i o n of t h e con-
i n t h e i n t e g r a l formula f o r Haar measure: i t must be t r u e t h a t 2 C dxdy 11 + (1x1 + y ) / 2 1 - ~= 1 , G V -1 C = 2 p l 2 w(p + q + 1 ) , G
where w(p) i s t h e "area" of t h e u n i t s p h e r e i n RP
.
F o r l a t e r u s e , we e s t a b l i s h a n o t h e r i n t e g r a l f o r m u l a .
We w r i t e a g e n e r i c
e l e m e n t g of G i n t h e form
t o e x p r e s s i t s "Bruhat decomposition".
LEMMA 5.4. Proof.
The J a c o b i a n of t h e map v k V(vg) i s p(A(vg))
Suppose t h a t
+ be
a n i n t e g r a b l e f u n c t i o n o n V.
d e n o t e t h e f u n c t i o n on G g i v e n by t h e f o r m u l a 2 (manv) = ~ ( a ) $ ( v ) (V,l) Then lies in H , on which G a c t s i s o m e t r i c a l l y , whence 1 , ~ 2 lVdv P (A (vg) ) tJ (V (vg)
+"
and t h e r e s u l t f o l l o w s .
.
2
.
By q0 we
'w'e now i n t r o d u c e t h e i n t e r t w i n i n g o p e r a t o r s .
There i s reason t o s u s p e c t
t h a t r.he r e p r e s e n t a t i o n s n
and n might be " e q u i v a l e n t " , and t h e i n t e r U*L u , -< twining o p e r a t o r s w i l l e x p r e s s t h i s e q u i v a l e n c e . We might a l s o s u s p e c t t h a t t h e Weyl group element w of G might p l a y a s p e c i a l r o l e i n a l l t h i s .
I t i s so.
It is evident that, i f f (mans) = then, i f we d e f i n e
W
f by s e t t i n g
u (m) (pa W
( a ) f (g) , 5 f (g) e q u a l t o f (wg), we have t h a t
f (mang) = f ( m a n g ) W 1 = £(ma wng)
-
we a r e n e a r t o e f f e c t i n g a t r a n s f o r m a t i o n from H to H , b u t we have a n u ,c u,-c -2 u n d e s i r a b l e f a c t o r of p ( a ) t o e l i m i n a t e , and f u r t h e r we c a n n o t " p u l l o u t t h e n".
An i n t e g r a t i o n o v e r N w i l l f i x b o t h t h e s e problems. We d e f i n e t h e i n t e r t w i n i n g o p e r a t o r A(w,
if a is a
c'
u,
a ) , a l s o w r i t t e n a s A(w,
u,
5)
by t h e formula A(w,
u,
A t l e a s t f o r m a l l y , A(w,
A(w,
a ) f ( g ) = lNdn ,f(ng)
u,
.
a) has the desired property:
u , a ) f (n'g)
by t h e t r a n s l a t i o n - i n v a r i a n c e
=
N
dn
w
f (nn'g)
of Haar measure on N.
By s i m i l a r change of
v a r i a b l e arguments we s e e t h a t
,
(A(w,~,a)fl(mg)=u(m)A(w,u,a)f(g) and
{A(w, P , a ) f l ( a g ) = l N d n f ( m a g ) -1 -1 = l N d n f ( a wa nag) 2 = JNdn p ( a ) f(a"wng) 2 = lNdn ~ ( a ) (pa) (a-1) = (pa
-1
) ( a ) A(w,
u,
f(wg)
a ) f (g)
.
The q u e s t i o n i s , of c o u r s e , t h e e x i s t e n c e of t h e i n t e g r a l . There a r e two d i f f e r e n t methods t o a t t a c k t h e problem of t h e e x i s t e n c e of t h e i n t e r t w i n i n g o p e r a t o r : one may work on t h e group K o r on t h e group V. s h a l l u s e both t h e s e approaches.
S t i l l working f o r m a l l y , we have t h a t
We
A(w,
u,
a ) f ( g ) = INdn f (wng)
.
= lVdv f
Thus, on K , we have t h a t A(w, u, a ) f ( k ) = ,fVdv f (vwk) = jVdv f (A-(v) N-(v) Ke(v)wk) = l V d v ( p a ) (A-(v) ) f (K-(v)wk)
I f 5 = 5 + i q , w i t h 5 p o s i t i v e , and i f f E H JVdv Il ( p a ) (A'(v)
m
us
c
,
t h e n , from Lemma 5 . 3 ,
f (K'(v)wk)
5
.
Il
The i n t e g r a l t h e r e f o r e c o n v e r g e s f o r smooth f u n c t i o n s f . The a l t e r n a t i v e approach t o t h e e x i s t e n c e problem i s t o w r i t e t h e o p e r a t o r a s a c o n v o l u t i o n on V .
We have t h a t
~ ( w ,p, a ) f ( v * ) =
jvdv
.
f(vwv*)
Now we w r i t e vw i n t h e form M(vw)A(vw)N(vw)V(vw); n o t e t h a t , i f v t = V(vw), then
whence
t
where
M(v W) = M(vw)
-1
,
t
A(v w) = A(W)
-1
.
I t follows that
t
= jVdv (dv /dv) u(M(w)
t
But s i n c e Lemma 5.4 t e l l s u s t h a t (dv /dv) = p(A(vw)) A(",
u, a ) f ( v
*
= jvdv ( p a - ' ) ( ~ ( v w ) ) = K
where
u-a1
*
f (V*)
-1 2
( p a ) ( ~ ( w)-' )
,
f (wf)
*
u(~(vw))-l f(w
,
.
(v) = (pa ) ( ~ ( v - l w ) ) p ( ~ ( v - l w ) ) - ' u,a I t i s c l e a r l y worth o u r w h i l e t o i n v e s t i g a t e more f u l l y t h e Bruhat decomK
position. LEMMA 5.5.
Suppose t h a t v i s v ( x , y) i n V .
Then, i n t h e second
.
c o o r d i n a t e s y s rem
l:
N(vw) =
2 2 1x1 ) / l y - 1x1 I * 2 -1 1 - x (1x1 - y) 2x
where
c = (y
and
d
and s o
=
0 d 0
-
M((oSv)w) = M ( w )
while t
I
e
and t h e r e f o r e
p (A((D'v)w)) =
Proof.
,
A(w)=a(t) , 2 -1 ( y - 1x1 )I21 = ~N(v)-'
where
=
,
spr P ( A ( V W ) )
9
.
I n Lemma 4.2, we showed t h a t , f o r t h e a c t i o n of G on
as,
we
have t h e formula ( 0 , O).v(x,
y)w = ( 0 , O)ov(x 2 xt = - ( l x l - y ) - L 2x
where
S i n c e ( 0 , 0)oman = (0, Oj, t h e element v ( x t ,
e x a c t l y t h e e l e m e n t v ' ( x , y ) , i . e . V(v(x, y)w). c u l a t i o n : i f vw = m a n v
,
t h e n mtatnt
=
,
Y
t
)
y t ) d e s c r i b e d i n Lemma 4.2 i s
*
t t t t
t
We now make an e x p l i c i t c a l -
t- l , whence, w i t h o u r c o o r d i n a t e s , vwv
* t
* S i n c e t h e m a t r i x r e p r e s e n t s an element of MAN, t h e upper r i g h t hand e n t r i e s r m s t be z e r o , and t h e lemma f o l l o w s . I t now f o l l o w s t h a t , on V,
t h e form
where Cl (v) = p(M(v J !
note t h a t Q
-1.
t h e k e r n e l of t h e i n t e r t w i n i n g o p e r a t o r i s of
r/2 - 512, Nr - r 9 K = 2 U -1 p9L w) ), and M ( w ) i s g i v e n by t h e p r e c e d i n g lemma.
i s even, i n t h e s e n s e t h a t R ( v ( x , y ) ) = R ( v ( - x , 11
ll
U
n e l s have been a n a l y s e d ( t h i s k e r n e l i s Horn(H )-valued, !J
but H
U
y))
.
We
Such k e r -
i s finite-
d i m e n s i o n a l and t h e r e a r e no problems i n checking t h a t t h e e a r l i e r theorems hold i n t h i s case).
It i s now p l a i n s a i l i n g t o show t h a t t h e i n t e g r a l d e f i n i n g A(w,
u , 5)
converges i f Re(<) > 0 ( t h e p o s s i b l e problems a t i n f i n i t y can be c o n t r o l l e d by m
t h e decrease of H -functions on V ( s e e f o r r m l a (5.3) and L e m a 5.3)). FurUrS t h e r , t h e a n a l y t i c map 5 ACw, u , 5 ) d e f i n e d f o r 5 such t h a t Re(5) > 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
-
+<
=
kc
;
moreover t h e asymptotic behaviour of t h e s p h e r i c a l f u n c t i o n s , d e s c r i b e d i n 55,
i m p l i e s t h a t t h e r e a r e no o t h e r e q u i v a l e n c e s of t h e $ a r e bounded.
and t h a t no o t h e r 4 5' C 1 I t may be shown t h a t t h e Gelfand spectrum of L (K\G/K) i s exhaus-
t e d by t h e i n t e g r a t i o n s a g a i n s t t h e bounded s p h e r i c a l f u n c t i o n s , a s d e f i n e d above, b u t we s h a l l n o t do t h a t h e r e . We need o n l y t h e obvious f a c t t h a t t h e map C, k f ( C ) i s a n a l y t i c i n o r d e r 1 1 t o prove t h e f i r s t r e s u l t on L (K\G/K), and hence on L (G). 1 The a l g e b r a L (G), equipped w i t h t h e i n v o l u t i o n
THEOREM 6.2.
',
i s n o t symmetric. 1 L e t f ' be a f u n c t i o n i n L (K\G/K) which d o e s n o t a n n i h i l a t e
Proof. all @
c.'
( 5 E T). and l e t f be f '
S i n c e ( f ' ) - ( ~ ) = 12(5)1- i f C E iR (be-
f".
c a u s e t h e a s s o c i a t e d r e p r e s e n t a t i o n s a r e u n i t a r y ) , i t i s c l e a r t h a t f is nonA
A
Choose a p o i n t w where f i s n o t r e a l ; such a p o i n t e x i s t s because f i s
zero.
a n a l y t i c and n o t c o n s t a n t ( i n f a c t , by t h e a p p r o p r i a t e v e r s i o n o f t h e RiemannA
A
We c l a i m t h a t f ( w ) i s i n t h e 1 T h i s i s c l e a r i f we c o n s i d e r o n l y L (K\G/K), and i f t h e r e were
Lebesgue lemma, f v a n i s h e s a t i n f i n i t y i n T). spectrum of f . 1 F i n C6 Q L (G) s u c h t h a t F PF
*
(f
- ? ( 6 ) ~ )= u,
*
(f
-
A
f ( w ) 6 ) = 6 , t h e n we s h o u l d a l s o have t h a t
a contradiction.
0
The n e x t r e s u l t i s l e s s t r i v i a l .
1 There e x i s t p r o p e r c l o s e d i d e a l s i n L (G) which
THEOREM 6 . 3 .
a r e n o t c o n t a i n e d i n t h e k e r n e l o f any i r r e d u c i b l e bounded r e p r e s e n t a t i o n of G. I t s h o u l d s u f f i c e t o prove t h e a n a l o g o u s r e s u l t 1 F o r i f I i s such a n i d e a l i n L (K\G/K), t h e n d e f i n e J by t h e
S k e t c h of t h e p r o o f . 1
f o r L (K\G/K): formula
1
*
f m f ) E I, 2 J should be a non-Tauberian i d e a l i n L'(G). J = {f
Let a: T
-
E L (G): P ( f l
fl,
1 f 2 E L (GI)
;
C be t h e map g i v e n by t h e formula
a(<)
(fexp(in5/2r)
-
l ] / [ e x p ( i n ~ / 2 r ) + 11)
2
c E T
With t h e a i d of t h i s map and t h e s p h e r i c a l t r a n s f o r m f +i f , we may r e a l i s e 1 L (K\G/K) a s a s u b a l g e b r a of t h e a l g e b r a A of a l l c o n t i n u o u s f u n c t i o n s on t h e closed u n i t d i s c
6
which v a n i s h a t 1 and a r e holomorphic i n t h e i n t e r i o r .
.
One then seeks non-Tauberian ideals in A.
These can be found with the aid
of work of A. Beurling [~eu],who, by using factorisation into "inner" and "outer" functions, showed that the space of all functions in A which vanish as fast as the function z t+
exp(Z/fz-
1) is
a non-Tauberian ideal. 1 The proof finishes by "pulling back" this ideal to L (K\E/K), and showing
that the "pull-back" is nontrivial. For this, it is necessary to have some 1 sort of characterisation of the space of spherical transforms of L (K\G/K)functions.
D
1 The problem of characterising the spherical transforms of all L (Kb/K)functions seems to be very difficult. One substitute for this characterisation is the result of P. Trombi and V.S. Varadarajan [TrV] already mentioned. One 1 considers the subspace of L (K\G/K) of those functions which, together with all their derivatives, vanish so rapidly at infinity that, even when multiplied by n a polynomial (which here means an expression of the form ka(t)kl k t ), they 1 1 P still belong to L (G). This space has been named 1 (G). The space I (GI is defined analogously. THEOREM 6.4.
The image of I ' ( G )
under the spherical transform
-
is exactly the space of all even functions in the strip T P T = E C: 15/2rl < l/p - 1/21 P which are analytic in the interior of T and continuous on the P whole of T , and whose restrictions to the vertical lines in the P strip belong to the usual Schwartz space on these lines, with uniform estimates holding on all the lines contained in T P Proof.
The idea of J.-L. Clercls proof [~le]of this theorem is as L
follows. The result must first be proved for I ( G ) . functions, I$
.
Next, certain spherical
say, are coefficients of finite-dimensional representations of G,
and for these L. Vretare [Vre] has proved a multiplication formula:
0r@ v
=
I,, c(v.
5. 5') I$5l
.
(This sum extends over a finite set whose size is bounded independent of 5 and v.) The product of an IP-function f with the spherical function I$ lies in a different space Iq(~), say, and one has that (cvf)-(r)
=
I,.
c(v. r, 5')
2(c1) .
Clerc first characterises the spherical transforms of the 1'-functions tain values of p, viz, those for which the products $ f, with f in I ' ( G ) , 2
in I (G), and then applies complex interpolation to fill in the gaps.
for cerlie But the
essential idea behind the proof is really to study the tensor products of the representations n with certain finite-dimensional representations. This 1,5 0 same idea has made its appearance in some other problems. We shall conclude by considering the Fourier transform of more general functions. For the sake of siinplicity, we shall restrict our attention to the representations nl,c, although the other families could be treated similarly. We should like to describe an analytic Fourier transform r (f), where f lies 1,c #. 1 in L (G), just as we have considered the analytic spherical transforms f(.) 1 for f in L (K \G/K). One problem arises immediately: the spaces on which the representations act vary with 5. 'There are two possibilities to consider. We might fix 195 two functions 41 and J, on K, say, and then extend these to functions $ and $-< 5 in H and H respectively. We could then deal with the functions 195 I,-< (K) (g) $5, g t-, ( 195 and define our Fourier transform by integrating against these. We should have n
J,-2
to impose the restriction that $ and J, belong to L'(K), $ 5 and
Jr
5
in order to ensure that
(P) and H (P'), when l/p = 5/2r + 112 and p' is the 195 1,-5 Our Fourier transform would be an expression of the form
belong to H
dual index.
obtained by integrating f against the function above, or rather a collection of such, for
+ and $I
may vary.
This way of doing things is not fine enough to obtain some analytic results, because restrictions on $ and J, are necessary, while one may wish to consider general $ and functions $ and
(I
tion for MAN in G.
(I,
2
for instance in L (K).
Further, one may ask why the
should be fixed on K rather than on V, or on some other secIf we fix $ and J, on K, then $
ly and antiholomorphically on V. in many situations, to allow $
5
and J, vary holomorphical5 -5 It turns out to be more canonical, at least
and J,
-5
to vary in this way everywhere.
I t is p o s s i l ~ l c dnd 11atur:ll t o
ill
II)W $
5
and JI
- r,
to vary with 5. The best
way to do thia is st.111dard in comp1c.x interpolation theory, where, given a function
ti
on a measure space
X,
one consideis functions of the form
If
the parameter z varies. We apply this idea.
@
lies in H
0 (O(Z
as
we d r f ine
"),
120
the analytic family of functions g - by the formula
We note first that
= oaC(a)
i.e. Q
belongs to H
. 195
@C(g)
.
(If we make the same definition for
5 @ will lie in H .) Moreover, if l/p = 5/2r + 112, then C us5 jvdv I@~(V)I' = jvdv I+(V)I (1+c/r)p 2 = jVdv I @(v) I , and 4
belongs to the appropriate 5 arises in this way.
space.
+
in H
u. ..o'
In fact every function in H
then
1,Z
(P)
There is a useful generalisation of this constructioni if we deal with vector-valued functions @ such that, on K. @(k)
=
y(k)@(e)
for some representa-
we produce are multiples of (P on K. Such functions 5 are in fact considered in the theories of generalised spherical functions and tion y of X , then the @
of the Eisenstein integral. Of course, in more general situations one may construct analytically varying sections of analytically varying vector bundles. We finish off this discussion by giving an application of the above con1 struction. If f E L (GI, and if (P and are in H (2), with il@l12(") = 1111(V) 1 PO = I , then
z + ( n 1.5 ( f )
&5.
J)
5)
is a continuous bounded function of 5 in the strip T, which is analytic in the interior of T.
Further, one has the estimate
.
= AfU,
(This estimate is much simpler than the estimate one obtains if one seeks to and J, defined earlier which were constant on K.) 5 -5 A different sort of estimate is available from the Plancherel formula for 2 2 L (G) (or for L (KB/K)). One has that (omitting the superscript V) 2 112 lJRdw(rl) l(nl (£1 +in, flin)l I use the functions $
1
C l JRds ).(I,
and 1.1
u(n)
n
(£1 uop21 lI2 ,TI 2 112 Inl (f lHS I in1
, rl
denote the operator and Hilbert-Schmidt norms.
If p be OP HS the Plancherel measure of G associated to the representations n , then 2 112 lpi~ I [ , d n u ( n ) .l(~~,~~(f) lin,, 1
"21
.
< lf12
It follows by a messy but elementary interpolation argument that, if f E L ' ( G ) and 1
Q
p
< 2,
then the function 5
* ( ~ ~ , ~ ( f65, ) J,-<)
is analytic on the interior of the strip T P
-
and on the edges of this strip, non-'tangential limits exist almost everywhere,
,
and moreover
) P'] l1Pt nl,e+ill(f) +e+in- J,-e+in S C(G,B) If! P This is a ~artialcharacterisation of the Fourier if 101 = 2r(l/p - 112). transform of a general LP-function, akin to the Hausdorff-Young inequality.
I J ~ d q(I
An easy corollary of this result is obtained by means of Cauchy's theorem. One integrates around the perimeter of the strip T and obtains that P c2.,-' l T 1 (f) dy,J, m, P5 C(G,P) lflp
-
c)I
.
This is the essential.step in the proof of the Kunze-Stein convolution theorem, which we state here, without proof. THEOREM6.5.
If 1 < p < 2 , then L ' ( G )
2 2 * L (G)C L (G)
.
7.
UNIFORMLY BOUNDED REPRESENTATIONS AND COMPLEMENTARY SERIES.
m
We recall that H
lJ,c satisfying the condition
is the space of smooth H valued functions f on G
f (mang) and that n
lJ
=
~ ( m ) (pac) (a) f (g)
,
is the right translation representation of G on this space. We
1195 have seen that, if we define 1.
by the formula
and if l/p = 5/2r + 112, then n
acts isometrically. So we have a strip of
lJr5 isometric Banach representations (n
.
: 5 E [-r, r] lJ,c In the preceding section, we saw that this analytic family of representations plays an important role in the LP-harmonic analysis of G. It would there-
fore be desirable to develop a calculus for representations which would permit us to ahalyse and synthesize bounded Banach representations of G.
Since Hil-
bert space theory is an essential feature of the known techniques for such calculi, it is natural to ask if we can realise these representations on a Hilbert space. We shall devote our attention to this question. First, since the ~'-s~aces on which the representations act isometrically are not Hilbert spaces, we shall have to modify them somewhat. n is rather suggestive: in R the potential spaces H E
The situation
-
HE;= If: * f E L'(R~)I P are very similar to the L -spaces, where llp = 112 + 61211. They behave in the same way with respect to rotations and dilations, and if 5 Z 0, then 'L Z H5 , while if 5
< 0,
then HE .'L
These are certainly good candidates for Hilbert
spaces on which the representations n
might act. In fact R.A. Kunze and lJ,5 E.M. Stein [KS~]showed that SL(2,R) has uniformly bounded representations on these spaces, that is, there exists a uniform bound for the operator norms
, as g runs over the group being represented. Various persons have OP subsequently considered uniformlp bounded representations: we mention here the Iln(g)ll
work of P. Sally [~al], R.L. Lipsman [~ilj,[Li21, [~i31, [~i41,E.N. Wilson
[Will, N. LhouC su oh], and L. Bamazi
am], all of whan attempted to develop
the work of Kunze and Stein [KS~] , [KS~], IKS31; some general Comuents on these representations may be found in [CoZ]. of the groups O(q) act uniWe shall show that the representations n P,c formly boundedly on the potential spaces ' H defined in terms of the sub-laplacian A on V, provided that 5 E (-r, r).
This result is best possible, in the
sense that no other nontrivial representations could be uniformly bounded, for the matrix coefficients must go to zero at infinity, and this excludes the other representations. After studying t;7e uniformly bounded representations, we shall turn our attention to the possibility that some of the representations TI r7
c
might be
unitarisable, F . e . , with an appropriate choice of inner product, they might become unitary.
This question has already been considered by other persons,
and in fact all the unitarisable representations of the groups SO(1, n) and SU(1, n), corresponding co the cases where F is either R or C, are known, and it seems that the case of Sp(1,. n), corresponding to the case where F = Q, has just been solved. We shall therefore only touch on the philosophy of our approach to these representations, after our analysis of the uniformly bounded representations. Before we state our theorem, let us give a provisional definition of the 5 space H This will be superceded later. Temporarily, we denote by H r,c * or just H , the closure of the subspace of H m of those functions which, when r,c restricted to V, have compact support, in the norm 1 8', given by the formula
'. "PE
I~U' IIA-''~ * f12(V) .
.
=
It follows from the proof of the theorem that ' H is actually the closure of m
H
P,c
in the same norm. THEOPZM 7.1.
'
the space H u,c
Of this, more anon. The representation II
r,c provided that 5 E (-r, r).
acts uniformly boundedly on
Proof. We first consider the actions of V, M, and A on H
1.1'
5
.
Since
is defined in terms of generalised differential operators acting on the
.
left and V acts on the right by translations, V acts unitarily. Next, l',c
(m) f =4''-A
P (m) (f - R ~ )= P (m) ( [b-'I4f
1 ORm)
;
i t follows t h a t M a c t s u n i t a r i l y .
The a c t i o n of A i s t r e a t e d s i m i l a r l y , b u t i s
a l i t t l e l e s s easy:
where s = e Z .
The necessary f a c t i s t h a t
whence, from the s p e c t r a l c a l c u l u s , A-5/4(f~DS) = s-512 (A-5/4 f)oD s
.
It follows t h a t
and A a c t s u n i t a r i l y . We claim t h a t i t s u f f i c e s t o show t h a t r (w) a c t s boundedly. For i f '495 t h i s i s so, then the o p e r a t o r s of the form r (g) , where g i s e i t h e r mawvwv' 1195 o r mawv, a c t boundedly, and
nn (mawv) l G An (w) lop Fc, S OP us5
,
b u t every element of G i s of t h e form manv o r manw ( s e e Lemma 4.3),
whence t h e
claim. To complete the demonstration, we use an elementary b u t i n t r i c a t e argument. 2 F i r s t , i f 5 = 0 , then we a r e done. For HO "is" j u s t L (V), and n a c t s uniA
t a r i l y on L ~ ( v )by Lemma 5.1.
v,iv
Next, we consider t h e c a s e where 5 = -2. I n o r d e r t o show t h a t r (w) UrS -2 a c t s boundedly on H , i t s u f f i c e s t o show t h a t IX r ( W ) ~ I IC~( L ) I ~ I - ' l G j < p j P,S wherc {Xj: 1 G j < p} i s a b a s i s of the subspace V of t h e Lie a l g e b r a -1 t h i s , we w r i t e everything o u t e x p l i c i t l y :
1.
For
(w)f(v) =u(M(vw)) (pa<)(A(vw)) f(V(vw)) , V,F where M(w) and A ( w ) a r e described i n L e m a 5 -5, and V(vw) i s given i n L e w a r
4.2.
We r e c a l l t h a t M(vw), a s a f u n c t i o n of v , i s homogeneous of degree 0 and
f u r t h e r i s C" on ~ \ { e } . On the o t h e r hand,
which i s homogeneous of degree - ( r + 5 j .
Thus we may w r i t e
t
t
(w)f (v) = m ( v ) f ( v ) = m f (v),
r
t q.5 where v t = V(vw), f (v) = f (v ) , and m E . v - ( ~ + ~(V). )
Clearly
t
.
Xj(n!J, c (w)f) = (xjm)tt + m ( ~J . f ) We consider t h e two p i e c e s s e p a r a t e l y .
Since d i f f e r e n t i a t i o n with X . lowers t h e degree of homogeneity by 1, X.m J 3 of t h e is homogeneous of degree - ( r + 5 + 1 ) . We may e s t i m a t e t h e
norm
f i r s t p i e c e a s follows:
t
t
We r e c a l l t h a t (dvldv ) i s p(A(v w)12, which is homogeneous of degree -2r i n v v i s homogeneous of degree -1, i n
and i n c ~ ( v \ I ~ ) ) . Next, s i n c e t h e map v t t+
(Kim) (v) i s homogeneous of degree
t h e obvious sense, t h e f u n c t i o n v t
'm
( r + (-2 + iq) + l ) , and i s of course i n C (Vt{e)). We m e t t h e r e f o r e e s t i m a t e t t t t 2 1 / 2 ( ( 4 ~U m ( v > f ( v ) I ) , where m
t
i s homogeneous of degree (-1 + i n ) .
But we showed i n S e c t i o n 2 t h a t
m u l t i p l i c a t i o n by f u n c t i o n s of negative homogeneity takes one space 'H
into
another (Proposition 2 - 1 6 ) , and we a r e i n e x a c t l y the r i g h t s i t u a t i o n t o be a b l e t o apply t h i s r e s u l t .
We conclude t h a t
t
It remains t o ostimaie mX: (f ) J
.
t* v t i s a
We n o t e t h a t the map t:v
t
diffeomorphism when r e s t r i c t e d t o ~ \ { e ) . It follows t h a t X.(f ) can be w r i t t e n J t a s (E.f) , where E i s a smooth v e c t o r f i e l d on V\{e). Since J j (f.DS)' = (ft).D1IS ,
w r i t t e n d i f f e r e n t l y , we have t h a t
t
t
11s
( E . ( ~ o D ~ )=) (11s) ( ( E ~ )~OID 3 whence E . ( ~ o D ' ) = (11s) ( E . ~ ) O D ' 3 3 -1 -2 (This i s a g e n e r a l i s a t i o n of the f a c t t h a t d/dx ( f ( x ) ) = -x
.
(df/dx)(x
-1
).)
The r e s t of t h e proof f o r t h e case where 5 = -2 i s now easy: i t i s c l e a r
,
and t h e usual homogeneity arguments w i l l complete t h e proof The proof f o r g e n e r a l v a l u e s of 5 c o n t a i n s no new i n g r e d i e n t s ; one a p p l i e s complex i n t e r p o l a t i o n and d u a l i t y arguments t o f i r s t prove t h e r e s u l t f o r t h e c a s e where
-2
<
C
< 2,
and t h e n a p p l i e s t h e above i d e a s t o t r e a t t h e g e n e r a l
case.
'.
It is c l e a r We now r e t u r n t o t h e q u e s t i o n of t h e d e f i n i t i o n of H m u,5 that, i f f E H t h e n we may w r i t e f a s t h e sum o f two H - f u n c t i o n s , one u,5 ' U,C of which v a n i s h e s n e a r w i n K and t h e o t h e r which v a n i s h e s n e a r e i n K . The
f u n c t i o n which v a n i s h e s n e a r w i s , when r e s t r i c t e d t o V , of compact s u p p o r t ,
.
(w) a p p l i e d t o t h e l a t t e r f u n c t i o n i s of u95 t h e same form a s t h e former, and so belongs t o H The theorem now i m p l i e s u,5 that f i s in H and t h e d e f i n i t i o n v h i c h we announced would be s a t i s f a c t o r y LI95 i s just that.
and s o belongs t o H
\l,<
Further, n
'.
',
T h i s completes o u r d i s c u s s i o n of u n i f o r m l y bounded r e p r e s e n t a t i o n s , and we now t u r n t o t h e q u e s t i o n o f complementary s e r i e s . We a t t e m p t t o d e s c r i b e t h e philosophy of A.W. Knapp and E.M. I f i t were p o s s i b l e t o f i n d a n i n n e r p r o d u c t B(
,)
.
S t e i n [K~s]
r e l a t i v e t o which n
use
acts
u n i t a r i l y , t h e n we should be a b l e t o d e f i n e a G-invariant map b from H to U ,5 H by s e t t i n g
u,-c
(b(f), f') T h i s map i s G - i n v a r i a n t
= B(f, f ' )
i n the sense t h a t
.
(g) f = n u , - 5 ($3 b f LI95 Thus we should have some s o r t of e q u i v a l e n c e between t h e r e p r e s e n t a t i o n s n 11,c and n T h i s would imply e i t h e r t h a t 5 was imaginary, and b would t h e n be LI,-5' t r i v i a l , o r t h a t 5 was r e a l , i n which c a s e b would have t o be a m u l t i p l e of t h e b n
i n t e r t w i n i n g o p e r a t o r A(w, LI,5 ) .
I n t h i s l a t t e r c a s e , a m u l t i p l e o f A(w, U , 5)
would br? p o s i t i v e d e f i n i t e o r s e m i d e f i n i t e .
Conversely, i f A(w, U , 5) ( o r a
m u l t i p l e of i t ) were p o s i t i v e d d f i n i t e o r s e m i d e f i n i t e , t h e n we could r e v e r s e t h e above argument and d e f i n e a G - i n v a r i a n t i n n e r p r o d u c t .
I n s h o r t , we f i n d
"new" u n i t a r y r e p r e s e n t a t i o n s e x a c t l y when t h e i n t e r t w i n i n g o p e r a t o r i s positive d e f i n i t e or semidefinite. One important "simplification" of the problem of f i n d i n g " a l l " r e p r e s e n t a t i o n s of G i s due t o R.P.
Langlands
a an].
the u n i t a r y
We s h a l l d e s c r i b e b r i e f l y
t h e idea. The i n t e r t w i n i n g o p e r a t o r s A(w, p , 5) have the property t h a t t h e o p e r a t o r
takes H
i n t o i t s e l f , and commutes with T It may be shown t h a t , 'for 1135 u,cg e n e r i c 5 i n C, n a c t s i r r e d u c i b l y . Therefore t h e r e e x i s t s a meromorphic u, 5 f u n c t i o n d(w, p, 5) such t h a t
from Schur's lemma. A(w,
u,
-5)
For most 5 i n R, A(w,
u,
5) i s i n v e r t i b l e .
i s p o s i t i v e d e f i n i t e i f and only i f A(w,
u,
I n t h i s case,
5) i s , and we might a s
well r e s t r i c t our a t t e n t i o n t o the c a s e where 5 l i e s i n R
+
.
The c a s e where A(w, 11, 5) i s not i n v e r t i b l e i s more i n t e r e s t i n g . suppose t h a t A(w, form ( 5
-
u,
5) be "normalised",
I f we
i . e . m u l t i p l i e d by f u n c t i o n s of the
y) t o e l i m i n a t e the poles where one would l i k e t o study t h e o p e r a t o r ,
then i t w i l l be t r u e t h a t
and Suppose t h a t 5
> 0.
Then u n i t a r y r e p r e s e n t a t i o n s could a r i s e from t h e repre-
sentations r o r r: i f A(w, u, 5) o r A(w, p, -5) were p o s i t i v e semidefiiJ,5 u,-c n i t e , by t h e procedure o u t l i n e d above. The matrix c o e f f i c i e n t s a s s o c i a t e d t o t h e u n i t a r y r e p r e s e n t a t i o n a r i s i n g i f A(w,
u,
-5) i s p o s i t i v e s e m i d e f i n i t e a r e
o f t h e form
The asymptotic formula f o r the m a t r i x c o e f f i c i e n t s ( c f . Lemma 5.6) i s
This means t h a t t h e m a t r i x 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 decay a t i n f i n i t y f a s t e r than (p
-1
a )(a(t)). 5
Langlands' c o n t r i b u t i o n c o n s i s t s i n showing t h a t any i r r e d u c i b l e u n i t a r y r e p r e s e n t a t i o n n ( i n f a c t the hypothesis of u n i t a r i t y can be g r e a t l y relaxed) i s e i t h e r tempered, which means t h a t the exponential c o n t r o l l i n g the vanishing a t i n f i n i t y i s l i k e t h a t which c o n t r o l s 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 "
o r the
r a t e of decay a t i n f i n i t y i s even f a s t e r ( i n which case we have " d i s c r e t e s e r i e s " ) , o r the r e p r e s e n t a t i o n may be embedded i n a p r i n c i p a l s e r i e s represent a t i o n .rr
11.5
whose r a t e of vanishing is the same of t h a t of
the r e p r e s e n t a t i o n s coming from
~r
u .-5
T.
This means t h a t
can be s a f e l y ignored, s i n c e e i t h e r they
a r e a s s o c i a t e d t o the r e g u l a r r e p r e s e n t a t i o n (whose decomposition i n t o i r r e d u c i b l e ~i s known) o r they occur i n some o t h e r n A t t h i s p o i n t , we make a c o n j e c t u r e .
CONJECTURE 7.2.
Suppose t h a t 0
u
9
5
.
< 5 < r.
Then the i n t e r t w i n i n g
o p e r a t o r A(w, p, 5 ) : H
--+ H i s p o s i t i v e d e f i n i t e o r semidef i n i t e u>5 up-5 i f and only i f t h e convolution o p e r a t o r with k e r n e l tr(K ) on V i s lJ95 p o s i t i v e d e f i n i t e o r semidef i n i t e .
REMARK 7.3.
I t i s easy t o show t h a t i f A(w,
u,
d e f i n i t e cjr semidef i n i t e , then. s o is convolution w i t h tr(K
5) i s p o s i t i v e Us5
). F u r t h e r , i f
5 = 0, t h i s c o n d i t i o n is necessary and s u f f i c i e n t . We now i l l u s t r a t e t h e problem of d e s c r i b i n g t h e complementary s e r i e s with a couple of examples.
EXAMPLE 7.4.
The class-one complementary s e r i e s .
The i n t e r t w i n i n g o p e r a t o r A(w,
1, 5) i s g i v e n by convolution with a s c a l a r
k e r n e l on V, which is nothing b u t a m u l t i p l e of t h e norm f u n c t i o n
The
F o u r i e r transform of t h i s is given by t h e formula
where C depends only on G.
This formula is t o be i n t e r p r e t e d a s follows: the
"Four.ier transform" of a c y l i n d r i c a l f u n c t i o n , of which the f u n c t i o n s tr(K
u,s
a r e examples, i s diagonal and constant on blocks, which correspond t o the eigenvalues of the Hermite o p e r a t o r ; T ) ( N ' - ~ ) ~ i s the Fourier transform of N'-~
a t the p a r m e t e r h on the kth block.
This formula w i l l be proved i n the
)
n e x t and f i n a l s e c t i o n .
<
If 0
5 / r, t h e n we have a m u l t i p l e of
r ( (p+4k-2-r)/4)/r(
(p+2-5)/4)
( t h e o t h e r terms are always p o s i t i v e ) ; i f k = 1, t h e n t h i s f a c t o r i s j u s t e q u a l t o one, b u t i f k when 5
>p
>
1, t h e n i t i s p o s i t i v e when 5
+ 2 and n e g a t i v e
+ 2.
I t f o l l o w s t h a t t h e class-one
complementary s e r i e s of t h e groups SU(1, n)
(which correspond t o t h e c a s e where F = C) "goes a l l t h e way" t o r , where the i d e c t i t y r e p r e s e n t a t i o n occurs. t o t h e c a s e where F = Q, p + 2
But f c r t h e groups Sp(1, n ) , which correspond
< r,
and t h e complementary s e r i e s s t o p s b e f o r e
one reaches t h e t r i v i a l r e p r e s e n t a t i o n .
T h i s i s a r e s u l t c f B. Kcstant f E o s ] .
Before we t a k e z p o u r n e x t example, we remark t h a t , even thsuzk ZSE groups S p ( l , n) have "property T",
i.e.
the i d e s t i t y represe-taci;~ i s iez-
l a t e d i n t h e u n i t a r y d u a l , t h e i d e n t i t y r e p r e s e n t a t i o n =a:; uniformly bounded r e p r e s e n t a t i o n s . SL(3, R ) f o r i n s t a n c e .
':o
a-,pr;a-iec!
5:;
T h i s i s n o t trae f o r zany t h e r grac-,s,
One should presumably d i s c u s s "?ro?srty '773", 3 7
which we mean t h a t t h e i d e n t i t y r e p r e s e n t a t i o n is i s o l a t e d i n c5s u r i f s l l n l p bounded d u a l .
See t h e a u t h o r ' s paper [Co2] f o r a f u r t h e r d l s c u s s i o z of - h i s
phenomenon, which i s n o t w i t h o u t i n t e r e s t i n harmonic a n a l y s i s .
EXAMPLE 7.5.
(Communicated by W. Baldoni S i l v a )
.
For t h e groups Sp(1, n ) , t h e r e i s a n a t u r a l r e p r e s e n t a t i o n of X, vhich i s j u s t Sp(n
-
l ) x S p ( l ) , which i s t r i v i a l on t h e "smsll" f a c t o r and which a c t s on
c ~ ( ~ -i ~ n )t h e "large" f a c t o r .
For t h i s example, i f n Z 3, t h e r e i s a comple-
mentary s e r i e s i n t h e " c r i t i c a l s t r i p ' '
(0, 4n
- 6),
and t h e n a n i s o l a t e d uni-
t a r y r e p r e s e n t a t i o n when t h e parameter C i s e q u a l t o 4n case, t h e r e p r e s e n t a t i o n n 1195
is reducible.
-
2.
In this latter
8.
AN EXTRAPOLATION PRINCIPLE.
We conclude with some a p p l i c a t i o n s of a theorem proved i n the t h e s i s of
F . Carlson (1914).
This theorem s t a t e s t h a t i f f i s an a n a l y t i c f u n c t i o n i n
> 01
t h e half-plane { < E C: Re(<)
and f (5) i s O(exp(kl< I ) ) , where k
moreover f (5) = 0 when 5 = 1, 2 , 3,
..., then
< a,
and i f
f is i d e n t i c a l l y zero.
The f i r s t a p p l i c a t i o n we o f f e r i s t o t h e c a l c u l a t i o n of F o u r i e r transforms We s h a l l t r e a t t h e k e r n e l s N ' - ~
on V. manner.
e x p l i c i t l y , i n a reasonably p a i n l e s s
This a n a l y s i s i s of i n t e r e s t not only i n t h e study oL the i n t e r t w i n i n g
o p e r a t o r s , b u t a l s o i n t h e f i n e r a s p e c t s of n i l p o t e n t harmonic a n a l y s i s . THEOREM 8.1.
Let the function a
aA,k(c) = 2
1-p/2 ,(p+q+l)l2
where
(5) i s t h e v a l u e of n (N'-') A,k X o f Section 3. Then a
be d e f i n e d by t h e formula h,k 1-
on the kth block, i n the sense
We s h a l l denote by c ( c ) , o r by vA(Nc-r the v a l u e of h,k )k, th nl (N'-~) on t h e k block. Proof.
F i r s t , by e v a l u a t i n g two
The proof c a n be d i v i d e d i n t o s e v e r a l - s t a g e s . i n t e g r a l s , we d e t e r n i n e r (N'-')
for two p a r t i c u l a r v a l u e s of 5, namely 0 and
a
2.
Next, we u s e t h e r e c u r r e n c e r e l a t i o n s o f Lemma 3.3 t o show t h a t a
(5) and ,k Third, we show t h a t t h e d i f f e r e n c e
c
(5) coincide f o r many v a l u e s of 5. X,k between the two f u n c t i o n s considered does n o t grow too f a s t a t i n f i n i t y . we apply Carlson's theorem. We consider N"'
Here a r e t h e d e t a i l s of t h e proof.
as 5 -approaches 0.
lim
(8.1) and
Last,
E-'
We f i r s t claim t h a t
=
o+ e+2-r
~ ( p )~ ( q )B(p/4, q/2)/2
1
6
[ o(p) w(q) B(p/4+1/2,q/2) (r-2) / 2 ] 6 P To prove the e q u a l i t y (8. I ) , where w(p) i s t h e a r e a of t h e u n i t sphere i n R (8.2)
1i m
eo+
A * N
=
.
-
we r e c a l l from the proof of Lemma 1.1 t h a t , i f f E C
C
(V),
then
,
1im =
lim
E+O+
JVdxdy N € - ~ ( x , y) f (x, y)
€--PO+
jgads 2 - I
f (0, 0)
It t h e r e f o r e s u f f i c e s t o e v a l u a t e t h e a r e a of Z.
l d ~ d wup-I wqel 2 W e now p u t v equal t o u and obtai:,
= r u(p) w(q) j j u 4
where w = l yl and u = 1x1.
Now
+
w2
-IT--
= r w(p) w(q) /2 /Jv2 =
w(p) u(q)
lo*/2 d9
,2
+
cos
p/2-1
-
l d ~ V" d ~ (?)
, t5zt
--- -- v:
---:+.
siz=
proving t h e e q u a l i t y (8.1). I n o r d e r t o prove t h e e q u a l i t y (8.2),
we r e c a l l [frctu ? i : ~ z s i z = : z
--.t-
zze
Remark 1.10) t h a t ~+2-r
whence
2
(A*N ) ( x , Y ) = E ( ~ - ~ - E1x1 ) N(x,Y) 2- r A * N = C 6, with
E-2-r
The c a l c u l a t i o n of C proceeds much a s above, and we l e a v e t o t h e r e a d e r t h e t a s k of v e r i f y i n g o u r claim. The e q u a l i t y (8.1)
is that
which i m p l i e s t h a t
from t h e Legendre d u p l i c a t i o n formula ( s e e , e.g., and t h e f a c t t h a t r = p
+
2q.
E.C.
Titchmarsh [ T i t ] , 1.86)
T h i s i s c l e a r l y eqiial t o Res(a
X,k
(5); 5 = 0 ) .
Analogously, from t h e e q u a l i t y (8.2), we have t h a t N2-r = ~ ( r - 2 ) w(p) w(q) B(p14 + 112, q/2)121 ~ - ,l whence i t follows t h a t
S i m i l a r c a l c u l a t i o n s now lead t o t h e conclusion t h a t c
1.k
(2) = a
A,k
(2).
The second s t e p of t h e proof involves an examination of recurrence r e l a tions.
We r e c a l l (Lemma 3.3) t h a t
where B(5) = -4(5
-
5-r-4 (A2 + B(5)n) * N ' - ~ = C(5) N 2 2) and C(S) = (5-r) (5-2) (5-p-2) (5-4)
F o u r i e r transform, we o b t a i n t h a t (nA(~)2 + B(i)ni(n))
nA(N
whence, f o r k equal t o 1, 2, 3,
S-r
...,
=
C(C) na(N
5-r-4
1
BY passing t o t h e
.
2 2 2 ( C - ~ ) / C ~ , ~ (= ; ) 41 ((p+4k-4) - (5-2) ) ~ ( 0 - l A, k On t h e o t h e r hand, it is immediate from t h e d e f i n i t i o n s of a and b that Ask A,k 2 aA,k(5-4)/a (3) = I l l b (5-4)/b (5) 1,k Ask X,k and t h e q u o t i e n t on t h e r i g h t hand s i d e i s an expression involving t e n gammai.e.
c
.
functions.
By using t h e r e c u r s i o n r e l a t i o n f o r t h e gamma-function
-
i t follows very e a s i l y t h a t (8-3)
a a,k ( ~ - 4 ) l a ~ , ~ (=i c)
A,k
(~-4)Ic~,~(~)
2 2 = 4 1 ~ 1 ((p+4k-4)
-
(5-2)
2
C(5)
-1
The t h i r d s t a g e of t h e proof i s t h e s t u d y of t h e growth of a a s 5 tends t o i n f i n i t y i n t h e l e f t h a l f plane.
Since N ' - ~
and c 19k A y k i s homogeneous, so
i s i t s Fourier transform; a
is homogeneous i n A by d e f i n i t i o n . I n what X,k follows, then we may assume t h a t 1 = 1. We s h a l l a l s o f i x k, a r b i t r a r i l y . It i s c l e a r from t h e formula (8.3) t h a t both t h e q u o t i e n t s involved a r e bounded by 1 when Re(5) and c
<0
and (51 i s l a r g e enough.
I f we can show t h a t a
a r e bounded i n t h e s e t S given by t h e formula X,k 0'
A,k
then it follows from the recurrence relation that a and c are b~unded A,k A,k in the set S, given by the formula S = (5 E C: Re(<) 9 -1, IIm(<)I
>
1)
.
We notice also that a continuous function which satisfies the recurrence relation is automatically bounded in the set
Ec
E C: Re(c)
< -1,
lIm(c)I
<
1)
,
by the same argument. On the one hand, since when Ibl tends to infinity,
I r (a
+ ib) l
%
(2*)'12
lbl a
-
112 exp(-T lb1/2) ,
uniformly for a in a compact interval (see, e.g., E.C. Titchmarsh
it], 4.42),
it is true that a is bounded in S and hence in S. X,k 0 in On the other hand, if f is a unit vector in the Hilbert space of n k A' th the k block, then the function (n f f ) is both infinitely differentiable A k' k and bounded. If Re(c) E [-5, -11 , then the distribution N'-~ is the sum of a compactly supported distribution of order at most 6 and a uniformly (in 5) integrable piece away from (0, 0) (see Lemma 1.1 and Remark 1.3).
It follows
from the analysis of the proof of L e m a 1.1 that S-r Ic = I(aA(N )fk, fk)l A,k c-r = llvdvN (v) (nA(v)fk, fk)I
< c, if 5
S
Therefore c is also bounded in S. X,k To finish off the proof, we apply Carlson's theorem. Both a and c Ask A,k are meromorphic functions whose only possible poles in the left half-plane are E
0-
simple poles at 0, -2, -4, -6, etc.; their difference d
A,k
-
= a A,k ~,k-~A,k is another function of the same kind. Moreover, we have the following inford
mation about d
First,d (2) = Oand Res(d (5); < = 0 )" 0 , i.e. A,k' A,k X,k dA,k has no pole at 0. Next, d satisfies the same recurrence relation as a A ,k A,k and c ((8.3)). Finally, d is bounded in the set S. A,k Ask We recall the recurrence relation: 2 2 (c-2) (c-4)(c-r) (<-p-2) dX,k(~-4) = 4((p+4k-4) (5-2) dA,k(5)
-
By letting 5 approach 2, we find that
.
Res(d
(5);
A,k i.e. there is no pole at -2.
5
=
-2)
=
0
,
< 2,
Now if Re(5)
... which
the factor (5-2)(5-4)
multiplies d (5-4) is nonzero; inductively, we find that there are no poles X,k at -4, -6, -8, But the recurrence relation actually gives us more than
... .
-p
(2 - p - 4k) = 0. X,k Another inductive application of the recurrence formula implies that this.
By putting 5 equal to 6
-
4k, we find that d
(8.4) d (2-p-4k-4n)=O Ask where iV denotes the set of natural numbers. We may now conclude the d
in the whole half-plane
{C,
E
,
is an analytic function in the left half-
X,k
plane, with many zeroes ((8.4)),
n E N
and which is bounded in the set S and thence
C: Re(<)
< -11.
Titcharsh [Tit], 5.81) now implies that d
A,k
Carlson's theorem (see E.C. is zero everywhere, as required
to complete the proof.
0
The second application of Carlson's theorem is in the decomposition of products of spherical functions. This application is due to L. Vretare I~re1, and is an essential part of J.-L. Clerc's treatment [~le] of the spherical transform, already touched on in Section 6. We shall merely outline Vretare's result. If one considers the space H lJ,
f (mang) = u(m)
c
of all functions on G such that (pal;)
(a) f (g)
,
one finds that for certain values of the parameters p and dimensional subspace of H
there is a finite-
invariant under the action of G.
u,c
When this hap-
pens, we have a finite-dimensional representation of G embedded in the principal series.
In N. Wallach's book [wall, one may find the following results
about these subrepresentations, which we sumnarise here as a theorem.
THEOREM 8.2.
If H
u, c
contains a nontrivial finite-dimensional
G-invariant subspace, then this is unique.
Consequently we may denote
this subspace unambiguously by V and by u the restriction of lJ.5' u,i n to V If there is no such subspace, then we put V equal Lt,S P,< u*5 to (01.
.
Every Cinite-dimensional representation of G occurs in this way.
The space V
1,-3r/2
contains a (K-f ixed)vector f -3./2
restriction to K is identically 1. If H contains a nontrivial subspace V then H contains 11,5 W,Z' 11, c-r the nontrivial finite-dimensional G-invariant subspace V V V,Z 1,-3r/2' The map f I-+ f f is a K-module isomorphism of the K-modules -3rl2 V and V f u,5 u , ~ -31-12' The representations n invade H in the sense that if we fix 1195 u,5' a representation K of K occurring in n , then there exists 5 in C such that, if 5 =
cK - nr, with m(~, (u
v.c
1
n in N, then
IK
=
m(K, ( n
v.~)IK)
* where m(~, p) is the multiplicity of the irreducible representation K
of K in the representation p of K. Proof.
Omitted.
We consider the so-called spherical principal series,i.e. the series of
, with 5 in C. We denote by f the function in H whose 195 5 195 restriction to K is identically 1. According to the above theorem, f lies in 5 if 5 is of the form -(2n+l)r/2, with n in N. For these values of 5, the V 195 spherical functions I$- representations rr
may be viewed as matrix coefficients of the finite-dimensional representations Finite-dimensional representation theory gives us a product formula for Ol,cCg) r) (g) when both 5 and v lie in the set -(2N + l)r/2. Analytic extrav
polation allows us to extend to the case where 5 is arbitrary. In order to understand better the finite-dimensional representations, we shall need a little more notation.
We shall consider only the cases where F
is C or Q, for simplicity. Let H and B be the subgroups of G which are given by the formulae H = (m(v, u):
u E D]
(cf. ( 4 . 3 ) ) , where D is the group of all (n-l)x(n-1) diagonal entries are of the form WS($)
+ isin($),
2'
(where
e
i E F and
$
E R , by which we mean
of course), B = {be: 9 E R ]
where b
diagonal matrices whose
,
is described in the second coordinate system as a matrix thus:
e
cos(0)
0 isin(8)
isin(0)
0
the central submatrix being an (n-l)x(n-1)
cos(f3) identity. The reader may wish to
consult Section 4 for the definitions of A and M. It is clear that both HA and HB are abelian subgroups of G.
They are in
fact so-called Cartan subgroups of G (maximal abelian subgroups of semisimple We notice also that H is a Cartan subgroup of M and
elements, in our case).
that HB is a maximal abelian subgroup of K. The matrix coefficients of finite-dimensional representations of G, when restricted to the groups HA and HB, may be written as finite sums of exponentials.
When HB is considered, these exponentials are all characters of the
compact group HB.
However, when one deals with HA, the exponentials are of a
mixed nature: real exponentials appear on the group A.
when restricted to HA, are o£ the form
of the representation a p,-(2n+l)r/2' f(h(q)a(t))
1P7q C(P,
=
Thus the coefficients
~ P $e4t J
q) e
where JI is a 'hultiparameter" and p is a multiindex, while q is an integer between nr and -nr. In particular, for the spherical functions
+-(2n+l)r/2'
where n E W, we
have that (8.5) where c(q)
'-(~n+l)r/2
(h($J)a(t))
depends also on n.
=
14 ctq)
eqt ,
By using finite-dimensional representation
theory, one could determine c(q)
explicitly.
If we consider the product I$ I$ of such spherical functions, then it must u v be a sum of matrix coefficients of finite-dimensional representations. In fact we may determine exactly which finite-dimensional representations intervene in
-
the tensor product of o and o in particular, the number of representa135 1, V' tions involved is at most the order of the Weyl group of the romplexification of G, a finite number.
More precisely, the only 5' which occur are of the form
5 + y, where y runs over a finite set
r
which depends on v but not on 5 .
Since the product of two K-biinvariant functions is again K-biinvariant, it is possible to find a formula (8.6)
050v =
CY
C(5, v: Y) 0
i+Y
-
This formula holds when 5 and v are in (2N + l)r/2.
It may be worth commenting
that the coefficients in the previous formula (8.5) determine the coefficients C(5, v; y) in the formula (8.6).
In particular, if we consider that y for
which the absolute value of 5 + y is largest, then the associated C-function is the product of two of the c-functions in the formula (8.5).
This may be inter-
preted as the statement that asymptotically, as t tends to +-, '(2n+l)r/2 (a(t)) nrt behaves like c(nr) e , and the asymptotic behaviour of the product is the product of the asymptotic behaviours. It is possible to find rational functions, denoted by C(c,
v; y),of 5
which interpolate the function C(5, v; y) determined by the formula (8.6). Let us now fix g in G, and consider the meromorphic function t. $<(g)
$v(g)
-
Eyfrc(c.
+5+Y (g) .
v;~)
One multiplies by the denominator of C(5, v; Y) to obtain an analytic function which is zero if 5 E -(2N + l)r/2.
Simple growth estimates show that Carlson's
theorem is applicable; it follows that the formula (8.6) holds for all 5 in C. It is an amusing exercise to show that the asymptotic behaviour of the spherical functions could also be obtained from Carlson's theorem and finitedimensional representation theory.
It is a natural question, at this point,
to ask to what extent harmonic analysis on the groups considered may be derived in this elementary manner. We should like to finish off by suggesting that the existence of discrete series may be established by this "extrapolation principle". We consider the finite-dimensional representations u
U, 5
of G.
Restricted
to K, these break up into a finite sum of irreducible representations of K. Certain representations of K occur with multiplicity one in a ; these are Y151~ the "highest weight representations", by which we mean the following: the restricted to HB breaks up into a sum of one-dimensional 5 representations, i .e. characters of HB. A certain number ( 1, the order of representation0
~ J I
IW C
the Weyl group of the complexification of G) of these are extreme points of the set of all characters which occur in this way.
There are certain represen-
tations of K for which at least one of these extreme points is again an extreme point; these representations we call the highest weight representations. There
are
IwCI/ IwKI
of these, where W
K
norm
is t h e Weyl group of K.
-
We look f o r c e r t a i n f u n c t i o n s f S i n V
v,5
of
one, which t r a n s -
form under HB by one of t h e extremal c h a r a c t e r s j u s t described, and l e t JI
5
be
t h e a s s o c i a t e d matrix c o e f f i c i e n t s :
R e s t r i c t e d t o HB, t h i s f u n c t i o n i s t h e extremal c h a r a c t e r of HB. I t would seem t h a t product formulae of t h e form
$5 $,"
= $S+Y
hold, and t h a t t h e s e formulae a l l o w u s t o i n f e r t h e e x i s t e n c e o f q5 which vanish r a p i d l y a t i n f i n i t y , and hence t h e e x i s t e n c e of d i s c r e t e s e r i e s .
Kore
p r e c i s e l y , such formulae h o l d when r, and v a r e n e g a t i v e i n t e g e r s , i n which c a s e y
i s a v-dependent n e g a t i v e i n t e g e r .
By " a n a l y t i c e x t r a p o l a t i o n " such formulae
c o n t i n u e t o hold when 5 is a p o s i t i v e i n t e g e r , and i n t h i s c a s e a s 5 i n c r e a s e s ,
'$
v a n i s h e s more r a p i d l y a t i n f i n i t y ; when 5 is l a r g e enough we o b t a i n square
integrable
+',
which correspond t o d i s c r e t e s e r i e s .
It seems necessary t o consider o t h e r groups along t h e way, B l a M . Flensted-Jensen
[FJl],
[ F J ~ ] , i n o r d e r t o work i n a n environment where HB i s non-
compact, s o t h a t t h e (not n e c e s s a r i l y u n i t a r y ) c h a r a c t e r s of HB a r e parametrised by C r a t h e r than 2.
REFERENCES
[Barn]
L. Bamazi, Bepre'sentations sphe'riques unifomement borne'es des groupes de Lorentz;
Analyse Harmonique sur Zes Groupes de Lie I I . Lecture
Notes i n Math. 739.
Springer-Verlag,
Berlin, Heidelberg, New York,
1979. [Beu]
A. Beurling, On two problems concerning linear transformations i n H i l -
bert spaces, Acta Math. 8 1 (1948), 239-255. [Call
CalderBn, Intermediate spaces and interpolation, the complex me-
A.P.
thod, S t u d i a Math. XXIV (1964), 113-190. [CaZ]
A.P. Calder6n and A. Zygmund, On singular integrals, Amer. J . Math. 78 (1956), 289-309.
[Cle]
J. -L.
Clerc, Transformation de Fourier sphe'rique des espaces de
Schwartz, p r e p r i n t , Universitg de Nancy I. [COW] R.R.
Coifman and G. Weiss, Extensions of Hardy spaces and their use i n
analysis, B u l l . Amer. Math. Soc. 83 (1977), 569-645. [Col]
M.G.
Cowling, The Kunze-Stein phenomenon, Annals of Math. 107 (1978), 209-234.
[CoZ]
M.G.
Cowling, Sur l e s coefficients des repre'sentations unitaires des
Analyse Hamonique sur l e s Groupes de Lie II.
g r a p e s de Lie simples;
Lecture Notes i n Math. 739.
Springer-Verlag,
B e r l i n , Heidelberg,
New York, 1979. [CoM]
M.G.
Cowling and A.M.
Mantero, Intertwining operators and representa-
tions of semisimple Lie groups, i n preparation. [EM1;2;3]
L. Ehrenpreis and F.I.
Mautner, Some properties of the Fourier trans-
fm on swisimple Lie groups. I , Annals of Math. 61 (1955), 406-439; 1 1 , Trans. Amer. Math. Soc. 84 (1957), 1-55; 111, Trans. Amer. Math. SOC. 9 0 (1959), 431-484. [I'Jl]
M, Flensted-Jensen,
Spherical functions on a real semisimple Lie group.
A method of reduction to the c~mplexcase, J . Funct. Anal. 30 (1978), 106-146. [FJ2]
M. Flensted-Jensen,
Discrete series for semisimple symmetric spaces,
Annals of Math. 111 (1980), 253-311.
[ ~ o l l G. B. Folland, A jiti,bia:~?~ial su!zltio?i fo12 a subeZLiptic 0 p c 7 z ~ ~ t 0 1 ; B u l l . Amer. Math. Soc. 79 (1973), 373-376. Folland. S u i ~ c z l i p t i cesthiatcs imd function spcces on niLp~t+??zt
I F O ~ ] G.B.
g?.~)i(;l.-, Arkiv f o r Mat.
[FoS]
F o l b n d and E.M.
G.B. 071
[He1 1
S
13 (1975), 161-207.
S t e i n , Zstimates far thc
ah
canp2e.x a?id analysis
the fici.~enbc~y group, C m . Pure Appl. Math. 27 (1974), 429-522.
. He lgason,
Difj*erlential Geactry, Lie Groups, and S p n e tz-ic Spaces.
Academic P r e s s , New York, San F r a n c i s c o , London, 1978. C. S. Herz.
Harmonic synthesis for suhg~~oups, Ann. Znst. F o u r i e r (Gre-
noble) 23 (1973). 91-123. R.A.
Hunt, On Lfp,ql
spaces, L'Enseignement Math. XI1 (19561, 249-276.
A. Kaplan, Fudanental soZutions for n c h s s of hypoelliptic pde gene-
rated by camposition of quajrutic folms, Trans. Amer. Math. Soc. 258 (1980),
A.A.
147-153.
K i r i l l o v , Unitary representaticns of nilpoten? Lie groups, Russian Math. Surveys 17 (1962), 53-104.
A.A.
K i r i l l o v , Elgments de
la ThEorie des Reprlsentations.
MIR, Moscou,
1974. A. W. Knapp and E.M.
S t e i n , Intertwining operators for semisimple groups,
Annals of Math. 93 (1971), 489-578. A. Korenyi and S. Vsgi, Singu2u.r integrals on hanagenems spaces and
sme p r o b h s of c k s s i e u t analysis, Ann. Scuola Nom. Sup. P i s a 25 (1971), 575-648. B. Kostant, On the existence and i r r e d u c i b i l i t y o f a certain s e r i e s o f
representations, Bull. Amer. Math. Soc. 75 (19691, 627-642. Kunze and E.M.
R.A.
S t e i n , Uniformly bounded representations and har-
monic anaZysis on the 2x2 reaz unimodukzr group, Amer. 3. Math. 82 (1960),
1-62.
&nee and E.H.
R.A.
S t e i n , Ihriform7g bounded representations II: Andy t i c
continuation o f the principaZ ::cr*ics of rcprcsentations of tho nxn cunplex un&iodular group, h e r . J . Math. 8 3 (19611, 723-786. R.A.
Kunze and E .M.
S t e i n , lhlifC>r??~%y !hdunb.? rcpreccntations I I I :
:nter~uiiizing ~ ~ C I * J ~ Qfor P S the principal series cn smisCr1 ic ;r;upa, Amer. J. Hath. 89 (1%7),
385-442.
bnglands, On the classificatic>?,sf i~>rc&zib!c
R.P.
~ ~ s : r i ? ~ : i o nuf: :
~ w a laZgz3raic gkdmps, p r e p r i n t , I .A. S., Princeton. H. Leptin, L!caL thcoly i n group algebras of locaZZy compact graoup.:, Invent. Math. 31 (1976). 259-278. R. L. Lipsman, Unifumly bounded representations of SLf2, C), Amer
. J.
Math. 9 1 (1969), 47-66. Lipsman, Harmonic analysis on S&fn,CI, J . 4unct. Anal. 3 (1969).
R.L.
126-155. Lipsman, Uniformly bounded representations of the Lorentz group,
R.L.
Amer. J. Math. 9 1 (1969). 938-962. Lipsman, An e x p l i c i t realisation of Kostant's principal series with
R.L.
app lieations t u uniformly bounded representatioiis, p r e p r i n t , Univers i t y of Maryland.
N. Lohou6, SZcr Zcs reprisentatians unifomement b o d e s e t 3e tA6orBne & c o m o t u t i a de Kunze-Stein, p r e p r i n t , Pniversiti5 de Paris 3C.XL. Pukanszky, Le~onssur h s Reprksent;a&i-ions
d s s g r m ~ e s . Iktnd, Pa*,
1967. S. Sadti. T m r w k p ~ $%t%&?mf opemtors on gpoups, S h o k t
Na&.
9. 22
(39.930), /109-419. Sally. Jr, Analytic continuation of the irrechrcibZe u n i t m y repre-
P.J.
sentations o f the universal covering group of SL/2,RI, Mm. h e r . Math. Soc. 6 9 (1967). [sit]
A. Sitaram. A n
analogue of the Wiener taaberian t h e o m for spherical
transforms on stmisimple Lie groups, t o appear, P a c i f i c 3. Math. [Ste]
E.M.
S t e i n . Interpolation of linear opemtors, Trans. h e r . Math. Soc.
83 (19561, 4 8 2 4 9 2 . [ S W ~ ] E.M.
S t e i n and S. Wainger, The estivation of an integral arising i n
mt l
tiyZiara tt'rzitsf,*: 11% i,i(;ns, S t u d i a Math. XXXV (1970), 101-104. I
[ SW2]
E.M.
S t e i n and S. Wainger, l'roblemc i n harmonic analysis relaxi! to
i??trnu:u?*L,Bull. Amer. Math. Soc. 84 (1978). 1239-1295.
E.M. Stein and C. Weiss, !i.:~.~.kct?'or:5, ibal>i&rAn,~Lysison ?uciidek?n
S~UP.:. rrinceton University Press, Princeton, 1975. R. S. Strichartz, Md?ti~lft,,*son
f?.dcii~~ So!?dl'c9 ~ spacds,
J. Math.
Mech. 16 (1967), 1031-1060.
E.C. Titchmarsh, Th< Y h c ~ r yaf Functidnc.
Oxford University Press,
Oxford, etc., 1978. P.C. Trombi and V.S.
Varadarajan, Sp/x?~.icillt ~ a n s f s i m son smis.impk
Lic groups, Annals of Math. 94 (1971), 246-303. I.
Vretare, b'lemsntcnj spherical function- of ::hmc t 1 . i ~spucec, Math. Scand. 39 (1976), 343-358.
N. Wallach, Il~~l?noni~: AnuZysis on IjLX7ojt?mo1~~ >i)accs. Marcel Dekker, Inc.. New York, 1973. G. Warner, Harmonic Analysis on Smi-Sinlpie Lie Groups. VoZs I and I I .
Springer-Verlag, Berlin, Heidelberg, New York, 1972.
Y. Weit, On the one-sided kriener's lizoorem for t h e motion group, Annals of Math. 111 (1980), 415-422. E.N. Wilson, UnifcrmZy bounded repraescntations for the L o r e n ~ zgroups, Trans. Amer. Math. Soc. 166 (1972), A. Zygmund, Trigonmetric Series. VoZ.
431-438. n,l I, English i r ~ n s l a t i o n , 2 cd..
Cambridge University Press, bndon and New York, 1978.
CENTRO IXTERKAZIONALE MATEMATICO E S T I W (c.I.M.E.)
CONSTRUCTION
DE REPRESEN TATIONS U N I T A I R E S D'UX GROUPE DE L I E
MICHEL DUFLO Universite Paris 7
RESUME Soit
:
G un groupe de Lie d'algGbre de Lie g
associer B certaines formes lin6aires g tions unitaires de
. La "mgthode des orbites" veut
sur g
une famille de representa-
G. Je construis de telles reprgsentations lorsque g est
admissible et bien polarisable. Elles sont param6tr6es par certainas reprgsentations projectives du groupe des composantes connexes du stabilisateur de g dans G. Lorsque G est alggbrique, l'ensemble des repr6sentations unitaires irr6ductible.s de
G obtenues par ce procLd6 suffit B dOcomposer L2(6).
Smary :
Let G be a Lie group with Lie algebra g ciate to some linear forms g
on g
. The "orbit method"
should asso-
a set of unitary representations of
I construct such representations when g
G.
is admissible and has a good polari-
zation. They are parametri2ed.b~some projective representations of the group of connected components of the stabilizer of
g
in G. When G
the set of the irreducible unitary representations of is large enough to decompose L2(6)
.
is algebraic,
G obtained this way
% ; B E DES P.!A?IERES :
Introduction. I. Le groupe mhtaplectique et la repr6sentation mgtaplectique. 11. Formes lineaires admissibles et bicn polarisables. 111. Representations des groupes de Lie rgductifs. IV. Techniques de rhcurrence. V. Extensions des representations des groupes de Lie nilpotents connexes.
VI. Construction des representations T g9T
.
VII. Applications.
Bibliographic.
Appendice : Parametrization of the set of regular orbits of the co-adjoint representation of a Lie group. (Un texte pr6parC pour une confhrence P 1"Universitg de Maryland, decembre 1978).
INTRODUCTION :
g
Soit G un groupe de Lie d'algsbre de Lie
. Notre but est
de construire un
ensemble de reprgsentations unitaires irrgductibles de G, assez gros dans un sens qua nous prgciserons plus bas. 11 est bien connu que cela peut se faire dans le cadre de la "m6thode des orbites" lorsque G est r6soluble connexe (cf. Kirillov C171 , Pukanszky C241 ), ou lorsque G
est r6ductif connexe
(il est facile d1interpr6ter les rgsultats d'Harish-Chandra [ I 2 1 dans ce cadre). Nous le faisons ici dans le cas g6nEral.
* Le groupe G
*
*
opere dans son algsbre de Lie g par la reprgsentation adjointe,
et dans le dual &* d'un 616ment g troduit dans
*
[a]
E
&*
de g par la repr6sentationco-adjointe.Le stabilisateur est not6 G(g)
.
et son algsbre de Lie &(g).
un certain revgtement d'ordre 2 de G(g).
C'est l'ensemble des couples
(x,m)
oC x
E
G(g),
m
~ ( ~ )sur g G(g).
la diffgrentielle est la restriction de ig Id 1 g(g)
(I,-])
Notons
l'ensemble des classes de repr6sentations unitaires T
I ( ] , - ] ) = -Id
.
, tels que x et m
aient mcme image dans le groupe symplectique de g/g(g).
X(g)
Notons le G(~)&
est dans le groupe
m6taplectique associ6 B l'espace symplectique g/g(g)
ment non trivial du noyau de la projection de
J'ai in-
, et
1'616-
Nous noterons
de G(&
dont
telle que
.
Une forme lineaire g
est dite admissible si X(g)
besoin aussi de la notion de forme g
E
est non vide. Nous aurons
g* bien polarisable
sera donnee au chapitre 11. Disons simplement que si les formes sont bien polarisables, et que si
. La d6finition
g est rGsoluble, toutes
g est semi-simple,
g
est bien
polarisable si et seulement si ~ ( g ) est une sous-algsbre de Cartan de g.
Dans ces notes, nous construisons,si g et si
r
X(g)
c
est admissible et bien polarisable,
, une classe de representations unitaires de G, notEe T
g,Ta
Nous dgmontrons les propriEt6s suivantes.
.
T
est isomorphe 1 celui de T En particug *T lier, T est irrgductible (resp. factorielle, factorielle semi-finie) si ~ Y T et seulement s'il en est de mSme de T (i)
Le commutant de
.
(ii) Si a est un automorphisme de l'on a pos6
a~
(iii)Si si
g'
Gg
't
.
= T
g'
g.T
entre T
Soit
=
g9T
- 1 , etc.. .) .
est admissible et bien polarisable, si T'
T ag,aT
(OG
c
~(g') et
T sont disjointes. g' ,TI dans X(g). L'espace des opdrateurs d'entrelacement
alors les repr6sentations T
g 9T
et
e t Tg,Tl est isomorphe 2 l'espace des opErateurs d'entrelacement et T'
(v)
: a~
o a
(iv) Soient T, T' entreT
G, on a
.
Supposons
T factorielle. Soit y
un Glement du centre de G
~ ( ~ ).gAlors T
( y , l ) l'blement correspondant du centre de
(y)
.
et
g,T
sont la multiplication par un m8me scalaire.
~ ( y 1,)
*
*
*
*
I1 est nature1 de demander si l'on obtient par ce procSd6 beaucoup de repr6sentations de G (du m8me, plus pr6cis6ment, quelles sont les reprdsentations de
G
ainsi obtenues). J'espSre revenir plus tard sur cette question. 3e me
contenterai de donner par des exemples un debut de rgponse. Lorsque G est connexe et localement algebrique, les representations T g9T
avec
.r irrbductible, suffisent pour dgcomposer L'(G).
Ceci est encore vrai lorsque G est connexe de type I. Plus g6n6ralement. si
G
est connexe (mais pas necessairement de type I), et si a
une fonction telle que avec
T
T
(a)
g .T irrsductible, alors
=
E
1
L (G)
est
0 pour toutes les representations T g9T
a = 0
.
Je ne demontrerai ces resultats que lorsque G est connexe et localement algebrique. En effet, ce cas fournit dejl (B mon avis) une bonne motivation de travail entrepris ici. D'autre part, il doit Stre etabli avant le cas g6nSral. Le cas gengral demande la comparaison de notre construction avec celles faites dans l'article fondamental de Pukanszky C251
, ce
qui deborde le cadre de ces
notes. Je donnerai cependant quelques applications aux groupes de Lie moyennables.
*
*
*
*
L'idBe de base pour definir les representations T est trss simple. En g appliquant, autant de fois qu'il le faut, la thgorie de Mackey aux sous-groupes invariants nilpotents fermes connexes, on se ramGne au cas oii g est rgductive. En appliquant une nouvelle fois la theorie de Mackey, 1 la composante neutre de G connexe. C o m e
cette fois, on se ramsne au cas oii G
g est bien polarisable, G(g)
est rdductif et
est un sous-groupe de Cartan
de G et l'on est dans une situation oii l'on peut appliquer des resultats drHarish-Chandra. Evidemment, cette idee n'est pas nouvelle et a dLjP Qtd appliquee avec succss dans de nombreux cas. Citons les groupes rtisolubles connexes (Auslander
-
Kostant [23 ), les groupes connexes B radical co-compact (Pukanseky [261), les groupes compacts non connexes (Kostant C191). Ce qui est important ici est d'avoir une formulation des resultats qui survive 5 l'application de la theorie de Mackey, et donc qui incorpore les obstructions cohomologiques qu'elle comporte. C'est prdcis6ment le r81e que joue le revztement ~ ( ~ )de g G(g).
*
*
*
*
Le groupe metaplectique et sa representation mdtaplectique interviennent de manisre essentielle dans la formulation et dans la dgmonstration de nos r6sultats. (I1 est peut-Ztre curieux de remarquer que la representation mgtaplecti-
que ellr-m6me n'est pas associEe I une forme 1inEaire bien polarisable sur I'algGbre de Lie du groupe symplectique, et n'est donc pas une de nos repre) . J'ai rassemblE dans le chapitre I un certain nombre de g,T r6sultats plus ou moins bien connus sur la reprgsentation mgtaplectique, pour
sentations T
faciliter la lecture de ces notes.
*
*
*
*
Une de mes motivations en entreprenant ce travail 6tait de donner un peu de consistance 1 des conjectures relatives B la formule de Plancherel des groupes alg6briques unimodulaires, EnoncEes dans une confsrence que j'ai faite 1 1'Universiti5 de Maryland. Pensant que cela pourrait ctre une motivation aussi pour le lecteur, j'ai adjoint en appendice le texte que j'ai Ecrit B cette occasion
-
en harmonisant les notations avec celles employEes ici.
Toutes les construcLionsdEpendent d'un choix d'un caractsre unitaire non trivial de
g-
. Nous noterons
i une racine carrEe de
toutes, et nous choisissons le caractsre x Ainsi si
V
e
-1
fixEe une fois pour
.
est un espace vectoriel r6el de dimension finie et si
son dual, alors V*
V
+
ix
V*
est
est canoniquement en bijection avec le dual unitaire de
par l'application qui 2
v
E
V*
associe le caractsre Tf : x
+
eV(X)
de
v . *
*
*
*
Tous les groupes de Lie considEr6s ici sont dEnombrables I l'infini. Lorsque
T est une representation d'un groupe de Lie dans un espace de Banach H , on suppose toujours H
sEparable et
T
fortement continue.
PrincipaZes notations : I. Si
V
est un espace vectoriel, on note
2. Si
V
est un espace vectoriel r6e1, on note
-
et v + v
son dual. V
E
son complexifi6,
la conjugaison complexe.
3. Si g
est une alggbre de Lie sur un corps cornmutatif, on note U(g)
son algsbre enveloppante, Z(g)
son centre.
4. Si G est un groupe de Lie, on note G
V*
Go
sa composante neutre. Si
opere dans un espace vectoriel V de dimension finie par une reprdsentation
et si v
E
V
, on
note
l'algsbre de Lie de 5. Soient
stable V 1 et
V
G(v)
le stabilisateur de v
G, on note 3
V2
V,
3
V2
. On note
g(v) -
l'algsbre de
dans V. Si g est Lie de G(v).
des espaces vectoriels, et
x
l'dl6ment de
x
E
GL(V)
GL(V]/V~)
laissant
qui s'en d6-
v~'v2 duit. Si tion 5
v1
g
est une forme lindaire sur V, on notera glvl sa restric-
.
6. Soit
H un sous-groupe fermg de
une reprgsentation unitaire de G Ind (R) H
G
d'alggbre de Lie
&
. Si
R est
H dans un espace de Hilbert H, on note G induite par R. Rappelons que l'es-
la reprgsentation unitaire de
pace de cette representation est l'espace de Hilbert form6 des fonctions
a
sur G, 2 valeurs dans H, qui sont mesurables, et verifient : 1
y
pour tout
[
(2,
E
G ,x
dx r
H
.
-.
G/H Le groupe
G
opere par translations2 gauche dans cet espace (voir [3] chapi-
tre V par exemple).
7. Si u 2 gauche sur
G
U(&)
--
, on note par la msme lettre l'opgrateur differentiel
qui lui correspond. En particulier, si X
g
,a
.
Cm(G) ,
CHAPITRE I
-
LE GROUPE METAPLECTIQUE ET LA REPRESENTATION METAPLECTIQUE
Soit V
un espace vectoriel reel symplectique. Nous donnons (par.1-5 .) une
description du groupe metaplectique et de sa reprdsentation mdtaplectique, sous la forme oG nous l'utiliserons. Cette presentation est due 1 Souriau C311
et Lion [203
au livre 1223
. Pour les demonstrations et
l'historique, nous renvoyons
. Pour tout sous-espace Lagrangien
un caractere Q~
L de VC , nous ddfinissons
du sous-groupe du groupe metaplectique stabilisant L et
donnons plusieurs proprietes amusantes de ces caractPres (par. 619.).
I. I . I n d i c e de MasZ;v :
Dans tout ce chapitre, V
est un espace vectoriel reel symplectique i-e. un
espace vectoriel rdel muni d'une forme bilingaire alternde non dEgdnErde que nous noterons B. La dimension de V
est donc paire. Nous la noterons 2d.
est un sous-espace de V, on note 'E
Si E
le sous-espace de V
Un sous-espace E *st dit isotrope si 'E =
E
. On note
Si L
E
A(V)
'E
, et
A(V)
L 1 o L2
@
L3
si E
est un sois-espace isotrope de V + E
. Ona
E
1'
, on
pose
A(V).
on considsre la forme quadratique
t(L (p,q)
E L
trois sous-espaces lagrangiens. Sur l'espace
On pose
oG
contient E, et lagrangien si
l'ensemble des sous-espaces lagrangiens de V
E L = (L n)'E Soient L, , L2 , L3
orthogonal.
L2' L3) = p - q
est la signature de Q
.
C'est
l'indice de Maslov, c o m e d6fini par M. Kashiwara. L'indice de Maslov
est donc une fonction 2 valeurs dans Z- d6finie sur A(V) NOUS noterons GL(V) Soient
x
A(V).
le groupe symplectique de V, i . e . le sous-groupe de
Sp(V)
B.
conservant
des sous-espaces lagrangiens de V , E un sous-espace
Lo, L1, L2, L3
contenu dans L1 n L2
isotrope de V
A(V)
x
+
L2 n L
, et x e Sp(V)
+ Lg n L l
.
On a les formules : (1)
t(LI.L2.L3)
= t(L2,L3.L,)
= -t(L 2 ,L 1.L 3
(2)
t(LI,L2,L3)
-
+
(3)
t(xL ,xL ,xL ) 1 2 3
(4)
t(L
t(L0.L2,L3) =
-
t(LO'L1.L3)
t(L0.LI'L2)
= 0
t(L ,L ,L ) 1 2 3
E E E ,L ,L ) = t(L L L ). 1 2 3 1' 2'3
On posera
I . 2 . Indice de deux sous-espaces lag~angiensorient& :
tln sous-espace lagrangien orients est un couple (L,e)
e
est une com-
.
A ~ L (Rappelons que tous les sous-espaoes lagrangiens
posante connexe de
sont de dimension d). orient&
oii
Notons
A(V)
l'ensemble des sous-espaces lagrangiens
de V. I.
Soient L = (L,e) nombre
E(L,L* 3
L' = (L',e'J
at
qui vaut
On peut choisir une base
telles que e l Alors
A
...
c(L,L1) = + I
g6n6ra1, posons
E
A
..., ed , et
que l'on a : L n L' = 10)
de L et une base f l ,
fk) j' si et seulement si f
= L
Nous allons dcfinir un
f 1. Supposons d'abord
el,
ed E e
dans A(V).
%(e
1
=
5
A
... A
jk
I
pour fd
6
e'
...,fd
de
f ,k
d
S
5
. Dans
.
L'
.
le cas
n L' , et choisissons une orientation de E. Par passa-
ge au quotient, on en d6duit des orientations eE et
e' de L/E et E
L'/E.
L'espace
E'IE
est canoniquement muni d'une structure symplectique, et l'on
obtient ainsi des elements L/E et L'/E E(L,L')
=
de A(E'/E).
€(L/E,Lq/E). (On v6rifie innnediatement que cette definition ne dB-
pend pas du choix de l'orientation sur E). s(L,L1)
(6)
On pose alors
=
d-dim L n L' i €(L,LV)
Soient L ,L ,L des elements de A(V) 1 2 3 proprietes suivantes :
Enfin on pose
. , et x
E
Sp(V).
L'indice
s a les
I . 3 . Le groupe m g t a p l e c t i q u e :
Soit x
E
Sp(V).
On definit une fonction sX
sur A(V)
2 valeurs dans les
racines quatriPmes de 1, en posant : sx(L) = s[L,xL>
(10)
03 L
E
A(V),
et L est une orientation de L. La definition ne depend pas
du choix de l'orientation. I1 existe deux fonctions
pour tout L, L'
E
A(V).
+
sur A(V)
Si L
E
A(V),
vsrifiant les relations
une telle fonction est complStement
determinee par le choix d'une racine carree de On note Mp(V)
l'ensemble des couples
est une fonction sur A(V)
sX(~)-I
(x,@), oii x
vgrifiant (1 1) et (12).
c
. Sp(V),
et on
$
On vCrifie que l'on munit (13) oii
est la fonction sur A(V)
L'application
1( I,]),
d'une loi de groupe en posant :
= (=',@")
( x , Ql)(x,$')
@"
Mp(V)
(x,@)
,-I)).
(1
-t
dgfinie par la formule.
x est un homomorphisme surjectif de noyau
fait un groupe localement isomorphe 1 Sp(V). Mp(V)
une unique topologie qui en
On verifie qu'il y a sur Mp(V)
est connexe. Si V
= (0)
, Mp(V)
=
Si V # (0) , on demontre que
- ;/2_2
.
Le groupe Mp(V)
s'appelle
le groupe m6taplectique (pour tout ceci, voir C221).
I. 4. Le groupe dfHeisenberg : On note 2 muni
V a ge
l'algPbre de Lie dont l'espace vectoriel sous-jacent est
du crochet
Cv + te, v' + t'el
=
On note N
B(v,v')e.
le groupe de Lie
simplement connexe d9alg2tbredeLie 2. C'est le groupe d'Heisenberg. I1 est de dimension 2d + 1 . Le th6orCme de Stone-Von Neumann affirme qu'il y a une, et une seule, classe d'gquivalence de reprssentations unitaires irreductibles de N triction au sous-groupe exp(5e)
dont la res-
est multiple du caractere exp(te)
Nous noterons T une telle representation et
+
eit.
3 ll'espace de Hilbert dans
lequel elle opCre. Soit L
6
pace L +
A L est associe un modPle concret de T. On note -L b l'esqe . C'est une sous-alggbre abdlienne de 2 On note BL le sousA(V).
.
groupe analytique correspondant, et x,(exp(v+te))
XLl'espace
= eit
xL
pour v + te c b -L
de Hilbert dans lequel
le caractCre unitaire de
. On pose
T L
N = Ind (X )
BL
tel que
et on note
B~
TL opgre (la dgfinition des representa-
tions induites est rappelee dans les "principales notations"). Les reprgsentations T et
TL sont gquivalentes.
Soient L, L'
A(V).
E
Come les representations TL et
il existe un opgrateur d'entrelacement unitaire de
%
TLl sont equivalentes, dans
xLl. Nous allons
en choisir un canoniquement. Soit dX une mesure de Haar sur L n L'. Soit a c $fL une element represents par une fonction continue sur N fonction FLVL(a)
pour tout
x
.
sur N
5 support compact modulo
BL
. On definit une
en posant :
N. Alors FLIL(a)
3'.
est dans
On dgmontre que l'on peut
choisir dX de telle sorte que FLVL se prolonge par continuits en un operateur unitaire de
dans
Soient L,, L2, L3
E
ML, qui
entrelace TL et TLt
-
On a (d'apres Souriau [311 et Lion C201).
A(V).
I . 5 . La repr6sentatiovr mktapZectique :
Le groupe Sp(V) X(V
+ te) = x(v)
Pour tout x
E
opSre c o m e groupe dlautomorphismesde 2 + te
Sp(V),
(x
c
Sp(V),
t
E
8).
I1 op2re donc dans N.
X -I T = T ox
la reprgsentation
par la formule
est equivalente B
T
d'aprGs le theoreme de Stone-Von-Neumann. I1 existe donc un opgrateur unitaire S'(x) S'
dans.
8
te1 que S1(x)T(n)S'(x)-I
pout tout
n c N
.
et
est une representation projective de Sp(V).
Supposons V # 0 Shale sentation unitaire ( 18)
= T(x(n))
S
~(x,@)~(n)~(x,@)-l
pout tout
(x,@) E Mp(V)
[30] a demontr6 qu'il existe une et une seule repre-
de Mp(V)
telle que
= T(x(n))
, et tout n
.
N
.
Nous appelerons cette repr6sentation la representation mgtaplectique. On a
(I-I) de
= 1
aR -
Lorsque V
=
0 nous adopterons cette formule c o m e definition
S.
(La representation metaplectique est aussi appelGe : "reprbsentation de SegalShale-Weil", "oscillat6r", "spinor", "harmonic").
xL
Nous voulons donner une description concrete dans l'espace le par. 4
. Soit donc
L
E
A(V).
Soient x
c
Sp(V),
u
c
dGfini dans
% . On pose
(AL(x)a) (n) = a(x-'(n)). pour tout n
N. Alors AL est un opErateur de
E
dans
yxL, multiple
d'une isomGtrie. On pose
( 19)
SL(x)
=
L'opbrateur (20)
I~A~(x)B-'FLSxL AL(x)
S1(x) L
~t(x)~~(n)s;(x)-l
pour tout n Soit
(x,$)
E
E
.
est unitaire dans
gL , et
l'on a :
= TL(x(n))
N. Mp(V)
. On pose
(21)
sL(x'$)
Alors
SL est la representation mGtaplectique de Mp(V)
=$(L)SL(x).
(Compte-tenu de (20) et (21),
dans %L
.
il suffit de v6rifier que SL est une reprgsen-
tation,ce qui resulte de (14) et (16)).
I. 6. Dlfinition des caract0res L'espace A(V
5
)
on note
vg
pl
:
est canoniquement un espace symplectique complexe. On note
l'ensemble de ses sous-espaces (complexes) lagrangiens. Si q1
-
le nombre de valeurs propres < 0
associge Ila forme hermitienne
1E
.4(VC), de l'application hermitienne
X + ~ B ( x , % )sur
1.
Si
_1
E
ACV,)
,
-
1
e s t une sous-algsbre abelienne de
opere par l a r e p r s s e n t a t i o n
dans l ' e s p a c e
T
Xm
representation. Par r e s t r i c t i o n
l'on a (22)
Si
(cf 141
dim H . ( l , . f )
= 0
dim H (I,^~WD) 91
= 1
J
E
, [ I 4 1 , ou
-
Sp(V), e t
Soit
(x.4)
E
2-module.
H.(l,V) 3 -
pour
si
-
de l a
On peut donc consi-
j
. On s a i t que
E
j # ql
-
v~
, on
note
S P ( V ) ~ l e s t a b i l i s a t e u r de
Mp(VIE son image rsciproque dans
MP(V)~
n
%
1271) :
e s t un sous-espace de
dans
cm
des v e c t e u r s
devient un
d e r e r l e s espaces v e c t o r i e l s complexes
. L'algsbre
!+
. Alors
done dans l e complexe standard
A*L@;1$ , e t
p r s s e r v e l a d i f f g r e n t i e l l e . Donc
(x,$)
Mp(V).
2 ,et
(x,@) opPre dans
E
dans
'f
. I1 opsre
on v L r i f i e que c e t t e a c t i o n
opere dans l ' e s p a c e
~%(t,g).
Come c e t espace e s t de dimepsion 1 , il opsre par un s c a l a i r e . Nous noterons ~ ~ ( x . 4 )ce s c a l a i r e
.
I. 7. CalcuZ de pl
Zorsque
-
-
Soit
L
A(V).
1
est rdeZ :
Nous a l l o n s c a l c u l e r
Pour c e l a , on remarque que
OD
gL
formules (23)
(19)
+
et
(21)
I
que l ' o n a , pour d e t xL
pour
a(l)
H , ( L , ~ ) (qui e s t d g a l 3
pl(x,@) = @(L)
-
-
.
1= L
E
e s t un espace de f o n c t i o n s
v o i t immddiatement que l ' a ~ p l i c a t i o na isomorphisme de
p1
I
1/ 2
de
cm s u r
dans
%/LC)
sur
(x, )
E
MP(V)~:
-
N, e t on
$ i n d u i t un
. 11 r e s u l t e
des
I.
8. CaLmZ
dc
1 est totazement cornpZexe :
krsque
Pl
-
I
On suppose que l'on a
= (0)
n
. La
forme hermitienne X
LB(x,X)
-t
est
sa signature, de sorte que q = ql , p = d-ql L'application de restriction induit un isomorphisme de SP(V)~ sur U(p,q).
non dGggn6rge. Notons
(p,q)
.
-
En particulier, SP(V)~
-
est connexe.
MP(V)~
-
(24)
est l'unique caractsre de
pa
Nous allons dgmontrer l'assertion suivante : tel que : 1(1,-1)
et
= -1
-
~ ~ ( x . 4 =) ~det(x 1) -
pour tout
(x,d)
MP(V)~
-
.
Ddm nstration : La premisre formule de (21) est Cvidente. I1 en rCsulte que
.
dCpend que de n
x
et dgfinit un caractere de
tel que
pl(x.@)' n
On Ccrit donc V V = V
I
+ V2
, on
~B(x,X)
Soient
A(V
=
1
, il
det(xl)"
(xl.@,) 2
E
pour tout
-
= U(p,q).
-
x
6
U(p)
ne
11 existe donc
(x,O t MP(V)~
suffit de l16tablir pour
-
Pour dCmon-
x U(q).
comme s o m e directe de deux sous-espaces symplectiques : pose
1.
-3
= In
est positif sur
tel que x = x E
=
-
trer l'CgalitC
L2
-
-
I
2
que
Sp(V),
P~(X,()~
Mp(VI)
+ x
V. 3
1, , et
, ( x ~ , $ ~ )t
, et
, et
5
on suppose que _1
Mp(V2).
Soit
+ L2) = @,(L,)
@(L1
+
-11
L2 ,
-
1 -2
nCgatif sur
=
(x,$)
a2(L2)
l'616ment de Mp(V) pour tout
L1 E A(V1),
) . L'application
2
X
On peut considhrer N 1 et
(les groupes dlHeisenberg correspondant P
N2
Mp(V2)
V I et V ) comme des sous-groupes de 2 N1
X
N2
-+
N
dans Mp(V)
.
est un homomorphisme de Mp(V,)
N
et l'application naturelle
est un homomorphisme surjectif. 11 r6sulte du th6orSme de Stone-
Von Neum;ann que, composCe avec cet homomorphisme, T est isomorphe 2
T, Q T 2
, et
T
S j
j
et
que, composee avec
,
(25)
S est isomor~heP
S, O S2 (oii
ddsignent les reprgsentations correspondantes de N
C'est une parrie de la dlmonstration de (22) que l'injection induit un isomorphisme de HO(i,
,u;)
x
j
et Mp(Vj)).
-$x
f?
-
SUT H (1.x)
Hq(L2,%)
X; +
9-
I1 rdsulte de tout ceci qu'il suffit de ddmontrer (24) lorsque V = V
1
et
lorsque V = V 2Supposons d'abord tation T
. On utilise une
71 = V1
. On note x1
autre rlalisation de la reprken-
l'espace des fonctions a
-
sur N
qui sont
cm et
qui vdrifient : a(n exp t E) = e-it a(n)
(n
E
N, t
E
%)
On note T1 la reprdsentation de N dans 2, obtenue par translations B gauche. Elle est isomorphe B T. Notons S1 la reprlsentation mltaplectique dans
-
. On
-
sait que l'on a, pour tout
06 c vlrifie cation a
+
(Voir [ 151 par eremple)
c2 = det(xl).
-
m
-
-1
rdalise un isomorphisme de xl/l gi
a(1)
que l'action cte
m
(x,@)
dans HO(l,q)
-
. D'autre
part l'appli-
- . On
sur g
est la multiplication par
voit donc
c , ce qui
ddmontre (24) dans ce cas. Supposons maintenant V dans l'espace
xi-
=
V2
. Alors
on peut rgaliser T (come ci-dessus )
par la reprdsentation T1 -
.
D'autre part, la dualitd de Poincarl montre que Hq(1,f)
~ " ( 1 ,0fn'l) 1
. L'espace
Il0(1,q) - s'identifie
eat isomorphe 1
1 l'espace des fonctions
at
telles que
-
dimension 1 , pour S-(.,@)a
=
-1
T i ( X ) a = 0 pour tout X E 1 . Comme cet espace est de a dans cet espace, il rbsulte de (26) que l'on a
, auec
ca
c2 = det(x-). -1
-
1
dans ce cas
. Comme
est
-1
1E
A(VC)
-
L'espace
c det(x ) -1
1
. C.Q.F.D.
I. 9. CaZcuZ de
Soit
dans Hq (1.f)
(x,+)
det(xi) = det(xl) , car sont des sous-espaces lagrangiens transverses, ceci Stablit (24)
donc la multiplication par est
L'action de
pl dans Ze cas g&n&raZ: -
. On pose
E1/E
E =
I
n V
. C'est
un sous-espace isotrope de V.
a une structure d'espace symplectique, et Y E C
-
en est
un sous-espace lagrangien totalement complexe. Soit x
i
SP(V)~
rons
(x-4)
Si L
E
c
A(V)
. Notons
Mp(VlE
xE
Soit
l'6lGment correspondant de
(xE,$)
est tel que L
depend pas du choix de
L
mule (12) appliqude P
$ I et 5
Soit E
(X
,$)
(x,@) E
E
MP(E'/E)
Mp(VI1
-
E
A(V)
. Alors
x
relevant xE
tel que
L
@(L)$(L/E)-I 3
.
est dsfini. I1 ne
E, c o m e il resulte dp la for-
JI
. On
le note
6
SP(V)~
, et
. On a
Conside-
E relevant x
un Element de M~(E'/E)
E, le nombre
Sp(E1/E).
$-'.
$
on choisit un element
(xE,$)
E
M~(E'JE)
l/EC -
allons dsmontrer la formule :
. Nous
Avant de faire la demonstration, remarquons que (27) ne depend pas du choix du representant (xE .$)
de xE. Remarquons que . "Ile(x
-
E
,$)
est calcule dans
le paragraphe 8. Remarquons que (23) est un cas partTculier de (27) qui a 6tL d6rnont1-61 part, malgr6 le double emploi, P cause de sa simplicitb.
Dknsonstmtia de (271 : On
choisit L
. Elle
+ Re -
E ' /
EX
%
tel que L contienne E. Notons
A(V)
l'alglbre de Lie
slidentifie 3 EL + Ee/E. Cette alglbre de Lie opPre dans
et on voit facilement, par restriction au sous-groupe analytique de
N dgalgSbre 'E
+
Re -
D'autre part, soient Notons pE(x,@)
, que
%LIE$
(x,@)
E
l'action de
est isomorphe come n -module 5 %ZlE. -E E MP(V)~ et (x ,$) 6 M~(E'/E) come plus haut. (x,@)
, dlduite de SL(x,@)
dans X;/E
par passage au quotient. Notons SL,E(~,$)
l'action de
(x,$)
dans
zLIE , '
m
par la reprgsentation m6taplectique. I1 r6sulte de (21) que l'on a :
Par ailleurs, il rdsulte facilement des formules (22)
-
ou bien on le voit en
.
1es d6mtrantr que H ~ ( L , ~ )est isomorphe H H (1IE , V I E 'km) La formule
s- E
(27) s'en dgduit. C.Q.F.D. De
(27) on d6duit les formules suivantes :
I . 10. Conpamison des
soient
.
1 1' E A(VC)-
pl -:
. Soit
(~$4) E Mp(V)
. 11 rgsulte de
n Mp(V) 1' -
-1
(28)
que l'on a : (29)
0G
pl(x,$)
-
--
El,ll
=
--
(x)det(xlllnll)~ll(x, )
-
- -
est un caractlre du groupe
I1 est intgressant de calculer
Sp (V)
-
n Sp (V)
-
B valeurs dans (+ 1 ) .
. C o m e nous n'utiliserons
E
-
pas le r6sul-
tat, je le ferai-avec peu de dltails. On remarque tout d'abord que si V est sonrme directe de deux sous-espaces symplectiques x,ettelsque,posant
l.=LnV.
-3
3s
'
VI et V2
1!=LtnV.
-3
3:
, stables par
(j=I,Z),l'onait
x I et
x2
sont les restrictions de x
2 V l et
V2. D'autre part,
(x). On est donc amend 1 calculer E ~ , (XI E~,~(-X) = E~,,(-I)E ~ Idans -1.1' les cas particuliers suivants : (i) Toutes les valeurs propres de x (ii) x n'a pas de valeur propre
< 0
sont dgales 2 -1.
.
Dans le cas (ii), x est dans la composante neutre de SP(V)~ n
-
Sp(VI1
-
, et
donc E ~ , ~ ~ ( x=) I. Dans le cas (i), on a
Soient
1 , i',1"c
E~,~'(X) =~~,~,(-l), et l'ona:
-
--
. Soit
A(VC)
x
Sp(V)
E
stabilisant
1 , L', et 1".
r
On a, d'apris (29) :
La formule (30) est compatible avec (39), de sorte que si on a ddmontrd (30) pour des paires 1 ,
1' et 1, 1", on
l'a ddmontrde pour
1
donc de ddmontrer (30) dans le cas particulier oii avec L
E
A(V))
et 06
1' n
=
A', 1".I1 suffit
est rdel (i.e.
- = LE
. Par des ddcompositions en s o m e di-
(0)
recte, on se ramene au cas oii dim V = 2. On doit donc examiner les cas particuliers suivants :
2
(i)
-1'
est rdel et
(ii)
-1'
est totalement complexe et
qlI = 0
(iii)
-1'
est totalement complexe et
qlt * 1
n
1' =
{01
-
-
Le premier cas rLsulte de (29). Le cas (iii) se ramine au cas (ii), car si 1'
est totalement complexe,
connexe, cf. paragraphe 8).
E
-1
,I'
=
I (eneffet
SP(V)~, = Sp(V)jl
-
-
est
I1 r e s t e 1 Etudier l e second cas et
. Cela
s e f a i t en cornparant l e s modsles
S1 -
S l I de l a r e ~ r E s e n t a t i o nmctaplectique, e t n k e s s i t e un peu de c a l c u l .
-
Remarque 1 : Supposons que bilise
2
et
x
Sp(V)
E
1' E
A(VC)
-
n ' a i t pas de v a l e u r propre r e e f l e , e t que
x
sta-
. Alors
+ q1* + d i m 1 1 1 n q1 -
1' =
0 mod 2.
( c e l a r E s u l t e d e (30) e t de l a r e l a t i o n
,L~ (XI -
E
= €I,
--
(-X) = 1 )
.
Remarque 2 : La formule (31) e s t compatible avec l e r e s u l t a t s u i v a n t , l a i s s 6 en exercice au l e c t e u r . S o i t
V
un espace v e c t o r i e l symplectique s u r un corps cornmutatif
de c a r a c t e r i s t i q u e # 2. Soient V, e t s o i t lequel
x
stabilisant
=
d e s sous-espaces lagrangiens de
L, L' e t L". Notons
V-
l e sous-espace dans
opPre avec l a vakeur propre gdndralisee -1 L- = L n V-
On pose
m
x
L, L', L"
dim L-/L-nLl
det x
L I L ~ L *det
,
etc...,
( i . e . V- = ker(Id+x)
e t on n o t e
+ dim L'/LlnLI
+ dim LI/LynL-
L I I L l n L a t d e t x L,llLttnL
=
.
dim V
).
CHAPITRE I 1
- FORMES
LINEAIRES ADMISSIBLES
ET BIEiV POLARISABLES
Dans tout ce chapitre, G
est un groupe de Lie reel d'algPbre de Lie
g
.
Nous dEfinissons les formes admissibles, les formes bien polarisables et terminons par quelques remarques sur la dgfinition des reprgsentations T g,r-
I I . I . Repre'sentation coadjointe :
g e
Soit
gf.
On note B
g
la forme bilineaire alternee sur g
dgfinie par
la formule Bg(X,Y)
(1)
= g(tX,Yl)
(X,Ycg).
I1 se trouve que le noyau de B est Sgal 5 gig) (cfC171) g/g(g) -
est canoniquement un espace symplectique. Le groupe G(g) g'
Nous emploierons les notations suivantes. Soit V d'une forme bilinEaire alternEe B, et soit H des automorphismes lingaires conservant B on note 'E
un espace vectoriel muni
un groupe operant dans V par
. Si
E est un sous-espace de V,
le sous-espace de V orthogonal. En particulier, V/V'
espace symplectique dans lequel H opsre. Nous noterons H" (h,m) E H x M~(v/v'),
couples
tels que h
et m
est un
l'ensemble des
aient mZme image dans
-
Nous dirons qu'un sous-espace L de V
est lagrangien si L = .'L
espace L de V est lagrangien si et seulement si L contient ,'v ;st lagrangien dans
Un souset L/V'
v/v'.
Compte-tenu de la description de M~(v/v')
'H
laisse in-
B
variant
sp(v/v'l
de sorte que
come l'ensemble des couples
donnee en T.3, on peut dgcrire
(h,$) , oii
h
E
H,
(I est une fonction
sur l'ensemble des sous-espaces lagrangiens de V, tels que si l'on pose
x
=
, et
hVIVl
@(LI#)
= @(L)
relations (1.11) et (1.12) L'application le noyau est {(I
(h,@)
+
pour tout sous-espace lagrangien de V, les
soient vBrifiBes.
h est un homomorphisme surjectif de JIV sur H dont
,I) ,(I,-])]
. L'application
(h,@)
-+
(($v~,@)
est un homo-
morphisme dans le groupe M~(v/v~). Soit
un sous-espace lagrangien (complexe) de
que hl
c
1. Nous poserons
V~
. Soit
(h,@)
c
H"
tel
(cf. par. I. 6) :
On a donc ((lompte-tenu de (1.28)
:
Si X appartient 1 l'algsbre de Lie
&
de H, et si XL
Tout ceci s'applique en particulier 1 G(g)
c
L , on posera
:
et 1 g muni de la forme bili-
-
ndaire B Dans toute la suite, le groupe G(~)& g 11 a Bt6 introduit dans [8].
joue un rGle fondamental.
11. 2. Formes admissibles :
Soit g
&*. On note X(g)
l'ensemble des classes de representations unitai-
res T de G( g ) g vgrifient les propristbs suivantes :
(7)
la differentielle de T
est un multiple de
On dit que g est admissible si X(g)
iglg(g).
est non vide.
I1 est immediat de voir que caractPre unitaire le
est admissible si et seulement s'il existe un
xg de(~(g)~)~
tel que
xg (],-I)
=
-I, et de diffbrentiel-
iglg(g).
S'il existe, un tel caract2re est unique, car rdciproque de G(g)O Supposons g
dans
(G(~)~)&
(qui est l'image
~(~)g)a au plus deux composantes connexes.
admissible. Alors
X(g)
est l'ensemble des classes de reprgsen-
(~(~)~)g est multiple de
tations de ~ ( g )dont ~ la restriction 5 Comme on a : ~(g)g/(G(g)~)~ = G(g)/G(g)O
, on
classes de reprEsentations projectives de
Xg ' s'identifie aux
voit que X(g)
G(g)/G(g)O
associees au 2-cocycle
dgfini par l'extension :
Remarque I : Disons que g
est acceptable si (G(~)~)~ a deux composantes connexes, et que
que g est entiere G(g)O.
si
ig\g(g)
est la diffErentielle d'un caractere de
On voit facilement que si g est entiere, alors g est admissible
si et seulement si
g est acceptable.
Remarque 2 :
- c % , stable par -
Supposons qu'il existe un sous-espace lagrangien b
g(g).
Rappelons la notation (5) :
Rappelons que tel que
pb est la differentielle d'un caractPre %(],-I) = -1.
0,
-
de
(~(g)~)~
-
On voit donc que g est admissible si et seulement s'il existe un caractere de
G(gl0
"de diffGrentielle pb + ig)g(g).
Remrque 3 : g donn6 dans &*
La notion d1admissibilit6 pour un
Go
ne d6pend que du groupe
'
Notations : Si cela est n6cessaire, on incorpore G dans la notation : X(g) = XG(g).
On note
x ~ ~ ~(resp. ( ~ )Xfac (g))
.
tibles (resp factoriels)
r
Soit de
r
le sous-ensemble form6 des 616ments irrlduc-
.
un sous-groupe du centre de G
. L'application
On notera X(g,r,)
y
(y, I )
-t
identifie
(ou XG,r(g,n))
r
ments dont la restriction 5 vide, on dira que g
est
et soit
r
n
un caractire unitaire
~(~)g.
B un sous-groupe de
le sous-ensemble de X(g)
est un multiple de q
. Si
form6 des 616X(g,n)
est non
n-admissible.
II. 3. Fomes bien polarisables : Pour un moment nous changeons les notations, et nous notons & une algibre de Lie de dimension finie sur c. - Soit g haut
E
&* , on d6finit B
comme plus
g
.
Une sous-algsbre
k
de g
est appel6e une polarisation en g si b est un
sous-espace lagrangien de 8. Soit G un groupe complexe d'algibre & et soit B que correspondant 5 une polarisation
b
en g
le sous-groupe analyti-
.
Alors Bg est un ouvert de Zariski de l'espace affine g + l'orthogonal de
2
kL
(oa
k1
dans &* ) .
Les conditions suivantes sont Gquivalentes (i) ~g = g +
b1
(ii) Pour tout (iii) Bg
X
E
2'
, 2 1
est ferm6 dans g
.
est une polarisation en g
+
h
.
est
b
Lorsqu elles sont satisfaites, on dit que
v6rifie la condition de Pukanszky
(cf. 1 2 1 ) si elle est r6soluble et si elle
Nous dirons qu'une polarisation est
v6rifie la condition de Pukanszky. Nous dirons que g est bien polarisable si
admet une bonne polarisation.
g
Lemne 1 : Supposons g
semi-simple. Alors un dl6ment
et seulement si &(g)
g* est bien polarisable si
g
est une sous-algPbre de Cartan de
5
.
De?monstration : Soit r
le rang de
. On
g
et seulement si dimg(g) en (cf
g
=
sait que g
r ,
et dans ce cas, les polarisations r6solubles
sont les sous-algsbres de Borel
. [ 7 1 p- 6 0 ) . Soit
sation r6soluble en
Gg est fermE dans &* sait que Gg
b
de &
g* tel que dim g(g)
g
g. C o m e
compact, de sorte que
admet une polarisation r6soluble si
b
g([b,b])
telles que = r
. Soit b
=
O
une polari-
B est un sous-groupe de Borel de G, G/B est
vdrifie la condition de Pukanszky si et seulement si
. Identifions
g et
est fern6 si et seulement si
g
$
par la forme de Killing. On
est semi-simple. Donc g est
bien polarisable si et seulement si g est semi-simple et r6gulier.
C.Q.F.D. Si g
est r&soluble, tout
g
gf
est bien polarisable
(cf [ 3 4 ] ). Dans
l'appendice, une notion de forme lin6aire "tres rdguliere" sur une algObre de Lie alg6brique g
est ddfinie, qui ggn6ralise les forme:
li'sres lorsque g
est semi-simple. Les formes trSs r6guliPres foment un ou-
semi-simples rggu-
vert de Zariski non vide, et elles sont bien polarisables (cf [ I ] ) Revenons au cas oG &
est une algbbre de Lie rdelle. Soit g
par la mcme lettre 1161ement de g* qui prolonge
5
g
€
&*
. On note
. Une polarisation en
g
(resp. polarisat ion v6rifiant la condition d r Fukenszky, resp. bonne po1ari.-
b
s a t i o n ) est une sous-alggbre
de
q u i a l e s mzmes p r o p r i e t e s pour
gc
11. 4. Remarques sur Za de'finition des repre'sentations Soit
G un groupe de Lie d1algPbre de L i e
missible et bien polarisable. S o i t
T
€
. Soit X(g) . Nous &
T
g9T
g
E
&*
g,
: un element ad-
voulons a s s o c i e r B c e s
de r e p r e s e n t a t i o n s u n i t a i r e s de g,T propriEtEs annoncees dans l ' i n t r o d u c t i o n .
G, verifiant l e s
J e veux i c i . d e c r i r e une m6thode n a t u r e l l e pour c o n s t r u i r e
T g*T
donn6es une c l a s s e
T
. Bien que
ce
ne s o i t pas c e l l e que j ' a d o p t e r a i i c i , pour d e s r a i s o n s que j e dgvelopperai p l u s b a s , e l l e donne une bonne i d d e de l a p a r a m e t r i s a t i o n employee. J e v a i s supposer, pour s i m p l i f i e r , q u ' i l e x i s t e une bonne p o l a r i s a t i o n g
qui s o i t r e e l l e , i.e.
que
e s t s t a b l e par
dans
.
n g
d'alggbre
et
t e l l e que
b = (b
G(g). On n o t e
B
l e .groupe
Bo
n g)C
-
. De
b
en
p l u s j e v a i s supposer
l e sous-groupe a n a l y t i q u e d e
G(g)BO. On demontre que
G
B est fermd
G , e t q u ' i l e x i s t e une r e p r e s e n t a t i o n u n i t a i r e , uniquement determin6e,
que nous noterons
? , de
B
( i ) La r e s t r i c t i o n de de d i f f e r e n t i e l l e (ii) Soit
.
t e l l e que l ' o n a i t :
$ B
est un m u l t i p l e du c a r a c t s r e u n i t a i r e
B0
iglb. x < G(g). S o i t
~ ( g ) g ~n a
8x1
On pose
T
alors
,
=
g,-r,b
r ~
(x,$)
( un ~r e p r 3 6 s e n t~a n t de
x
dans
~T(X,+).
G (7). Compte-tenu de l a d B f i n i t i o n d e s reprdsen-
= Ind
B
t a t i o n s i n d u i t e s , rappelde d a n s l e s " p r i n c i p a l e s n o t a t i o n s " , r 6 a l i s 6 e dans un espace de f o n c t i o n s q u i v g r i f i e n t les r e l a t i o n s s u i v a n t e s .
a
sur
T
est P,T& G , '5 v a l e u r s dans l'espace d e T,
pour tout
X
6
b , oii
pb(X) =
-
l'on a posE
- 71 tr(ad
X tb) pour
%-
X
E
b.
Remrque 1 : La ddfinition (I I) pour
X
E
&(g)
,
coyncide sur g(g)
avec la dsfinition (8), de sorte que,
les relations (9) et (10) sont compatibles.
Remrque 2 : Rappelons que
pb(x.@)'
-
=
det(Ad x
)
~1%
, de sorte que la difference entre
( 9 ) et la formule qui ddfinit les reprPsentations induites est l'absence des
valeurs absolues. On epssre que la classe de T est independante de b , et que son commug,~,b tant est isomorphe 5 celui de r Lorsque ceci est vrai, il est legitime
.
de noter T cette classe. Lorsque est nilpotente, c'est prbcisement P,T la m6thode invent6e parKirillov pour decrire les representations unitaires irreductibles des groupes de Lie nilpotents connexes. Je vais expliquer maintenant pourquoi je prEjPre adopter une autre mgthode. (i) MGme dans le cas idgal considi5rE ci-dessus, oii il existe une bonne polarisation rdelle G(g)-invariante, dante de
b .
on ne sait pas si T
g , ,b ~
est indiipen-
Les r6sultats les meilleurs dans cette direction sont ceux
dlAndler I I] qui a demontre que c'est vrai quand- g
est alggbrique.
( i i ) En gEn6ra1, il n'p a pas de bonne polarisation rselle. Tant qu'il y a
des bonnes polarisations
G(g)-invariantes,
ce n'est peut-8tre pas
fondamental, car il existe des proc6dEs (dans le style "thEor6me de Borel-WeilBott") pour extrsire des representations de G
vgrifiant
( 9 ) et (10)
. Mais
G du
faisceau des fonctions sur
cela devient trSs vague.
(iii) En gdnEral, il n'y a pas de bonne polarisation G(g)-invariante.
(Un exemple en est le produit semi-direct de
SL(2,g)
avec un groupe
dlHeisenberg de dimension 3. On trouvera un exemple d'un groupe alg6brique complexe connexe sans polarisation rdelle invariante dans [ I 1 ) . La msthode adoptde ici, esquissde dans l'introduction, et d6taillEe ci-dessus, est certainement moins slsgante, mais elle est beaucoup plus simple, de sorte qu'on arrive B la mettre en oeuvre complPtement.
CHAPITRE I I I
- REPRESENTATIOIiS
DES GROYPES DE L I E REEUCTIFS
II. I . Intr--duction : Dans tout ce chapitre, G
est un groupe de
Lie dont llalgDbre de Lie g
est rgductive, g un element de g* bien polarisable et admissible, T
6
X(g).
.
Nous allons construire la reprfisentation T Lorsque G est connexe, et grT 7 irrfiductible, nous obtenons de cette manisre exactement les "reprEsentations irrgductibles tempEr6es de G dont le caractere infinitEsima1 est regulier", et notre mfithode dans ce cas se reduit P dire laquelle de ces reprfisentations (dfij2 construites par Harish-Chandra) nous choisissons d'appeler
.
T Dans le cas gfinEra1, il faut calculer l'obstruction de Mackey 2 fitendre g9-r une reprgsentation de ce type de GO 1 G. Comme Kostant l'avait d6jl fait dans le cas compact, on utilise une g6nbralisation (due P D. Vogan) du thgorDme de Kostant-Borel-Weil-Bott.
flotations : La classe de repr6sentations Gventuellement associEe 2 bien polarisable, et
Si G(g) tOre
xg
sera notee T
T E X(g),
est connexe, tous les elements de
. Nous
de G(&
Nous poserons Cartan de g
=
. On
'
g.T
noterons alors
X(g)
T
P.
g
c
,
k'..
4
g* , admissible et
T~ s'il est utile g.T
OU
sont des multiples du caracG
(ou T ) la reprgsentation g
g(g). D'aprOs le lemme 11.1. , c'est une sous-algPbre de
note
A c
h* -"
l'ensemble des racines de
c
1
6
le groupe de Weil
correspondant. Si a
le sous-espace radiciel correspondant. On identifie
E
h*
A
%
par rapport
, on
note
a
%
1 un sous-espace de
le sous-ensemble de
A
form6 des
On pose
A*
=
{a
6
E
A
%a
tels que
.
k
semble des racines de
a
-g
+
A, ua > 0) et
AC = A
+
c
k
5
. Bien que
n AC
: c'est l'en-
K ne soit pas
n6cessairement compact, les representations unitaires irreductibles de sont de dimension finie et parametrees par leur poids dominant y (dominant par rapport A
A:
A+)Un element y C
E
K
if
, dominant par rapport
iL*
, est le poids dominant d'une representation irreductible unitaire de
si et seulement si c'est la differentielle d'un caractsre de nous noterons On note
p
K
T. Dans ce cas
la representation correspondante.
&Y (resp.
Alors l1Plement
pC) + p
la demi-some des elements de
-
2pC
de
iL*
A+ (reap. .):A
est dominant pour 8 :
(car 11 est
A+ dominant) et differentielle d'un caractere de T (car
rggulier et
1
g
est admissible). On sait qu'il existe une et une seule representation unitaire irreductible
T de G ayant les propriEt6s suivantes. intervient dans (ii) Soit y
6
caractere de T, vLrifiant
it*
, dominant
{y+2pC
T.
pour
,y+2pC)
A:
< (I.I+P,
et diffgrentielle d'un
v+P)- Alors
&Y
n'in-
tervient pas dans T. Nous noterons
T cette representation unitaire irrdductible de G. R
G est compact et connexe, T g de poids dominant ig p . Si
est donc la representation irrdductible
-
L'existence d'une representation T vdrifiant
(i) et (ii) est due 5 Harish-
Chandra et 5 Schmid, l'unicite 5 Vogan (cf r 3 7 ] , [ 2 8 ] ,[353
details. les r6ferences donn6es dans r91).
, ou pour plus de
(i) Soit a un automorphisme de G. On a : a ~ =g T
ag
(ii) Soit Gcrivant g(gl) = si g' r
g'
E
E*
, bien
, on ait a'
t' + 2'
polarisable, admissible, et telle que, = 0
. Alors
T = T si et seulement g g'
Gg. (iii) Les repr6sentations unitaires irrgductibles de G obtenues
par ce procddd sont les reprgsentations unitaires irrdductibles de carre integrable modulo le centre de G. L'assertion (i) est evidente. Tout le reste est dG B Harish-Chandra (cf 1371)
I I I . 3 . Se'rie fondamentale pour Zes groupes c..nnexes : On suppose dans ce paragraphe que
ce qui signifie que On note
T, A. H
On note M'
est le centralisateur de
[m', m']
dans g
est fondamentale,
.
l,a,& . Alors
H.
le centralisateur de A
loin un groupe M), et E' r = -
t
les groupes analytiques dlalgCbre de Lie
est Egal l
G(g)
&
h
G est connexe, et que
dans G
(nous aurons B considgrer plus
m' * r + 2
son algPbre de Lie. On Ccrit
et oii s est le centre de
m' . On
note R et S
oij
les sous-
groupes analytiques correspondants de sorte que M' = RS. 0 Posons r = g(+
.
s = gls
. On verifie
facilement que r
est un element bien
polarisable admissible de L* , verifiant les conditions du paragraphe III.2., &e sorte que nous avons d6jl defini la representation irreductible : T Come
de R.
g est admissible, il existe une (et une seule) representation de
dont la restriction B
R est :T
, et
M;,
dont la restriction 1 S est le carac-
tPre de diffcrentielle is. Nous poserons m' = glm' Nous noterons
. Alors
m'
est admissible, M1(m') = M1(m') 0
la reprgsentation de
D'aprGs le lemme 1 , le stabilisateur de
MA M1 u Tm,
d6finie ci-dessus. dans M'
est M'(m')MA
=
H.
M' La representation In%,(T U1 0 On choisit un element
,)
X
E
est donc irreductible. Nous la noterons
5 tel que si a
a(a) = 0. On pose
U' et on note
U'
=
E
A
verifie a(X) = 0, alors
+ a
g
B(x)>o acA
le sous-groupe analytique correspondant. Alors
sous-groupe ferme de
M'U'
est un
G(c 'est un sous-groupe parabolique "cuspidal", au sens
de 1371). 11 eat connu sue la classe d16quivalence de la representation ~ndf;,~~(~:@~d,,)
X
ne depend pas du choix de
(i)
Tg
T est irreductible. g Soit a un automorphisme de
(ii)
Soit g ' c g*
(iii) que g(g)
. On la note
G. On a
Tout ceci est d3 2 Harish-Chandra
-
T = T , si et seulement si g' E Gg. g g' au moins quand G est de centre fini.
est particulierement profond. Lorsque G s s t ae centre fini,
on en trouvera une dLmonstration d a m [123:Dans
, et
une demonstration dans [ I 0 3
le cas gSnGral, on trouvera
une dans [35]
.
Les hypothPses sont celles de 111. 3. On fixe une polarisation totalement complexe, i.e.
-h,
et telle que
Nous noterons
bnb= h
%
-b
%
. On
note 2
5
le radical nilpotent de
en g ,
, contenant
b
.
l'espace de Hilbert dans lequel opere la representation le sous-espace des vecteurs
g -mc?dule, et, par restriction un 2-module.
5
b
est une sous-algsbre de Bore1 de g
(dCfinie au paragraphe 3), et. J$ est un
.
a~ = T g ag
un element bien polarisable, admissible, tel
soit fondamental. Alors
Le resultat ( i )
.
cm
. Alors
T
g
%OD
Pour j h -
N- , on considPre les espaces vectoriels ~.(n,q) 3 -
E
x--,
normalise g, et opere dans
Si
y
, on
-
note
H.(n,T) 3 -
Y
& opsre dans Hj
. Comme
l'algebre
(=,r).
le sour-espace propre g6nLralisL correspon-
dant. On note qb le nombre de valeurs propres < 0 de l'application hermitienne associee B la forme X + ig([~,X]) sur b
.
dim H. (n,r$) J
-
ig+pb
= 0
. On a
-
dim H (g,f)ig+pb = I . qb -
D6monstration :
On choisit une involution de Cartan 8 l'ensemble des points fixes de dant. On note
8
, K
de g
telle que
h.
On note
k
le sous-groupe analytique correspon-
xf le sous-module form6 des vecteurs cn
8h =
kfinis de
Km . C'est
.
un sous-&-module simple de ;6e On peut donc aussi considLrer les h-modules f H.(n,# ) . Leur structure est complGtement dCtermin6e par Vogan dans C361 J
.
-
En particulier, on a dim H~(g,$)ig+pb
# 0 si et seulement si j
dans ce cas, la dimension est I (~361,-th, 6.10).
=
lb , et
-
Le lemme 3 est donc corollai-
re du leme 4 ci-dessous. C.Q.F.D.
kme 4 : Soit T une reprhsentation de G dans un espace de Banach
-
centre de G opsre scalairement, et telle que Z(&)
gm.Alors
-
l'injection
H~(~.z~) H~(g.lP).
xf
-
2,telle que le
opere scalairement dans
induit un isomorphisme de frmodules
1. Rappelons I'homomorphisme dlHarish-Chandrade
glbments de e l que Z(&) de y
.
A
.
Z(&)
dans les I1 existe y E h*
Notons le Z -. Z invariants par W~ 5 opgre dans %" par le scalaire Z + kb), et llorbite W y
St%)
-
E
est bien dCtermin6e (cf 1 7 3 ) . D'aprSs le lemme de Casselman et Osborne
151, Hj(?,f)
est un module 9-f in;, et les poids de
h qui
interviennent
.
On a un resultat analogue sont tous de la forme w(y) + pb , avec w E W 5 pour H.(n,$), de sorte qu'il suffit de d6montrer que pour tout n E t* 3 5 01 llinjection 74 induit un isomorphisme H. (n;Kfln
xf
+
1
-
+
~~.(~.6~-
2. On se ramSne imgdiatement au cas oG g est semi-simple, de sorte que l'on a (parce que
h
r
=
k n 2 , et
1
et dont le terme E
P*9
Hochschild-Serre) pour
t
est une sous-algSbre de Cartan de
est fondamentale). On pose
spectrale, aboutissant 5
De &me
que
H (n
bin
* -'
est H (n
4-4
2
, dont
= k
%
n
n
k
. I1 existe une suite
les flgches sont des t-morphismes,
,x" s ~ ~ ( ~ 1 % ) ),~(suite spectrale de
.
H*(?,f) r)
. I1 suffit donc de dgmontrer que
est un isomorphisme. Comme
A*(%/%)
la flgche naturelle
admet une filtration, par des
1%-
modules de dimension I triviaux. et compatible avec l'action de
5, un raison-
nement standard montre qu'il suffit de prouver que pour tout B
5'
E
t*
la
fl2che naturelle
est un isomorphisme
3. Montrons que
xm est 6gal B
pour la restriction de
T 5
On choisit une base de
-k, XI,..-,Xm , et
l'espace des vecteurs de
3, cm
K.
orthonorm6es pour le produit scalaire
une base de g , XI....,Xm,Xm+,,..Xn,
-(X,eX)
, oii
(
.
)
est la forme de
2 m 2 A = Z X I , AC = Z X 1 I "
Killing de 8. On pose L'opbrateur D
et DC
Q = A - 2 A c . m
.
et donc opere scalairement dans X Notons les extensions self-adjointes de A et , AC operant dans f Cl
est dans
m
(on sait que sur pour tout k
.A
,
E
des vecteurs Cm
Z(&)
et
AC
et D~
sont essentiellement self-adjoints). Alors
ont mEme domaine de d6finition. C o m e l'espace
pour G (resp. pour K) est l'intersection des domaines de k (resp. D ), notre assertion est d6montree.
definition des D~
Le lemme 4 est donc consequence du lemme 5 ci-dessous. C.Q.F.D.
Leme 5 :
Soit T une repr6sentation de K l'espace des vecteurs
cm
teurs k-finis. Soit
r,
Z
du centre de K
E
dans un espace de Banach X. Soient
K~
de
le sous-espace de
%= . On suppose qu'il
tel que K/Z
'f
form6 des vec-
existe un sous-groupe discret
soit compact, et la restriction de T 5
Z
soit scalaire. Alors la fleche naturelle
Elj (s,$)R
+ Hj(nc,f)n
est un isomorphisme.
D6monstmtion : L'espace de K
xf est dense dans
intervenant dans
. Soit 6 xf, ct soit n6
une representation irreductible le sous-espace de
K opere de manisre isomorphe P un multiple de 6
%,
contenu dans
m
%
, et
z6
dans lequel est ferme dans
admet un suppl6mentaire ferm6 invariant. De plus,
le nombre de 6 pour lesquels on a : hj(s,$)r theoreme de Kostant [I91
. Alors
X
f 0 est fini d'aprBs le
.
I1 suffit donc de prouver l'aseertion suivante. On suppose que pour toute representation irrdductible S de K
Hj(n -c' %6 ) TI
=
0
. Alors
H.(n
J -c
,TJr=
de dimension finie, on a 0
.
Come
K/Z
~ * n4b
-c
. On
f
=
0
. On
o p s r e de m a n i s r e semi-simple dans l e complexe
c o n s i d e r e un blgment w
Notons, pour t o u t dw
t
e s t compact,
6
,
v6
L
Le problGme e s t d e c h o i s i r l e s
'+I (AJ v6
d o n t l e s composantes s o i e n t l e s
v6
k
dans
AJEc 8
t e l que
dv6 =
d a n s l e complexe
fl
=
v, =
6'd$ . L 1 0 p 6 r a t e u r
On y t r o u v e e n p a r t i c u l i e r que dans Notons l e c
6,n
. Choisissons
k , e t notons encore
.
w6
u r ~ ' + ~ n ~ @ y ~
(A*
@, f
-c
l'action de
dd* + d*d. Dans
U e s t i n v e r s i b l e s i e t seulement si
donc c h o i s i r
.
% . On a
A* n
TDD, commutant 1
dans
e t a y a n t l e s p r o p r i 6 t b s s u i v a n t e s . Posons I'opErateur
dw = 0
.
-
commutant 1 l ' a c t i o n du commutant de
sur
t e l que
de t e l l e s o r t e q u ' i l e x i s t e
+I
d*
n
w
@
-C
de d e g r 6
I 1 e x i s t e un o p 6 r a t e u r
zc O f )
l a composante d e
w6
peut c h o i s i r
(A'
6
H (n # )
j r ' 6 n
(AJ
& I
. On
= 0
h,
peut
i] e s t c a l c u l 6 e t d 6 f i n i d a n s [ 191.
zc @%)n
, 0
e s t un s c a l a i r e .
un p r o d u i t s c a l a i r e d b f i n i r i b g a t i f , i n v a r i a n t l e p o i d s dominant de 6
6
. On ' p e u t
choisir
d*
de t e l l e s o r t e que : 2csVq = (6+pc, 6+pc) Soit
(1 )I
( r + p c , C+P,).
une norme d g f i n i s s a n t l a t o p o l o g i e d e
Casimir de
k
bl6ment
de
u
opere dans
N
6
par l e s c a l a i r e
6
A*% Q
v.
(6+pc, S+pc)
~ * BV n , dont l e s composantes s o n t n o t e e s
Z(6+pc, 6+pc) pour t o u t
-
N
I1 u611
. Appliquant
~ 1 v61 11 < m . Posons N , . Donc v 6
v = Cv
. On a
.
C.Q.F.D.
@
, de u6
s o r t e qu'un
, vErifie
:
<
c e c i 1 u = d*w
6
L ' o p L r a t e u r de
,
on e n c o n c l u t q u e I r o n a :
C(6+pc, 6+pc)
N
11 v6 11
<
pour t o u t
Rerni~rqu~ : Lorsque
G
e s t compact, l e lemme 3
e s t un r e s u l t a t d e Kostant 1191,
q u i e n t r a ' i n e l e thbork-me de Borel-Weil-Bott.
Lorsque
h = t , c'est
un r 6 s u l -
t a t d e Schmid, q u i e n t r a i n e l a " c o n j e c t u r e de Langlands" s u r l e s r 6 a l i s a t i o n . s d e s s e r i e s d i s c r s t e s r291.
111. 5 . F o m s Zingaires standard : Nous ne supposons p l u s que g standard
, i.e.
pour t o u t
a €A.
nous supposons que, a v e c l e s n o t a t i o n s ( I ) , on a :
2
Ceci implique que Gprgsentation T g
"
dans
Go G(g)Go
On c o n s i d e r e l ' a l g e b r e
l ' e s p a c e de
sons que
-c
. On
pose
b=
S
h
-g
o:,
. C'est
+2
'dOet e l l e e s t s t a b l e par
une p o l a r i s a -
~(g).
xm
E
y
G ( ~ ) &, e t t o u t
E
Go
,
ern . Euppo-
laisse stable
d d d u i t que
%? ,
t e l l e que,
l'on a i t :
Go Go -I S(x,@)Tg ( y ) s ( x , @ ) - l = Tg (xyx )
(2) Alors
.
cde
(x,$)
0
e t i l r d s u l t e du lemme 2 que son s t a b i l i s a t e u r
et l e sous-espace d e s v e c t e u r s g s o i t une r e p r 6 s e n t a t i o n u n i t a i r e de G ( ~ ) Ed a n s
S
pour t o u t
,
1 = 1 ga
t i o n t o t a l e m e n t complexe, Notons
vcr #
e s t fondamental. On a d d f i n i (paragraphe 111. 3) l a re-
de
e s t dgal B
G
G est connexe. Dans c e p a r a g r a p h e , nous supposons
.
e t , a v e c l e s n o t a t i o n s du paragraphe 4 , on en ( d e dimension 1 ) H
G ( ~ ) &o p s r e dans l ' e s p a c e
qb
(n,f)
-
-
ig+pb
-
'
Lemne 6 : I1 e x i s t e une, e t une s e u l e , r e p r d s e n t a t i o n u n i t a i r e d e vdrif i a n t (2) au c a r a c t e r e
,
r e l l e que l ' a c t i o n de
Pb -
pb
-
de
G ( g ) - i n v a r i a n t l a g r a n g i e n de
Nous n o t e r o n s
s
l a r e p r 6 s e n t a t i o n de g
dans
H
(1,x ) ig+% m
qb -
(Rappelons que l e c a r a c t e r e de espace
G(g18
~ ( ~ 1d a 6 n s )e, s o i t dgale
-
~ ( ~ e) s tg d d f i n i pour t o u t sousc f . c h a p i t r e 11. formule ( 2 ) ) G ( ~ ) &d d f i n i e dans l e lemme 6 .
De'mvnstration LIU leme 6 : Soit S'
une repr6sentation projective de G(g)
unitaires, et vCrifiant
dans
(~*f)~~+,,~ . Alors on definit -
S (x,@) g
qb -
sg(x,$)
par des opdrateurs
(2). Une telle repr6sentation existe. Soit c(x)
le scalaire qui s'en dgduit representant l'action de x f ,
%, t
G(g)
dans
par la formule
4x1-' pb(x,@)s'(x). Comme c'est l'unique possibiliti., on voit que l'on a demontri. l'existence et =
.
l'uniciti. de la reprgsentation S g Supposons que x soit dans Go(g).
Alors
S (x,$) doit stre proportionnel g Go Go 3 T (x). C o m e on a G (g) = H, l'action de T (x) dans g 0 g est par dgfinition la multiplication par le caractere de (9pe)ig+pb qb diffcrentielle ig + pb On a donc : -
.
pour tout
(x,$)
6
G(g)5 g
.
En particulier, Sg(x,@) Notons
Z
est unitaire pour
(x.4)
E
~(~)g.
le centralisateur de g
dans G. Alors le groupe Z G(g)O est & d'indice fini dans G(g). Si x E Z , on a Sg(x,$) = pb(x,$) = @ (=?I). & Le groupe IIS~(G(~)~)IIest un sous-groupe fini de 0 , et donc trivial.
B
Donc S
est unitaire. C.Q.F.D.
g
Soit
T
6
X(g).
Soient x
E
G(g),
y
E
GO, (x,g)
E
G(~)&
representant x.
On pose
Go
(4)
(T @ SgTg )(xY) = T(x,$)
@
Go
S (xP$)Tg (Y) g
-
C'est un opgrateur unitaire dans l'espace produit tensoriel de l'espace de T et de celui de XY
et q u e
TCO . TCO
g iO S
On verifie (grzce 3 (2) et ( 3 ) ) qu'il ne depend que de
g
est une repr6sentation de G(g)Co.
Sa restriction P phe B celui de
Go
T
est un multiple de
TCO g
, et
son co-tant
est isomor-
.
On pose G Go T = Ind G(g)G0 (T asg Tg g *T
.
(i) Le comnwtant de T est isomorphe B celui de T g9T (ii) Soit a un automorphisme de G. Alors a~ = T g.= %,aT (iii) Soit g' E g* un 616ment admissible, bien polarisable, et
-
standard. Si g' pour tout T'
E
4 Gg,
les repr6sentations T et T grT g' ,T'
X(g').
(iv) Soient T, T' et T'
sont disjointes
E
X(g).
Les espaces d'entrelacement entre
T
,et T
et Tg,Tl sont isomorphes. ~ Y T
Dt5monstmtion : Cela rCsulte du lemme 2 et de la th6orie de Mackey.
Rermrque I : Supposons G compact. I1 est facile de voir que les reprgsentations T g.T avec T E x ~ ~ ~ foment ( ~ ) l'ensemble des representations unitaires irreductibles de G. 11 est facile de voir que cette description du dual unitaire de
G est Cquivalente B celle de Kostant [191. Enfin, il est facile aussi de voir que T est isomorphe B la representsg*T tion obtenue par translations B gauche dans l'espace des fonctions a , cm sur G, B valeurs dans l'espace de t
,
qui v6rifient :
Xa = 0
(7)
X
pour
-
,
E
11
ol
e s t c o m e au d6but du paragraphe.
Nous ne nous s e r v i r o n s pas de ce r e s u l t a t q u i e s t 21 comparer aux formules (9) e t (10). c h a p i t r e 11.
Remarque 2 : I1 a r r i v e que l ' o n a i t
G(g) = Z
g -
G(g)
(nous e n v e r r o n s un exemple important
0
au paragraphe 7 ) . Dans ce c a s l a c o n s t r u c t i o n s e s i m p l i f i e enorm6ment. On a : G T = Ind (T gsT zgGo
r
o l l ' o n a not6
@
TCO g
B
G T~O)
l a representation de
Go
ZgGO
d 6 f i n i e p a r l a formule :
Go
( ~ 6 ~3( x~Y )= ~ ~ ( x , l ) T (Y) g
(9) pour
x
Zg
E
III. 6. Lk
,y
E
Go
.
cas gkne'rai! :
Nous ne f a i s o n s p l u s dlhypothPses r e s t r i c t i v e s , n i s u r c e l l e s expliqu6es e n 111. 1 . Rappelons que v c M
v dans
le c e n t r a l i s a t e u r de
M(m) = G(g), e t on remarque que On pose
g) 8
C n arA = va>O correspondant. A l o r s MU = (
Bien que l ' o n a i t
, et
z*
de
x. S o i t
L
on n o t e
m M(m) = G(g), M(m)-
U
et
nombre Soit
I)
g
sauf
m = g]m
. On a
l e sous-groupe a n a l y t i q u e de
(x,$) r G(g)g
. I1 r e s u l t e
, (x,$) 2
G.
X(g), e t E
m
M(m)-
. Alors
X(m)
. En
effet,
des reprgsentants L +
2 est un sous-
de l a formule (12) du c h a p i t r e I que l e
( L + ~ ) $ ( L ) - ' ne depend pas du c h o i x de
T r X(g). On d C f i n i t un 6ldment
G
G ( ~ ) &ne s o n t p a s e n g6nEral Ggaux.
un sous-espace l a g r a n g i e n d e
espace l a g r a n g i e n d e
-
m est s t a n d a r d .
e s t un sous-groupe f e r n 6 d e
x c G(g), e t s o i e n t
g
e s t d e f i n i e n ( 1 ) . On n o t e
son algPbre de Lie,
G,
I1 y a cependant une b i j e c t i o n n a t u r e l l e e n t r e soit
G, n i s u r
E
L. On l e n o t e
X(m) e n posant
$$-I.
pour tout
(x,@) c
representant x
M(m)-
m
. Cela ne ddpend
pas du choix de
(x,I#I)
E
G(~)S
.
La reprdsentation
a dt6 dafinie au paragraphe 5. On pose :
m, c
Leme 8 :
Les reprdsentations T de G ont les propriGt6s g7 7 (v) de l'introduction.
(i), (ii),
(iii),
(iv),
Le leme 8 sera demontre dans les paragraphes suivants.
I I I . 7 . D6monstmtion du l e m e 8
lorsque
G
est connexe :
Dans ce paragraphe, nous supposons que G est connexe. Les notations sont celles du paragraphe 6. Rappelons la mdthode dlHarish-Chandrapour construire des representations de G. On note M'
le centralisateur de
.
On pose m' = g ] ~ ' Soit Ei connexe, on a
El = G(g)
=
a
dans G, 2' son algPbre de Lie,
le centralisateur de
&
dans G. Comme G est
De manisre analogue au paragraphe 3, on
M1(m').
choisit une sous-algsbre 2' de g , on note U' correspondant, de sorte que M'U'
le sous-groupe analytique
est un sous-groupe parabolique "cuspidal"
de G. De manisre analogue B ( l o ) , on definit un element
pour tout
(x,$')
E
Fl'(m')E1
(x,I#I) de x dans G(~)&
. Cela ne dEpend . On
F'
de X ( m l )
en posant
pas du choix du representant
dEfinit la reprdsentation Tm' ~' ,C'
come au paragraphe 5 (la remarque 2 de ce paragraphe s'applique).
de M'
1i 4
On pose
T'
=
G IndM.Ul: T (
,C,
gtT
Ldtnti.
@ Id",).
.! : u'.
La repr6sentation T' ne depend pas du choix de Les propridtes g¶T (i), (ii), (iii), (iv), de l'introduction sont verifiees pour les representations T' g9.T
.
Dcltnonstrzl t i m :
Tout ceci est essentiellement dO B Harish-Chandra. Donnons quelques ddtails. Tout d'abord, X(g)
est de type I, et la construction de T'
g
sommes boreliennes de reprgsentations. oii T
E
commute aux 9'C
I1 suffit donc de considCrer le cas
irr X (g), ce que nous faisons ci-dessous. est une repr6sentation irrdductible de M',
Dans ce cas T ~ ' m' ,O'
de carre
integrable module le centre. Montrons que que
T'
m',
u'
ne d6pend pas du choix de
, (x,$')
(x,I$) c G(~)& de
ne dgpend pas du choix de u'. Cela revient B demontrer
Tm*,c+
le nombre
t
M1(m')-
m'
, et donc que si x
E
G(g),
, et si L est un sous-espace lagrangien
$J(L+u')$'(L)-~
ne depend pas du choix de 2'. I1 faut donc
voir que I$(L+u_') ne depend pas du choix de (12) du chapitre I, et de ce que 2'
u'.
Cela dsulte de la formule
est invariant par x.
La premiPre assertion du lemme est un resultat dlHarish-Chandra (cf 1371). La propriEti (i) signifie que T' est irreductible. Lorsque LG,Gl est de gJ centre fini, cela r6sulte de la thdorie dlHarish-Chandra 1121 . Le point cld est que y tel que
a
est non nu1 pour toute racine a reelle, i.e. pour tout
alt
= 0 (ceci vient de ce que
ga
est non nu1 pour tout
Cependant, le r6sultat n'est explicit6 que lorsque trouvera Ir fait que
T'
3
a
a E
E
A
A) .
est fondamental. On
est irreductible, complGtement explicite, et sans
grT
I'hvpothi-se dl. centre fini, avec une d6monstration diffdrente, dans 132i
.
(ii) est Bvident. (iii) et (iv) signifient que des repr6sentations T et T grT g' ,T' avec T et T irrgductibles, sont dquivalentes si et seulement si les donnees (g,~) et
(g',~')
sont conjuguses dans G, ce qui est bien connu
(cf C371). (iv)
est facile 5 vsrifier. C.Q.F.D.
Pour demoncrer le l e m e 8 (lorsque G est connexe) il suffit d'btablir le resultat suivant.
Les reprssentations T' et T sont Gquivalentes. g*T gJ
D&mmstration : Compte-tenu du lemme 9, on peut choisir
-u c u'.
On a alors U' = (M n U')U
u' de telle sorte que
. D'aprPs
L'on ait :
le th6orPme d'inductionpar 6ta-
ges, il suffit de demontrer que l'on a :
06 l'on a pose
U"
=
.
M n U'
Rappelons que , c o m e G est connexe et ~ ( g )= ~ ( 1 )= M'(ll) = H = Z
B
G(g)O
h
fondamental, on a
. Les representations P
et
"'SO
sont induites 5 partir des reprGsentations correspondantes des groupes
. Appliquant encore le th6or6me d'induction
et Z M;)
g
=EM0
par Stage, il suffit de
demontrer que l'on a
D1apr6s la formule ( 9 1 , les deux representations ci-dessus ont &me tion 5
Z
25
,5
savoir ~(x,l)6t Id pour
trer que l'on a :
x
E
Z
E
restric-
. I1 suffit donc de dgmon-
Mo
Mo
-
Tm = IndMlv"(Tm' 8 Id",,) 0
Hais ceci est vrai par definition msme de
2
(paragraphe 3 ) .
C.Q.F.D.
Remarquc :
h
fondamental, nous avions defini T. deux fois g (paragraphe 3 et paragraphe 5). Le lemme 9 montre que les deux definitions
Lorsque G est connexe, et
coincident.
111. 3. Fin de l a d5monstration du Zemne 8 :
On emploie les notations du paragraphe 7. C o m e les constructions faites commutent aux sommes borGliennes, il s'uffit de dCmontrer le lemme 8 quand T est factorielle. Nous supposons ci-dessous que T est factorielle. Come
G (g) est d'indice fini dans G(g), if existe une sous-represenHO tation irreductible dans la restriction de T B On en choisit une, Z
-
et on l'appelle
T ~ . On note
G ( &
.
le stabilisateur de T, dans G(glg -r 1
et G(g)=
son image dans
G(g)
1
qui prolonge T]
de G(~)$ 1
.
. On choisit nne repr6sentation projective que nous noterons encore T,, et qui v6rifie
les relations : T,(29) = T]
(13)
n
,~(~1:
I
.9
Soient x, x'
(S)T~(~) et ~~(92) = -rI(FIT,(%I
pour tout
~ ~. ( ~ 9 E
et soient 2, 2' des representants dans G(~)$
G(g)T
I
I
.
On pose
On a ainsi defini un 2-cocycle sur G(g)T
/GO(g). 1
I1 existe une unique representation projective de G(g)T
/GO(g) 1
, que
nous
noterons
T2, telle que iron ait
(15) pour
-1
T (xx') = c(x,x')
2
x, x'
E
(T~ B il) = T
Ind
. Elle vgrifie
1 T~(x)T~(x')
.
G(gIT 1
.
On considsre la reprdsentation T de Go C o m e le lemme 8 est valable g9=] pour Go, d'aprss le paragraphe 7, cette representation est irrgductible, et son stabilisateur dans G est le groupe G(g)T
Go
I
.
Nous allons construire une reprdsentation projective de G(g)T
Go
, que
nous
1 noterons T I , et vdrifiant les propri6tbs suivantes
(17)
T 1 (xx')
= c(x,xl)T (x)T.(xl)
1
pour
1
x, x'
E
G(gIT 1
.
De plus nous montrerons que l'on a (18)
T~ g,T
=
G (T2 @TI) Ind G(dT Go
-
I Le leme 8 rdsulte imm6diatement du cas particulier des groupes connexes, de
la formule (Is), et de la thdorie du Mackey.
Construction de
T1 :
duit le groupe M I
. On intro!ZYT o1 la reprgsentation de ~ ~ ( m ) ~
T I est tout B fait similaire 2 celle de
La construction de
=
M n Go
. On note
dbfinie de manisre analogue 2 m
T~
o (formule 10). Le stabilisateur ~(m)m
'"1
dans M(m). On utilise '?I encore (10) pour dbfinir une reprgsentation projective, notde encore ul, de
de
dans M(m)-
est l'image rdciproque de G(g)
m
, et verifiant les proprietes analogues 1 (13) et (14), avec le m8me 1 cocycle c.
M(m)<
Nous definissons une representation projective, que nous noterons Dl 6S T O g g
, du groupe M(m) O1MO , en posant (01@
Mo
SmTm ) (XY) = al(B) 8 Sm(B)Tm
Mo (y)
Nous dEfinissons une representation projective, que nous noterons R, dans
.
l'espace de T , du groupe M(m), MI Si y E M1 , on pose M, m*"l I , on definit R(x) de la manisre suivante. R(y) = T ~ ' (y). Si x E M(m), ' "1 1 Rappelons que T est induite 1 partir de la representation a, @ SmTm0 m. Dl du groupe Ml(m)MO Un Lllment de l'espace de TM 1 est donc une fonction m,'J a sur MI , 1 valeurs dans l'espace de 'J, ':B T~S , verifiant eertaines
.
relations. Pour une telle fonction, on pose
pour tout y de R(x)
E
Ml(m)MO- On vsrifie que lorsque x
coyncident. Enfin, si x
R(xy) = R(x)R(y).
.5
M(m)
E
Ml(m),
et si y
E
MI
les deux definitions
, on pose
On vErifie sans difficult6 que l'on a R(yx) = R(y)R(x),
, y E MI, et R(xxt) = c(x,x')R(x)R(xl) pour x,xl E M(m)u "1 1 Rappelons que la representation T est induite par la representation M. .%*TI T ~ @ :Idu ~ de ~ MIU Elle eat rialisee dans un espace de fonctions sur Go M 1 valeurs dans l'espace de T Soit a une telle fonction. On pose, si m, u, pour
x
E
M(m)
.
.
x
6
, T 1 x( = R(X)(X~~X) = M *1 Go T,(xy) = TI(x)Tg (y) pour x E G(g) O1
G(g)
pose
pour tout y r Go
, et l'on
Les formules (15), (16), (17) sont faciles 1 verifier. I1 reste B demontrer la formule ( 18)
.
.
Pour cela, nous remarquons que, d'aprss le thdorsme d'induction par btages,
T~
g ST
est induite 1 partir de la reprdsentation (p@Sm
Mo
Tm ) @ Idu du groupe
M(m)M
U. D'autre part, par construction de T I , il est facile de voir que la 0 G reprlsentation Ind ( ~TI)~est 8induite 1 partir de la representsG(g) Go tion T~ 0 ( U, 8 Em TmM0I
IdU du groupe
@
ddmontrer que l'on a : M(m)MO (20) IndM(m)4a(~2
Mo
e
(c, Q sm T~ 1)
M(m)
M U. I1 nous suffit donc de
4O
Mo
UB s~T,,,
=
.
Pour cela nous ddcrivons un op6rateur d'entrelacement entre ces deux espaces.
o est isomorphe 1 la reprdsentation induite par
I1 est facile de voir que
.
la reprdsentation -r2 EI ol de
X @%
@
2
c @ Sm
prdsentation
3 , oC l'on
.
B(e) pour f
E
~(m)g ,
1 valeurs dans
est donc une fonction sur ~(m)"
a. not8 %!
l'espace de: T
l'espace de
La fonction
=
(~~(y1-lC$
8E
m
@
M(m)-
Un lllment 6 de l'espace de la re-
2 @ 3 , en
H valeurs dens
l'espace de
T2
0,
,
f3 vdrifie
4(8)-1
.A
,
0 Id)B(?)
6 on associe une fonction a sur M(m)MO
posant, pour
x
E
M(m),
y
E
Mo
, ji
E
,
~(m)m
reprdsentant x :
On vlrifie que l'application
+
a
est un opdrateur d'entrelacement entre
les deux representations figurant dans (20). C.Q.F.D.
Remarque : Les formules (15), (lb), (17) calculent l'obstruction de Mackey 1 dtendre la
-
reprlsentation TLO 1 son stabilisateur dans G. Elle est isomorphe 1 1' g9T ] l'obstruction qu'il y a 1 ltendre r 1 1 son stabilisateur dans G(~)E
.
Peut-on dlcrire de manisre aussi simple l'obstruction de Mackey pour les au-
tres reprdsentations irrdductibles de
GO
, plus
particulisrement les reprd-
sentations irrdductibles tempdrdes dont le caractere infinitdsimal n'est pas rdgulier ?
CHAPITRE I V
- TECHNIQUES DE
RECURRENCE
Dans ce chapitre, nous Qtudions ce que devient une forme lin6aire admissible, ou bien polarisable, quand on la restreint h un idgal, ou h llalgGbre de Lie du "petit groupe" dl? la th6orie de Mackey. Nous n'utiliserons les rdsoltats ci-dessous que dans le cas d'un idQal nilpotent. Cependant, en vue d'applications ultGrieures, j'ai trait6 une situation plus gQn6rale.
4. 1. Formes bien polarisables : Dans ce paragraphe, g
est une algebre de Lie complexe de dimension finie, et
g est un dlQment de g*.
Larune 1 : Soit q un idQal de g
contenu dans ker g. Soit
g'
l'dl6ment de g/q
obtenu par passage au quotient. On suppose 1 r6soluble. Alors
g est bien
polarisable si et seulement s'il en est de msme de g'.
De'monstration :c'est Qvident. Ci-dessous, on considsre un idQal h = glh
. Nous notons
1
de g
. On pose
1
=
.
gll , h = g(1)
L et H les groupes analytiques correspondants.
Leme 2 : La forme g est bien polarisable si et seulement s'il en est de m8me de et de h. Dans ce cas il existe une bonne polarisation l'on ait : b =
bn2
+
b
en g
b n h.
Dans la dGmonstration, nous aurons 1 utiliser le resultat suivant.
1
telle que l'on
Lemo 3 :
(&+L)'
= g +
On a L(l)og
ou
(&+L)'
est l'orthogonal de
h+L
dans
9*DEmonstmtion : [24]
Voir
p. 500.
DEmonstration du Zeme 2 :
t
I. Soit
une sous-algebre de g
existe une bonne polarisation en
t.
algebre de
g
. Posons
contenue dans
C'est une bonne polarisation en
la msme propri6t6 pour
b
2. Soit
t
t = glt
. Soit
t
. On suppose qu'il b
une sous-
si et seulement si elle a
g.
un sous-espace lagrangien de g
. Les conditions suivantes
sont Equivalentes :
.
bc
(i)
b
(ii) (iii)
n
est lagrangien dans
1.
bn&
est lagrangien dans
&.
Quand l'une de ces conditons est vbrifiee, on a
b=b
n 1_ +
b n 1. (cf
C71
p. 57).
2
Supposons que
b
n
soit une bonne polarisation en
est une bonne polarisation en
1
, et
b n
&
une bonne polarisation en
telles que c normalise a. Alors 5 +
3 . Soit E
r* , et
telle que
1, 5 une bonne polari-
c
est une bonne polari-
g. (La condition de Pukanszky vient du lemme 3 ) .
satinn en
r
. Alors
contenue dans
REciproquement, soient 5 une bonne polarisation en sation en h
k
g
r
posons
b
n
une algebre de Lie r6soluble, soit s
=
rls
. Alors
s
un ideal de 2, soit
il existe une bonne polarisation
2 soit une bonne polarisation en s(cf
C333).
b
en r
4. On suppose
g
2 ,et
r d s o l u b l e de
g . En a p p l i q u a n t
r = glr
on pose
1 e t 3 P 17algPbre
b
polarisation
r
b i e n p o l a r i s a b l e . On n o t e
bn5
t e l l e que
l e p l u s grand i d e a l
. Soit 5
une bonne p o l a r i s a t i o n en
b
v o i t que
+
2 , on
g
a une bonne
s o i t une bonne p o l a r i s a t i o n en r .
On considPre l l a l g P b r e g ( r ) . On pose
= ~ ( r n) Ker r . C ' e s t un i d g a l de
K J
g ( r ) . On pose gl = g ( r ) / q , i1 = I ( r ) 1% -
, et
n E a i r e s obtenues p a r passage au q u o t i e n t s u r I1 r d s u l t e du lemme 1 e t d e 2 que
g;
~7
et
l e s formes li-
gl,ll
on n o t e
.
e s t bien polarisable.
g1
b
Nous a l l o n s ddmontrer q u ' i l e x i s t e une bonne p o l a r i s a t i o n
t e l l e que
en g
s o i t une bonne p o l a r i s a t i o n en 1. On l e demontre p a r r e c u r r e n c e s u r l a
b n -
. C'est
g
dimension de
dim
clair s i
g
=
0.
On suppose l e r d s u l t a t 6 t a b l i
pour l e s a l g g b r e s de L i e de dimension s t r i c t e m e n t i n f e r i e u r e P dim
. Alors
i) dimsl < dimg
bl
n
rdciproque de
ll
b -I
il e x i s t e une bonne p o l a r i s a t i o n
dans
g ( r ) . D'aprPs 3 , c o m e
1
i i ) dim
. Notre
gl
=
dimg
. Cela
, g'
= gig'.
g
. Soit
Alors
C ' e s t une sous-algPbre r e s o l u b l e de l'orthogonal par rapport P Alors
3'
i l en r d s u l t e que
g
et
implique que
(b' n
I(r)) +
5 une bonne pola-
e s t de dimension au p l u s un,
r
Bs
=
[i,I] . A l o r s 5
g' l e c e n t r a l i s a t e u r de 5 dans g,
s
g e s t soinme d i r e c t e de -
b une bonne p o l a r i s a t i o n en g. S o i t -
sla.
1
l'image
e s t resoluble il e x i s t e
g. D ' a u t r e p a r t , posons 5
e s t un i d g a l semi-simple de s = g ls
b'
g
a s s e r t i o n e s t d6montrde dans c e c a s .
contenu dans l e c e n t r e de
soient
en
5 e n r normalisde p a r b'. D'aprSs 2 ( a p p l i q u e B r ) ,
+ 5 e s t une bonne p o l a r i s a t i o n en
r i s a t i o n en
b
--I
b'
l l . Soit
s o i t une bonne p o l a r i s a t i o n en
une bonne p o l a r i s a t i o n b' -
. On
:.
considPre deux c a s
t e l l e que
g
5
l a projection de
5, e t l'on
. Choisissons
a
a
1
c
b
e s t a u s s i une p o l a r i s a t i o n pour
s . Come
5
sur
5'
a , 06
une p o l a r i s a t i o n
g'
et
a'
c
. Soit
. ddsigne
5
pour
5' e s t r d s o l u b l e ,
5' e s t une sous-alg6bre de Bore1 de 5 ([73 p. 6 0 ) . On a
donc
5
=
a' , c e
b
Comme i l e x i s t e une bonne p o l a r i s a t i o n bonne p o l a r i s a t i o n en
h
5. On suppose g d e r i v a t i o n s de
5
+
n
b
n
f
g
, r6soluble,
g1
2,
est
5
4
e t stabilisant
g,
s o i t une
n
e s t une bonne p o l a r i s a -
4
une algPbre de L i e de
g. Nous a l l o n s montrer que
g
. &
,et
e s t b i e n p o l a r i s a b l e e t que
r e s o l u b l e de d e r i v a t i o n s de
b
2.
l e p l u s grand i d e a l r 6 s o l u b l e de
en 4. Notons que
. D'aprPs
t e l l e que
bn h
bien polarisable. Soit
a une bonne p o l a r i s a t i o n s t a b l e p a r On n o t e
en
1, i l r e s u l t e de 2 que
b=b
e t que
bns
=
s , e t n o t r e a s s e r t i o n e s t d h o n t r 6 e dans c e cas.
une bonne p o l a r i s a t i o n en
t i o n en
2
qui e n t r a e n e que l ' o n a :
, stabilisant
81
On raisonne p a r r6cul;rence s u r l a dimension de g
gl, g,
on d e f i n i t
cornme
opBre cornme algBbre '
. Come
en 4, on c o n s i d s r e
deux cas. i ) dans
g1 <
dims
. Alors
i i ) dimgl = dimg a l g i b r e de Cartan de
. La d6monstration e s t analogue
g e s t rgductive , e t g(g)
g(g) -
1 et
h
en
d
2 une sous-
g contenant
sont b i e n p o l a r i s a b l e s . S o i t
p o l a r i s a t i o n en h . D'aprPs 5, appliqu6 2
-a
et
.
6 . On suppose que
sation
4
~ ( g ) e, t n'importe q u e l l e sous-algPbre de Bore1 de
convient
e s t une sous-
8. Le c e n t r e de g e s t de dimension au p l u s 1
opPre t r i v i a l e m e n t dans c e c e n t r e . On peut donc i d e n t i f i e r a l g s b r e de
a c e l l e f a i t e en 4.
1
, stable
par
5
5. D'aprPs 2 ,
et
1 , il
c+5
c une bonne
e x i s t e une bonne p o l a r i -
e s t une bonne p o l a r i s a t i o n
en g . C.Q.F.D.
Corollaire du Zemme 2 : i ) On suppose lement s i
h
1
r 6 s o l u b l e . A3ors
e s t bien polarisable.
g
e s t b i e n p o l a r i s a b l e s i e t seu-
ii) On suppose g/L r6soluble. Alors seulement si
g est bien polarisable si et
1 est bien polarisable.
Les cas les plus importants sont les suivants : 1
=
[g,g]
,
et
=
5
(le plus grand idgal nilpotent).
I V . 2. Fonnes admissibtes :
Dans ce paragraphe, G est un groupe de Lie dlalgSbre de Lie 8 , g r &*
r un
,
un caractsre unitaire de T.
sous-groupe du centre de G,
Leme 4 : Soit 9 un ideal G-invariant de g
contenu dans ker g, tel que le sous-
groupe analytique Q correspondant soit ferns. On pose G' = G/Q
r'
=
r/r n Q
, et
11C16ment de &'*
on note g'
tient. On suppose que g est On note 11'
le caractPre de
obtenu par passage au quo-
ryadmissible. Alors
r
, 9' = g/q ,
q est trivial sur
I' n Q.
bbtenu par passage au quotient. Alors g'
est
q'-admissible. Plus pr6cisement, l'application ~(g)&
, et
x
-+
~'(g')~' est isomorphe P
(x,l)
identifie Q
G(~)~/Q
a un sous-groupe de
. La composition avec la projec-
tion ~ ( g -+) G1 ~ ( g ' ~ ~ 'donne un isomorphisme de X(gl ,n')
sur X(g,n).
De'monstration : C'est 6vident.
1 est un id6al G-invariant de g . On pose 1 = glh . Le groupe G(1) opSre dans 1 en conservant
Dans la suite, h
=
~(l), h =
H son rev8tement B deux feuillets H que de
r
n'
dans H, et
E
T. On note L
G(1)-
le caractPre de
nl(v, '1) pour y
=
=
1
. On note
T'
g(l
,
B1. On note
l'image rbcipro-
I" d6fini par la formule
fn:~)
le sous-groupe analytique d1algZbre de Lie
I.
De manisre analogue au l e m e 3, on a :
Leme 5
:
on a : ~ ( l ) ~ =g g
+
( h + ~ ) l , et
.
~(l)(h) = ~(g)~(i)~
Wmonstration : Cf 1241 p. 500. Rappelons que x x
G(g). E
et
De msme
G(1),
et
(x,$)
E
Soient L' de
h
G(~)E est H
H(h)- h
ddcrit cornme un ensemble de couples
(x,@)
avec
est ddcrit c o m e un ensemble de couples
(x,$)
avec
c o m e un ensemble de triplets (x,$,8),
avec x
E
G(l)(h),
H(h). un sous-espace lagrangien de
. Alors
L' + L"
et soient (x,@)
et
1
et L"
un sous-espace lagrangien
est un sous-espace lagrangien de g (x,$,8)
. Soit
des reprdsentants dans GCg)g
et
x
E
G(g),
H(h)- h
respectivement. On peut considsrer le nombre
11 rdsulte des formules (12) qu'il ne ddpend pas du choix de L' et L". On le note
8-I
.
Leme 6 : La forme g est
q-admissible si et seulement si h
ce cas, dtant donnd
T
E
XG,r(g,~]),
est n'-admissible. Dans
il existe un unique dldment
c
s , r ,(h,n')
tel que l'on ait :
pour tout
(x,$)
L'application
T
E -+
G(g)- g et tout reprdsentant
c est une bijection de XGlr(g,r;)
servant le type. De plus,
(x,$,B)
1 est admissible.
de x
dans
sur s , r , (h,';')
H(h)- h
.
pr6-
DImonstration : 1 . Supposons
H(h)-
h
a + (a,l)
et l'application
. Notons
5 , on a
B
H(h)-
h
identifie A
l'image rEciproque de
, et
= BA
est connexe, et A/A n B triction de -r
-C
XG,r(g,n).
E
. Crest un
est admissible. Posons A = (L(l)-)O1
Montrons que h de H(h),
g n-admissible, et soit
1 A n B
A/A n B
G(g)
sous-groupe
5 un sous-groupe de
dans
H(h)-
est homdomorphe 1
simplement connexe. Comme g
. DraprPs le lemme (h+i) . Donc A n B
h
est admissible la res-
fournit un caractPre de diffdrentielle igll(g).
Ce caractere se prolonge uniquement en un caractPre
X de A de difflrentiel-
le iglL(1). Ddfinissant a
6
c sur A
par la formule
c(ab) = a(a)~(b)
pour
A, b E B. Ceci ne dlpend pas des choix faits et fournit la representation
cherchk dans
s,rt (h.Tl') -
I1 existe un unique prolongement de et donc
x
2
(L(l)O)- 1 tel que
~(1,-I) = - 1 ,
1 est admissible.
2. Supposons
(I)
(I), on pose
h nr-admissible. Soit
dgfinit un GlGment de X (g,n). G,r C.Q.F.D.
0
r XH,r,(h,~').
Donc g est
La formule
n-admissible.
CEAPITRE V
- EXTENSIONS DES REPRESENTATIONS DES GROUPES DE LIE NILPOTENTS
Dans ce chapitre, U u , et -
est un groupe de Lie nilpotent connexe d'algibre de Lie
u un element admissible de u*. Nous rappelons la construction de
Kirillov de la reprEsentation unitaire irreductible associ6e 1 u, et le calcul de l'obstruction de Mackey B Etendre une telle representation. Ce chapitre est essentiellement destine B fixer les notations.
-
V. I .
La thdorie de KiriZZov :
On sait que U(u)
esf connexe. D'autre part, l'application sur son image dans U(U$
isomorphisme de U(u)
est admissible, c'est dire que U(u)
iulu(u)
, de
x + (x,l)
est un
sorte que dire que u est
est differentielle d'un caractire de
-
(Remarquons que le centre Z de U
est connexe, et qu'une forme lineaire u'
sur 2 est admissible si et seulement si, notant 2 l'alghbre de Lie de Z, iu'lz
est differentielle d'un caractere de
Z).
I1 existe des polarisations reelles b en u, i.e. des polarisations telles que
b = (b n u)C. Soit b -
b
analytique d'alghbre
xb -
une telle polarisation, soit B le sous-groupe
nu.
I1 est fermd, et il existe un caracthre unitaire U de diffdrentielle iulb n On pose T = Ind (X ) .
u.
Lemme I : (Kirillov
[ 173)
La reprdsentation T
u ,b
u ,b
B b
.
est irrEductible, et sa classe ne depend pas de b.
On la note TU. Si u et u'
sont deux elements admi'ssibles de
et seulement si u'
c
UU
.
, on
a T = T u u'
si
V. 2. Ope'rateurs dfentreZacement :
%-
l'espace de la representation induite T u ,b polarisation reelle. Comme les reprdsentations T et u ,b lentes, il 'existe un operateur d'entrelacement $ + Notons
-
. Soit b'
une autre
TU,bl sont equiva-
. I1 existe un choir
canonique d'un tel op6rateur d'entrelacement. Nous le noterons
Fb',b . I1 est
caract6ris6 par la propridti5 suivante : il existe une mesure de positive invariante sur
B'/B n B'
(voir Lion [ 2 1 1)
telle que, pour tout element y
SU
:
un groupe d'a~tomor~hismesdeU
le rev8tement
-
stabilisant u. On peut donc d6finir
de A .
On choisit une polarisation rgelle rateur %(x)
de
xb -
dans
b
en u. Soit x
Yxb eli posant, pour -
On pose
C'est un operateur unitaire dans
pour tout y E U. Soit
(x,@) c
U, et tout vecteur
.
V. 3. La repre'sentation Soit A
E
AE
. On pose
xb, et l'on -
a
A . On d6finit un op6-
E
a
E
xb-
et y
E
U
:
AE dans
(i) Sb est une reprEsentation unitaire de
-
b,
(ii) Sb ne depend pas de
-
dans le sens suivant : si F
entre-
-
lace T et T alors F entrelace Sb et Sbl. u $11 u Nous noterons SU la representation de AE dans l'espace de TU
-
ainsi d6fi-
.
nie
pour tout y
.
h
U, (x,r$) c A-
E
(La representation
. On trou-
SU a Qtd ddcrite de manisre diffdrente dans [ 9 ]
Vera dans El31 une bonne explication 1 l'existence d'une representation de AE verifiant (iii). La prdsentation adoptde ici est due B Lion C213).
V. 4. Extension des reprdsentations des groupes de L i e niZpotents :
Dans ce paragraphe, G est un groupe de Lie d'algsbre de Lie ideal nilpotent G-invariant de g dlalgGbre de Lie Nous noterons
u
,u
= G(U)~ , 3 =
.
analytique d'algsbre de Lie q du groupe (connexe)
U(u)
x
L'application
de U(u)
Soit
=
r
!' un sous-groupe du centre de
r
dans H, et on pose
de G I . On note
nl
tient de caractsre n'
,Q
, et
femd.
le sous-group est un isomorphisme Q est la composante
.
H/Q.
qui a G m e restriction 5 de
eSt un
de diffdrentielle iulu(u).
Le groupe Q est donc fermd et invariant dans H
gl = h/q , G ,
On suppose U
x + (x,l)
U
On pose
*.
~ ( u )n ker u
sur son image dans G(u)E
neutre du noyau du caractsre
u
, U le sous-groupe analytique de G
un Bldment admissible de E*
h = g(u) , H
g,
G. Soit
=
r'
U(u)/Q.
le caractsre de de
xu. On note r'
n U(u) que
TI
TIU(u)/Q
r un caractsre unitaire de
r,
l'image rCciproque
C'est un sous-groupe de centre qui provient par nassage au quo-
qui prolonge
r
X, , et
tel que
~'(y,fl) =+~(y) pour y
6
T.
Soit T I une reprssentation de ql
ple de
. On dgfinit une
G I dont la restriction B
representation
de la maniere suivante. Soient x
G(u)U Soit
(x,$)
E
, notee G(u)
et
T I 8 SU TU du groupe y
H un representant de x. On note encore
E
T I est un multi-
E
U. (x,JI) son image dans
GI. On pose :
On vErifie que cette definition ne dCpend pas du choix de
x,y,(x,$)
, et
que
cela donne une representation.
Lenm 3 : i) Le srabilisateur de
TU dans S
est le groupe G(u)U.
I1
est fermi5. ii) Soit T, une repr6sentation de GI dont la restriction 5
rl
est un multiple du caractere ~ 1 , ~On . pose
La restriction de T 5
r
est un multiple de
est portCe par l'orbite (sous G) L'application G I et G
de
TU
q
. La restriction de
T 1 U
dans le dual unitaire de U
.
TI + T est une bijection des ensembles de representations de
decrits ci-dessus. Cette bijection induit un isomorphisme des espa-
ces d1op6rateurs d'entrelacement et des commutants.
D6monstration : Tout cela rCsulte des lemmes I et 2 et de la thdorie de Mackey (cf 183). C.Q.F.D.
CHAPITRE 6
REPRESENTATIONS
est un groupe de Lie d'algebre de Lie g
Dans ce chapitre, G groupe du centre de
- CONSTRUCTION DES
G,
lineaire T-admissible, T
17 un caractsre unitaire de
I'
,g
E
, r
T
g7=
un sous-
g* un forme
X(g,d.
E
Nous allons construire les reprgsentations T par recurrence sur dim g. g9-r Supposons d'abord d i m g = 0. Alors g = 0, G(g) = G, et l'on pose
T x = I ) pour tout x g9 7 duction sont verifiges.
E
G. Les propribt6s (i) l (iiiii) de l'intro-
On suppose la construction faite pour tous les groupes de Lie de dimension strictement infdrieure, de telle sorte que les propri6tEs (i) 1 (iiiii) soient vbrifi6es.
4;
. Le sous-
est fermd et invariant ; de plus u
est admissi-
On note 2 le plus grand idgal nilpotent de g groupe analytique U
de
G
ble (ch. IV lemme 6). On emploie les notations
. On pose
d6duit de
1 et 2,que
h
=
& , 3, , G I , r l , '7,
du paragraphe V. 4. De plus, on pose h = glh , et on note
El
u
g;
, etc...
l'Ql6ment de
par passage an quotient. I1 rssulte du chapitre IV, lemmes
gl est bien polarisable. I1 rdsulte du chapitre IV, lemmes 4 et
6 , que gl est I-,-admissible,et qu'il y a une bijection canonique T entre XG,r(g,n)
et
X Gl'rl
T~
+
(g,,?ll).
Nous allons consid6rer deux cas. i) On suppose que l'on a : dim& au plus I, u
est injectif,
=
dim g -1
. Alors u
est rdductive de centre 2
est de dimension
. On definit
T g*T
cornme au chapitre 111. ii) On suppose que l'on a : dim
gl < d i m g . Alors l'hypothcse de recur-
rence nous permet de construire la representation
de GI. D'aprSs la g~'~,
proprietc (iiiii), sa restriction h
rl
est le caractere
. qI. Le lemme 3,
chapitre V, nous permet de poser :
Remarque : Si ,& est reductive, mais si
u n'est pas injective, nous avons defini deux
.
fois T~ une fois dans le cas ii) ci-dessus, et une fois au chapitre 111. g.~' On verifie facilement que les d e w definitions coincident.
ThgorGme I : Les representations T. verifient les propri6tEs (i) h (iiiii) de l'introg *T duction.
D ~ m o m t m t i o n: Les proprietds (i), (iiii) et (v)
resultent immediatement du lemme 8, chapi-
tre 111, dans le premier cas, de llhypothPsede recusrence et du lemne 3 chapitre V dans le second cas. La restriction de T 1 U est portee par l'orbite de g9T T dans U. Cela resulte de la construction de T au chapitre 111 lorsg9.I que g est reductif (ler cas), et de la formule (I) sinon. Soient g' E g* , DQmontrons (iii).
T'
E
X(g')
, U'
= g'lu
les restrictions 5 u'
E
U
. Supposons
et T non disjointes. Alors g,r g',~' sont non disjointes, et il en resulte que l'on a
Gu. Quitte h remplacer g'
T
par un conjugue, on peut supposer que l'on
a:u = u'. La (Sfinition de
(
1
Si g #
gl , GI , etc... ne depend que de u. Si g
resulte du lemrne 8
=
g,
,
, chapitre 111.
gl , il resulte du lemme 3, chapitre V, que T
et T g19~1 g; ST; sont non disjointes. L'hypothese de recurrence montre que g1 et gi sont
conjugugs par Quitte
G,, e t donc par
B remplacer g '
u = u ' , g, = g; g(u) + 2.
. I1
G(u).
par un conjugug, on peut supposer que l ' o n a :
en r g s u l t e que
g
et
g'
ont mZme r e s t r i c t i o n B
11 rLsulte du lemme 5 , chapitre 4 , que
&me G-orbite.
C.Q.F.D.
g
et
g'
sont dans l a
CRAPITRE V I I
- APPLICATIONS
Comme je l'ai dit dans l'introduction, je ne donne ici que des applications simples de la construction de representations T
.
Elles sont de deux org *T dres : une classification de l'ensemble des representations unitaires irrgductibles d'un gnoupe de Lie moyennable de type I, et une application B la representation r6guliZre des groupes de Lie localement algsbriques connexes, avec en particulier la classification des reprgsentations irreductibles de carre integrable.
V I I . I . Groupes de Lie moyennabZes :
Dans ce paragraphe, on suppose que G est moyennable. Ceci est equivalent aux deux conditions suivantes : GIGO
est moyennable, et le radical resoluble de
G est cocompact. Les resultats de ce paragraphe sont essentiellement contenus dans l'article de Pukanszky [ 2 6 ]
.
!Thdor2me 1 : On
suppose G moyennable. Soit I un idgal primitif de la c*-algZbre de G.
I1 existe une forme lin6aire admissible et bien polarisable g T E
x ~ ~ ~ tels ( ~que )
I soit le noyau de T g9T
&*, et
dans c*(G).
Ddmonstration : On raisonne par recurrence sur la dimension de
G. Lorsque la dimension de G
est nulle, le resultat est evident. On suppose cidessous que la dimension de g est strictement positive et que le resultat est etabli pour tous les grou-
r
pes de Lie moyennables de dimension strictement inferieure. Soit
u
le plus
grand ideal nilpotent de &
, et
soit U
le sous-groupe analytique correspon-
, il
dant de G. D'aprBs le th6orBme 4.3. de Gootman et Rosenberg [ I l l
existe
une representation unitaire irrEductible T de G, de noyau I, dont la restriction B
U est portee par une quasi-orbite transitive de G dans le dual
unitaire de U, car U u .=
u*
est de type I et car G/U est moyennable. Soit
un element admissible tel que l'orbite de la reprgsentation TU de U
porte la restriction de T B U
. D'apres I'
tations de ce leme, et posant taire irreductible T, de GI
, il
= { 1)
, dont
le l e m e 3, par. V.4., avec les noexiste une representation uni-
la restriction 1
rl
nI , et
est
telle que l'on ait : T = Ind G(u)U
(TI @
su'fu)
.
C o m e dans le chapitre VI, on considere deux cas.
ler cas : dim&] = dim g
. Alors
g est reductive de centre
u,
et
est semi-
simple compacte. C o m e le groupe des automorphismes de ~ 1 % est fini, modulo les automorphismes interieurs, la restriction de T B
GO est port6e par
l'orbite sous G d'une representation unitaire irreductible de Go
. Cette
es t de la f o r eO : (oP g est une forme lineaire g admissible bien polarisable sur g, cf. par. III.]., exemple). I1 resulte du representation de Go
par. 111. 5. qu'il existe
T
E
Xirr(g)
tel que l'on ait T = T
g,T
' et
donc I est le noyau de T g 9T
2 h e cas :
dim
&,<
dim g
. Come
GI est mopennable (come quotient d'un sous-groupe
ferm6 de G) il existe un Clement admissible bien polarisable T~
E
frr(g,)
tels que T I et T 81 J 1
aient &me
gl
E
g;
et
noyau dens la c*-alg2bre
de
GI. Comme la restriction de
TI 1
T I est le caractere
p l , il en est
T , ce qui implique que l1on a E ~'~~(g~,~l])11 existe gl '1 donc un dldment admissible et bien polarisable g E g* , et T E xirr(g) de mfme de
tels que les dldments G I , g l , T] ddfinis ci-dessus soient les mgmes que ceux qui scat ddfinis au chapitre VI 2 partir de on a (cf
. chapitre VI) T g,T
T et T g9T
et de
T
. Par d6finition,
:
G 1 G(u)~ 'Ig,,T~@
= Ind
C o m e les reprCsentations T I et de
g
U '
T ~ )*
T
ont mfme noyau, il en est de mzme gl "C1 , dlapres les thdorEmes de continuit6 de Fell. C.Q.F.D.
CoroZZaire : On suppose
G
moyennable. Soit
T une representation unitaire irrdductible
normale de G. I1 existe une forme lindaire admissible bien polarisable g
E
g*
et
De'monstration
T E X
irr
(g)
telS que
T
=
T
g ST
:
En effet, une reprdsentation unitaire irrdductible de
T dans la c*-alg5bre de G est Gquivalente 5
G ayant mfme noyau que
G.
C.Q.F.D.
Remarque 1 : Ce corollaire, joint au thdor5me 1 du chapitre VI, donne une parami5trisation du dual unitaire des groupes de Lie moyennables de type I.
Remarque 2 : Pukanszky C261 a obtenu ces rdsultats dans le cas des groupes connexes. La d6monstration donnde ici n'est pas fondamentalement diffGrente, mais llemploi du thdorEme "marteau pilon" de
[ I l l permet un expos6 plus simple et un dnonc6
plus general. Notons que la paramdtrisation du dual unitaire et la definition de 11admissibilit6d'une forme lindaire, sont diffdrentes dans
C261.
Remarque 3 : Lorsque G est connexe et moyennable
- ou plus
gdndralement lorsque G vdri-
fie une certaine hypothsse (H) ddfinie par Charbonnel et Khalgui C61
, on
peut rdaliser toutes les representations T corn "reprdsentation holomorg9-r phes induites". C'est-&dire qu'on peut trouver une polarisation b en g telle que T soit rdalisde, par translations 1 gauche, dans un sous-espace g9T approprie de l'espace des fonctions vdrifiant les relations (9) et (10) du paragraphe 11.4. (Ceci est implicite dans [I61 et dans C231).
On voit donc que
toutes les representations unitaires irrsductibles normales d'un groupe moyennable connexe (ou plus gdndralement de type (H)) sont obtenues pas induction holomorphe (ceci est bien connu pour les groupes resolubles connexes
-
cf C21
et [241, et est demontre pour les groupes moyennables connexes simplement connexes dans L231).
Remarque 4 :
Supposons G moyennable et connexe. Soit g bien polarisable, et soit
r
E
&* un dlement admissible et
Pukanszky a demontrd C261 que la
E
representation irrdductible T est normale si et seulement si G est un g7-r g sous-ensemble localement ferm6 de g* et si T est dimension finie. Notons aussi que, dans le cas gzndral, une application immddiate des rdsultats de Thoma [331
donne des conditions ngcessaires pour que G soit de type I en
fonction de la structure de G(g) polarisable.
, pour
chaque g
E
g* admissible et bien
VII. 2. Groupes de Lie algdbriques : Nous nous i n t e r e s s o n s dans c e p a r a g r a p h e aux groupes de L i e connexes l o c a l e ment a l g g b r i q u e s , ( c ' e s t - 2 - d i r e
l o c a l e m e n t isomorphes au groupe d e s p o i n t s
r e e l s d'un groupe a l g Q b r i q u e a f f i n e d e f i n i s u r
g).
Pour l e s n E c e s s i t 6 s de l a r e c u r r e n c e , nous c o n s i d e r e r o n s une c l a s s e p l u s v a s t e de groupes.
Dgfinition : Un groupe d e L i e
G e s t d i t presque algebrique s ' i l v e r i f i e l e s conditions
suivantes
i . I1 e s t l o c a l e m e n t a l g e b r i q u e .
ii. I1 e x i s t e un sous-groupe l e centre de
G',
on
.sit
de
G'
ZG, G
0
=
G
d ' i n d i c e f i n i t e l que, notant
G'.
Nous a u r o n s b e s o i n d e s lemnes s u i v a n t s . On n o t e t e n t de
g et U
Leme 1 : S o i t I, e t
G
G
un groupe de L i e p r e s q u e a l g e b r i q u e . A l o r s U (i.e.
l e s o r b i t e s de
G
e s t de t y p e
dans l e d u a l
:
Les deux r e s u l t a t s s e d e d u i s e n t immediatement du c a s oii i l s s o n t dus 2 Dixmier e t Pukanszky ( c f . e . g .
Leme 2 : S o i t
G
s o n t localement f e r m Q e s ) .
U
D6monstratiun -
2 l e p l u s grand i d e a l n i l p o -
l e sous-groupe a n a l y t i q u e c o r r e s p o n d a n t .
opGre r g g u l i e r e m e n t dans
u n i t a i r e de
z ~ '
G
G
[ 2 8 1 p . 85-86).
un groupe de L i e p r e s q u e a l g g b r i q u e . S o i t
ment a d m i s s i b l e . Le groupe
G,
e s t connexe, oii C.Q.F.D.
u
E
$
un Q l e -
d e f i n i au paragraphe V. 4. e s t p r e s q u e alg6-
Dlmonstruti,-n : I1 suffit visiblement de le ddmontrer quand
G
de le ddmontrer quand
est connexe et simplement connexe. Soit
groupe discret du centre de G d'un groupe algdbrique. Alors xes. Donc
I'G(U)~
tel que G(u)/I'
=
~ ( u )n ker u
D'autre part
et ~(u)?
, oti Q
briques. Rappelons que l'on a G I = G(U)?/Q
3
G(u)
sont presque alg6-
est le sous-groupe ana-
G , est localement algebrique, il suffit de montrer que
correspondant soit algebrique. Pour cela il suffit que le sous-
que localement isomorphe h
G . Soit
Z
un groupe alg6bri-
le centre de G , et soit T
G est localement isomorphe B
maximal de Z. Alors
ment isomorphe B un groupe algdbrique G soit unipotente. Alors
ZO est contenu dans U, et U/Z0
est unipotent,
C.Q.F.D.
est uo groupe de Lie presque alggbrique,
groupe fermd du centre tle
est locale-
tel que la composante neutre ZO de
est unipotent.
Dans la suite, G
le
. Rempla~ant T
G/T x T
par un groupe uni~otentabllien de mEme dimension, on voit que G
U
G, et
11
r
un caractere unitaire de
est un sous
r . On
note
A
G
n
-
G
tel que le sous-groupe
connexe d'un groupe unipotent est alglbrique. Soit donc G
donc
G1
correspondant h 2 soit unipotent, car tout sous-groupe de Lie
groupe U
Z
est
. La condition ii/ est v6rifide pour
est localement isomorphe h un groupe alggbrique G analytique Q
un sous-
a un nombre fini de composantes conne-
est d'indice fini dans G(u).
lytique d'algebre
r
soit dgal B la composante neutre
G/r
localement algebrique. On voit donc que G(u)
Pour montrer que
est connexe, et pour cela,
G
l'ensemble des classes de reprdsentations unitaires irrdductibles de G
r
dont la restriction 1
est le caractere
~1
. Come
G
est de type
I, il
de mesures bordliennes sur l'espace bordlien existe une unique classe p 11 G standard G,, tel que la reprEsentation Ind (n) soit quasi-equivalente h A
- 1;:
r
I
la representation de Plancherel de
G
r;
T dyn(T) n'
.
. On
dira que p
T
est la classe de la =sure
Z'h'hkor6me 2 :
.
Le complementaire dans
, 06
T g9-C
et .r
g
G
n
de l'ensemble des representations de la forme
parcourt l'ensemble des formes n-admissibles bien polarisables, ~ ~ ~ ~ ( ,~ est , n de )mesure nulle pour
l'ensemble
UP
On raisonne par recurrence sur la dimension de G. Le rdsultat est evident G
pour les groupes discrets. On suppose la dimension de
strictement positi-
ve, et le resultat gtabli pour les groupes presque alggbriques de dimension A
infgrieure. Notons irreductibles de
l'ensemble des classes de reprdsentations unitaires
ur:
U
dont la restriction B
r
n U
est
q. Notons
.
mesure de volume fini dans la classe de Plancherel de U l'image de u* -
.
v
E
*!
G I , rl, q 1
de
G
il suffit de voir que
rU
est fermd dans G(u).
un sous-groupe de
G Z
il suffit de prouver que
.
a :
r
Ts(u) E"). n U, on dEfi-
est ferm6. En effet,
Comme 'I n U = r n U(u),
contient la composante neutre ZU
est fermg. Divisant par
il suffit de le montrer quand
groupe alggbrique. Alors
G
est un ouvert d'un
ZU/U est fini et le resultat est clair.
.
. On peut
la mesure de Plancherel correspondante sur G
'h
I ,nl
former la representation RU
pour tout u
n
, on
rl
,.
est ferm6 dans G. I1 suffit de le faire quand
est connexe et simplement connexe. Comme U de
n
c o m e au paragraphe V. 4. Le groupe TU(u)
du centre Z
dans U
-
notons v
s de U /G dans
, admissible pour la restriction de q 2
il suffit de voir que
Notons
.
ri
Pour chaque u
G
11
sur U /G. On choisit une section bordlienne
(de sorte que pour toute orbite w
nit
, et
une
y
E
Gn/6, notant
de
u
=
G
s(u),
ddfinie par la formule :
nous poserons R~
=
RU
.
Compte-tenu du l e m e I, une lggere gdndralisation de Kleppner et Lipsman C181
(qui considerent le cas
r
=
G (11) montre que la representation Indr(n)
est
quasi-6quivalente 5 la representation
Pour d6montrer le thGorPme, il suffit donc de demontrer que pour tout u
E
2*
T , avec El y T 1 qui est de mesure nulle pour
c o m e ci-dessus, l'ensemble des representations de la'forme
.
-rl
E
pn,
Xirr (gl,nl) a un complementaire dans G , "I i
. En effet, si
ments
gl et
ments
gl et
l'on choisit
g
6
g
et
T
E
xirr(g, )
tels que les 616-
correspondant d6finis au chapitre VII soient egaux aux 616T]
ci-dessus, on a, dlaprPs le chapitre VI : G Ind (T Q SU TU) = T G(u)U gl,~] g*T
G La representation Ind (n)
r
est donc desintSgr6e en reprEsentations T
g , ~'
ce qui prouve notre assertion. Nous considErons deux cas.
l e r cas : dim g, = dim g Pour le groupe Go
. Alors
g
est reductive de centre 2
.
le th6orSme 2 est vrai : il est d6 1 Harish-Chandra [ I 2 1
lorsque [G ,G J est de centre fini et a Et6 6tendu au cas g6n6ral par Wolf 0 0 C381.
, on
Appliquant come ci-dessus le r6sultat de Kleppner et Lipoman [I81 ramen6 B l'assertion suivante : soit g soit
o
6
irr XG ,rnG(g,nl) (06 0
n'
E
g*
un 616ment
est la restriction de
Toutes les representations unitaires irrgductibles de G
la forme T~ gs?
.
n-admissible , et 11
B
r
n GO).
dont la restriction
de la representation T g $0 Ceci r6sulte du paragraphe 111. 8.
1 Go est portee par l'orbite sous G
est
sont de
< dim
2Zme cas : dim -1g
g
. On
applique l'hypothsse de recurrence (grgce au
lemme 2). C.Q.F.D. On trouvera dans l'appendice des id6es qui devraient Stre utiles pour decrire prdcisement la classe de la mesure de Plancherel. J'espsre revenir sur cette question. Je donne ci-dessous un resultat partiel, gbn6ralisant des r6sultats bien connus de Harish-Chandra (pour les groupes semi-simples), Moore et Wolf (pour les groupes nilpotents) et Charbonnel
Thlorr3me - --
o our
les groupes rbsolubles).
3 : (Les notations sont celles du th6orPme 2).
Une representation unitaire irreductible T de G dans
G Indr(n)
risable, g
6
intervient discretement
si et seulement s'il existe un Element 17-admissible bien pola-
&*
tel que G(g)/r
T soit isomorphe
5
soit compact, et
T
E
x ~ ~ ~ (tels ~ ,que~ ) ~
T
g *T
D6monstmtwn : D'aprSs le th6orSme 2, il suffit de demontrer l'assertion sui"ante. Soit
g c g*
T c xirr(g,n)
- Alors
ment si
G(g)/r
un 616ment
T
n-admissible bien polarisable. Soit
G Ind (11)
intervient discrstement dans
r
g . ~
si et seule-
est compact.
Nous d6montrons cette assertion par rdcurrence sur la dimension de G. Elle est est claire pour les groupes de dimension 0 (rappelons que, par hypothsse, ceuxcci ont un sous-groupe abelien d'indice fini). On suppose donc la dimension de
G
strictement positive, et le rgsultat htabli pour tous les groupes
presque algebriques de dimension infgrieure. On pose
u =
glu, et on emploie
les notations du chapitre VI. On considsre deux cas.
l e r cas : dim g -1
=
dimg
. Appliquant
la mEthode de Kleppner et Lipsman c o m e
dans la dEmonstration du thdorPme 2, on voit que
T g9.r
ment dans
IndG (nj
r
intervient discrete-
si et seulement si les deux conditions suivantes sont
i. Soit
TCO
une repr6sentation irreductible de Go dont le g*c Gorbite porte la restriction de T B Go (voir paragraphe 111.8). Alors g,T intervient discretemene dans Ind O (nl), 05 n' est la restriction gYu rnGO de n B r n Go.
TCO
Cela implique, d'apres Harish-Chandra (cf [ 3 7 3 ) , que Go(g)/rnG0(g)
est
compact. On suppose donc que Go(g)/TnGo(g) est connexe (et donc
0
tion 1
r
Comme ~(g)&
intervient discretemerit dans la representa-
T(G(~)~)&
induite par la reprssentation de
est le caractere
GO(g)
XP(~)).
est l'unique element de
ii. La repr6sentation T tion de G(~)$
est compact. Cela entrafne que
n , et
dont la restric-
la restriction 1 (,G(g) )&
0
le caractsre
contient un sous-groupe abelien d'indice fini, on voit que cela
implique que le groupe
G(g)/rG(g),,
est compact.
Les conditions i. et ii. ensemble sont dquivalentes 1 la compacite de G(g)/F.
2dme cas : dim g < d i m s -I
.
I1 rssulte de la demonstration du theoreme 2 que T intervient discreteg9T G ment dans Indr(n) si et seulement si les deux conditions suivantes sont realisees. i. Soit w ii. T
l'orbite de
TU dans U
on a : ;({w))
> 0.
Indr 1 (nl). 1
Etudions la condition i. Soit D
r
. Alors
intervient discrstement dans
gl '*I
engendre par
n
n U, et soit
fi
le sous--groupe de U, connexe, algebrique, son algebre de Lie. Alors
D/r n U
est
compact, et l'ensemble des formes lineaires sur 2 qui sont admissibles pour la restriction de
q
1
r
n U
est une reunion localement finie de sous-
espaces affines
dl
l'orthogonal
2i
(indexee par les caractsres de D/rn U)
de
d
dans *
parallSles 1
. On en deduit une partition
ensembles isomorphes (pour la structure borelieme) P
de U en sous-
U R / ~, et la mesure de
-3
Plancherel sur U:/U est equivalente 1 l'image d'une mesure finie sur -J Bquivalente 1 la mesure de Lebesgue.
gi
On voit donc que la condition i. est equivalente 1 la condition i'.
. =A'
it. Le sous-ensemble G0u est ouvert dans u + dl Cette condition est encore 6quivalente 1 lt6galit6 gu l'orthogonal, 1 la condition
(gull =
d.
Mais on a (gu)'
tion i. est donc dquivalente 1 la condition:il'. Le groupe
=
, et, passant
5
~(g). La condi-
uXg)/rn
U(g)
est compact. Etudions la condition ii. Par l'hypothhse de recurrence, elle est 6quivalente
1 la compacit6 de G(gl)/rl
. D'aprPs
le lemme 5, paragraphe IV. 2., ceci est
6quivalent, 1 la compacit6 du groupe G(g)U(u)
/l'U(u) = G(g) /rU(g)
.
Donc i. et ii. ensemble sont 6quivalente.s 1 la compacitd de G(g)/r
.
C.Q.F.D.
Remarque : Les groupes presque alg6briques unimodulaires pour lesquels il existe des representations unitaires irrsductibles intervenant discrstement dans G Ind (TI) ont une structure trss particuliere (voir l'article d'Anh cit6 dans
r;
l'appendice)
.
BIBLIOGRAPHIE
M. ANDLER. Sur des representations construites par la methode des orbites. C. R. Acad. Sc. Paris 290 (1980) 873-875. L. AUSLANDER et B. KOSTANT. Polarization and unitary representations of solvable groups. Invent. Math. 14 (1971) 255-354. P. BERNAT et al. Repr6sentations des groupes de Lie rbsolubles. Dunod, Paris 1972.
J. CARMONA. Repr6sentations du groupe de Heisenberg dans les espaces de (0,q)-formes. Math. Ann. 205 (1973) 89-112.
W. CASSELMAN-M. S. OSBORNE. The E-cohomology of representations with an infinitesimal character.Compositio Math. 31 (1975) 219-227.
J. Y. CHARBONNEL et M. S. KHALGUI. Polarisations pour un certain type de groupes de Lie. C. R. Acad. Sc. Paris 287 (1978) 915-917.
J. DIXMIER. AlgZbres enveloppantes. Gauthier-Villars, Paris 1974.
M. DUFLO. Sur les extensions des representations irreductibles des groupes de Lie nilpotents. Ann. Sc. Ecole Norm. Sup. 5 (1972) 71-120. M. DUFLO. Representations de carre integrable des groupes semi-simples rEels. Sem. Bourbaki exp. 508, 1977-1978. T. J. ENRIGHT. On the fundamental series of a real semi-simple Lie algebra. Their irreducibility, resolutions and multiplicity formulae. Annals of Math. 110 (1979) 1-82. E. C. GOOTMAN et J. ROSENBERG. The structure of Crossed product c*algebras : a proof of the generalized Effros-Hahn conjecture. Invent. Math. 52 (1979) 283-298. HARISH-CHANDRA. Harmonic analysis on real reductive groups. Annals of Math. 104 (1976) 117-201. R. HOWE. On the character of Weil'srepresentation. Trans. Amer. Math. SOC. 177 (1975) 287-298. N. E. HURT. Proof of an analogue of a conjecture of Langlands for the Heisenberg-Weyl group. Bull. London Math. Soc. 4 (1972) 127-129.
M. KASHIWARA et M. VERGNE. On the Segal-Shale-Weil Representations and Harmonic Polynomials. Invent. Math. 44 (1978) 1-47.
M. S. KHALGUI. Sur le's caractPres des groupes de Lie B radical cocompact. Preprint 1980.
A. A. KIRILLOV. ReprEsentations unitaires des groupes de Lie nilpotents. Uspekhi Mat. Nauk 17 (1962) 57-110.
.
A. KLEPPNER et R. L. LIPSMAN. The Plancherel formula for group extensionsI1. Ann. Sci. Ec. Norm. Sup. 6 (1973) 103-132. B. KOSTANT. Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. of Math. 74 (1961) 329-387. G. LION. Indices de Maslov et representation de Weil. Publ. Universite Paris 7 , NO 2 , 1978
G. LION. Extension de reprEsentations de groupes de Lie nilpotents. C. R. Acad. Sc. Paris 288 (1979) 615-618. G. LION et M. VERGNE. The Weil representation, Maslov index and theta series. irirkhzuser , Boston 1980.
R. L. LIPSMAN. Orbit theory and representations of Lie groups with co-compact radical. Preprint, Maryland 1980. L. PUKANSZKY. Unitary representations of solvable Lie groups. Ann. Sc. E.N.S. 4 (1971) 435-491. L. PUKANSZKY. Characters of connected Lie groups. Acta Mathematica 133 (1974) 81-137.
L. PUKANSZKY. Unitaiy representations of Lie groups with co-compact radical and applications. Trans. Amer. Math. Soc. 236 (1978) 1-50. I. CATAKE. Unitary representations of a semi-direct product of Lie groups on d-cohomology spaces. Xath. Ann. 190 (1971) 177-202. SCHMID. Some properties of square integrable representations of semisimple Lie groups. Ann. of Math. 102 (1975) 535-564.
W.
2
W. SCHMID. L -cohomology and the discrete series. Annals of Math. 103 ( 1976) 375-394.
D. SHALE. Linear symmetries of free boson fields. Amer. Math. Soc. 103 (1962)
149-167.
J.M. SOURIAU. Construction explicite de l'indice de Maslov et applications. Fourth international colloquium on group theoritical methods in physics, University of Nijmegen, 1975. B. SPEH et D. VOGAN. Reducibility of generalized principal series representations , Preprint 1978.
E. THOMA. Uber unitare Darstellungen abzahlarer ciiskreter Gruppen. Math. Ann. 153 (1964) 111-138.
C341
M. VERGNE. Construction de sous-algPbres subordonn6es 5 un E l h e n t du dual d'une algPbre de Lie r 6 s o l u b l e . C. R. Acad. Sc. P a r i s 270 (1970) 173-175 e t 704-707.
C351
D. VOGAN. The a l g e b r a i c s t r u c t u r e of r e p r e s e n t a t i o n s of semi-simple Lie groups. I Ann. of Math. 109 (1979)
C361
1-60.
11. P r e p r i n t 1977.
D. VOGAN. I r r e d u c i b l e c h a r a c t e r s of semi-simple Lie groups 11. Duke Math. Journal 46 (1979) 805-859. G. WARNER. Harmonic a n a l y s i s on semi-simple Lie groups New-York 1972.
C381
. Springer-Verlag,
J. WOLF. The a c t i o n of a r e a l semi-simple Lie group on a-complex manif o l d 11. Unitary r e p r e s e n t a t i o n s on p a r t i a l l y holomorphic cohomology spaces. Mem. Amer. Math. Soc. 138 (1974).
APPENDICE : PARAMETRIZATION OF THE SET OF REGULAR ORBITS OF THE COADJOINT REPRESENTATION OF A LIE GROUP
(Un t e x t e p r d p a r g pour une confgrence 1 l ' U n i v e r s i t 6 de Maryland, dgcemb r e 1978). We c o n s i d e r o n l y a l g e b r a i c groups and a l g e b r a s . It i s l i k e l y t h a t a l l t h e r e s u l t s can be f o r m u l a t e d f o r non a l g e b r a i c groups a s w e l l , b u t t h i s i n t r o d u c e s v a r i o u s k i n d s o f c o m p l i c a t i o n s which I p r e f e r t o avoid i n t h e s e l e c t u r e s .
I s h a l l d i s c u s s t h e s e s c o m p l i c a t i o n s i n t h e l a s t paragraph.
k
I. Lie aZgebras over an a2gebraicaZ.q closed field Let
k
b e a s above,
Lie algebra,
g*
s t a b i l i z e r of
g
Ho
we d e n o t e by Suppose
g
&*
6
G an a f f i n e a l g e b r a i c group d e f i n e d o v e r
t h e d u a l space of and by
say t h a t
is regular ( i .e. the o r b i t
gs*
if
g
g
and
g'
G
for a l l
g
. IJe
E
, 11
=
[s,gl
G(g) t h e
i s an a l g e b r a i c group
H.
i s of maximum dimension). S(g)
i t s maximal t o r u s . We
i s r e g u l a r and
S(g)
of maximum d i -
S(g)
2
i s an open
and
S(g')
are
(GO, Go)).
s
such t h a t
S(g)
i s conjugate t o
u s e t h e f o l l o w i n g n o t a t i o n s : H = c e n t r a l i z e r of
H' = n o r m a l i z e r o f
-h*el P* .
we d e n o t e by
H
are strongly regular?
Let u s f i x a t o r u s . S with Lie algebra
W = H1/H
g*
( o r i t can be proved i n an e l e m e n t a r y way) t h a t
[Ill
G,
6
g its
k,
t h e s e t of s t r o n g l y r e g u l a r e l e m e n t s . It f o l l o w s
c o n j u g a t e (even by a n element of
in
g
its Lie algebra. I f
g(g)
is strongly regular
g
s e t , and t h a t i f
S
If
t h e connected component of 1 i n
mension. We d e n o t e by from
.
&
G(g)O i s c o m u t a t i v e [ b ] . We d e n o t e by
Then
of ckaracteriskic 0 :
. It
S
in
G,
2=
i s clear that
L i e a l g e b r a of b o t h
g
=
h b E.
We i d e n t i f y
B
and
8'
g* and
S
,
Lema 1 :
(i) Every G-orbit in
h* , every
(ii) In
* n %*
-h
a (h) W
gs*
&
intersects
.
regular element is strongly regular. The set
is equal to the set of regular elements h a 0, where a is an homogeneous polynomial on W
(iii) There is a natural bijection of
&*
such that
&* , defined
onto
(&*
n
below.
g)/~'.
Corollary :
k(g*)G
is isomorphic to k(h*)H'
Definition of
nW
.
:
The dimension of p is even, say 2d. We choose a non zero exterior 2d-form
w on E. On E , we consider for every h Bh(X,Y)=h(fX,Y])
(X,Y~~).Wedefine
(dl)-' Bh that
nW(h)
A
...
A
Bh = n (h)w W
E
&*
the 2-forms
Bh
such that
~ ( h )bytheformula W
(there are d factors Bh, so
is the Pfaffian).
Example I : Wc suppose G
semi-simple and connected. Then g
E
< is strongly regular if
and only if it is regular, and semi-simple (when identified to an element of
-g
using the Killing form). This is the case if and only if G(g)
subgroup of G. Here W k(&*lG
k(&*lW
is a Cartan
is the Weyl group, and the corollary read :
, which is due to Chevalley. Note that the most interesting
*G part in Chevalley's theorem is the isomorphism kCg 1
k~~*]'
, but
this
does not generalize in the situation of the corollary.
Remark I :
Let us say that g r
$
is very regular if there exist 4,.
...,$ , elements
, whose
of k(LlG, defined at g dant, with
p
=
dim G(g).
differentials at
g are linearly indepen-
The set of very regular elements is open, and non
empty by a result of Rosenlicht. It is contained in the set of regular elements. Moreover, if
g is very regular, G(g)
*
Thus, for the very regular elements in proves that k(l*)H
h
centralizes G(g)O
* , we
n%
have G(g)
c
C61
.
H. This
with Galois group W.
is an extension of k(h*)H'
Remark 2 : If g is very regular, the set a subgroup. Consider the set of regular, and such that G(g)
Z(g) g
E
of semi-simple elements of G(g) is
&* which are very regular, strongly
has the maximum number of connected components
(among very regular and strongly regular elements). Then this set is.open, and for g
in this set the
C(g)
are mutually conjugate in G. This follows
from [ll]or
it can be proved in an elementary way.
Remark 3 : If G
is connected, so is H
Dixmier the heart of g
. When
G
. Thus the heart
is connected, k(g*lG of
of g with Galois group W. If moreover
h
is called by
is an extension of the heart
G is solvable, W = 1 .
Remark 4 : Corresponding to the restriction mapping from kCg*lG into k[h can define a "Harish-Chandra mapping" from
u(&)~
ves the choice of a "system of positive roots" of
into ' ) h ( U
* 3H) , one
. This invol-
g in 2* , but the result
is independant of this choice.
ExanrpZe 2 : G is the semi-direct product of k*
x
k* by a unipotent group whose Lie al-
x, y, z, w
gebra has a b a s i s (t,u) ( t ,u)w
5
u - ~ w . Let
a,b
= z
. The
a c t i o n of
2 (t,u)z = u z
,
be t h e canonical b a s i s of t h e Lie a l g e b r a of
s=k
k* x k*. We can choose
a
h = . Then -
k a + kb + kz + kw
i s not. For g e n e r i c
i s unimodular, but t h a t t h e subgroup
[x,y]
i s given by : ( t , u ) x = t u x , ( t , u ) y = t - l u y ,
k* x k*
E
with b r a c k e t s
k* x (21)
of
,
g
. Note
that
g
is conjugate t o
C(g)
k* x k*.
2. Real Lie algebras :
a. StrongZy regular o r b i t s : We consider a Lie group
with Lie a l g e b r a
G
p l e x i f i e d Lie algebra. We assume t h a t
G
&
. We denote by %
t h e com-
has a f i n i t e number of connected
G o ; t h e connected component of I , is t h e a n a l y t i c sub-
components, and t h a t group with Lie a l g e b r a
& of an a l g e b r a i c group G
E
with Lie a l g e b r a
g
.
S
Lem 2 : Let
g
and
gy) of
g'
*
%,s (Go, GO).
be i n t h e same connected component ( f o r t h e Hausdorff topolo-
. Then
n E*
k'e denote by
T(g)
gular i n
(recall that
&*
S(g)
and
S ( g f)
a r e conjugate by an element ,of
t h e maximal compact subgroup of
G(g)O
, when
g
i s re-
G(g)O i s commutative). From lemma 2 , we g e t :
.Theorem I : TI,
There e x i s t a f i n i t e number groups of
G
and only one Consider one
Tj
such t h a t i f
n g*
6
+ *,, n g:
of connected compact a b e l i a n sub-
i s conjugate i n
,T(g)
T j ' T We introduce some n o t a t i o n s : t . j -3
.
,Hj=Z(T.) G J
g c <,s
g
..., Tq
,H!=N(T.) 3 G I
such t h a t
T(g)
,g T
t o one
i s t h e L i e a l g e b r a of
, p . = [ ~ ~ , g ] -3 i s conjugate t o
G
isthesetof h.
j' -J
t h e Lie a l g e b r a of
H. J
.
g -
. On
and on h. -3
E~ we choose the quotient measure, which is defined by a
. As
differential form w.
J
r
= r uj
j
on : h
above we define an homogeneous polonomial
,andweidentify
-J
. There correspond Haar measures on
and on H j
We choose Haar measures on G
-g*
and
*
*
h.@p..
-J
-J
Lema 3 : (i) The intersection of a G-orbit in :g Thus g):/r
is isomorphic to
J
(ii) g; n :h
is the set of h
-3
T.
J
* (Lj
(h) # 0, and
T(h)
=
T j
i
-J
n g.)/H! -1 J
: h
-3
*
with : h
1
is a non-empty
h!-orbit. 3
.
which are strongly regular, such that
'
Remark I :
*
*
g. n h.
-3
is the union of a cercain number of connected components (for the
-3
Hausdorff topology) of a Zariski open subset of h): -J
tal" (i.e. the dimension of T
j
. If
T
j
is
"fundamen-
is maximal), then it is Zariski open.
Remark 2 : The case q
=
1
is specially interseting. This happens for instance if g
is solvable (cf. [ 3 ] ) , or more generally if the semi-simple part of g is compact. This happens also if g has a complex structure, or if
g is reduc-
tive with one conjugacy class of Cartan subalgebra.
b. AdmissibZe orbits Let g e G (g)g
(*)
g . Then
. We
:
G(g)
denote by
has a canonical two-fold covering group (*)
(~(~)g)~ the inverse image in G(~)&
defini au chapitre 11.
of G(g)O,
and
by
(1, - I )
the non trivial element of the kernel of G(~)&
+
G(g).
We say
of (~(~)~)g with diffeXg If it exists, such a character is
that g is admissible if there is a character xg(l, -I)
rential ig and such that unique and unitary. If g
= -I.
X
is admissible, we denote by
)
irreductible classes of unitary representations of ~(g)g to G(~)~)&
is a multiple of
xg .
gi
the set of
whose restriction
.
Ye denote by &* the set of admissible elements in The purpose of aj these lectures is in fact the description of /G. Let X be an element of it. such that a(X) is non zero for every non zero root a of t in -J 5 and let P be the set of non zero roots which are positive on X. We denote
%
by
p
by
Rj
-1
C (dim g q a , with evident notations. We denote 2 a e ~ C= the set of h &; such that p + ihlt. is the differential of a the element
-3
character of Tj. The set R does not depend on the choice of P, and is the j union of affine spaces, translated from the orthogonal tf of t. in h? -J -3 -J The connected components of R are indexed by elements 1 of a lattice L.. j J 1 We denote by L' the set of 1 c L such that n is not zero on R j j j j ' Let 1 E L' Then R.1 n g. is non empty, and an union of connected compoj J -J 1 nents (for the Hausdorff topology) of a Zariski open set of R j '
.
.
'
Theorem 2 :
The set g*./~ is isomorphic to ( I -3
Y
1
(Rj n g:)/~.) -3
J
/Wj
(where
W. = H9./H.). J J J We use theorem 2 to describe a measure of
gj/~. To simplify, we suppose
that g is unimddular (if not we have to use measures with values in a suitable line bundle, cf. [ 3 ] ) . Recall that we have chosen measures on G
.
and
on H We put on T the normalized Haar measure, on t? the cocresponding j j -J Haar measure (for the duality given by exp(i< g, X > ) ) . On each R.I ( I - E L.) J J we put the translated Haar measure. This gives a measure on R. , and thus J
. We multiply
on the open subset R. n g:
J
3
26 = dim..
3
and n = 3 . 1 4
this measure by
...). The result
1
(2~)-~ln. (where
is a measure on R. n 3
J
gi
which is
HI-invariant, and depends only on the Haar measure on G. Each HI-orbit in 3
3
: h
-3
has a canonical invariant measure (we choose the normalization described
ij (Rj " $)/H;
in [ 2 ] p. 20). On denote it by
lJj
, we
=
put the quotient measure, and we
-
c. PZanchereZ f o m Z a :
The group G
is as in a. Moreover we suppose it is unimodular, and provided
with a Haar measure. Let ments of &*
.
. For each
be the set of strongly regular admissible eleR
G be the unitary dual of G
E
E:/~
, we
choose an element g ~ 1E Q
.
Conjecture : There exist an injective mapping (gn, T) . into G , with the following properties.
+
(i) (Infinitesimal character). If u ponding element of krg*] T
(u)
E
TgnST from
U(g)
, and
.
n U GIG if O
(gn)
is the corres-
is a multiple of G(ig).
n
E
g z / ~ and T
Then Cl is tempered, T is of trace class, and for (I gnsT support in a sufficiently small neigborhood of 0, we have
^
Xirr
(under the isomorphism defined in C51), then
(ii) (Kirillov's character formula). Let
where
. Let
is the Fourier transform, $(X)
T (exp X)dX gn'T
Bn , and
E
6
c:(~)
the canonical measure on Sl j(X) =
I d(expdX
(iii) (Plancherel formula). For each j = I , ...,p
irr X (g ) with
,
X)
there exists a func-
tion
for
on
n
j
.
u
s:j/~
xLrr(%)
with v a l u e s i n
10,
such t h a t
m[
4 E c~(G). ( i v ) (Plancherel formula, continued). Suppose
then
pj(gQ,')
= ( (G(&)/G(&)~))''
i s fundamental,
T
j
dim
T
.
EmnipZes :
I . , Assume
g i s n i l p o t e n t . Then
i s t h e d i r e c t product of a compact con1
GO
nected a b e l i a n group by a simply connected n i l p o t e n t group. I f t e d t h e conjecture reduces t o K i r i l l o v ' s t h e s i s [ 8 ]
. If
G
G
i s connec-
i s n o t connectea
then t h e c o n j e c t u r e i s s t i l l v a l i d , a s i t i s seen using [4] and [5]. 2. Assume t h a t
i s connected and s o l v a b l e . Then, except perhaps f o r t h e
G
f a c t t h a t a l l r e l e v a n t o r b i t s a r e tempered, t h e c o n j e c t u r e i s t r u e ( c f . [ 2 1 ch. 9 f o r ( i i )
and [3] f o r ( i v ) ) .
3. 4ssume t h a t
G
a ~ d (iv)
i s compact. I f
,
i s connected ( i i ) i s proved i n C91
G
can be obtained i n the same way. Then t h e case of disconnected
G
can be reduced t o t h e connected case. Let u s e x p l a i n t h e c o n s t r u c t i o n of
T 8.T
G'(g)O
is a Cartan subgroup of
t h e Bore1 subalgebra of
a
ding t o roots character
is
p
of
c it*
%
Go.
Let
. Here, t
which c o n t a i n s
such t h a t
G ( ~ ) Esuch t h a t
since
g
is regular
be i t s Lie a l g e b r a . Let
t
(ig, a ) < 0
,
b
be
and t h e r o o t spaces correspon-
. There
i s a well defined
p ( x ) 2 = ( d e t Ad,&jb (XI ) - I
G(g)-stable, and t h a t t h i s i s e s b e n t i a l l y t h e d e f i n i t i o n of
(note t h a t
G ( ~ ) E ) .Then
Tg,T i s t h e r e p r e s e n t a t i o n by l e f t t r a n s l a t i o n s i n t h e space of f u n c t i o n s
on with values i n t h e space of
-r which s a t i s f y $(xy)
b
= p(y)-l~(y)-l$(x)
$
for x
E
G
,y
and p * X = 0
E
G(g)
for X
(note that P(~)-'
T (y)-l
in the nilradical of
b
is well defined on G(g))
. (By
,
the Borel-Weil theorem,
ifG is connected,T is the dual of irreducible representation with lowest g weight ig + p). 4 . Assume that
g is reductive, and G in the Harish- Chandra class (which
means here that Ad x
is for any x
c
G
an inner automorphism of
9 )-
Then, the conjecture is true : (iii), (iv) and the irreducibility of T g rT are dueto Harish-Chandra C71 , and (ii) to C121 Note that one has to be
.
careful when comparing our parametrization of the relevant part of G with Harish-Chandra's. Ours seems more natural (although more complicated) because it does not involve a choice of positive roots as for instance in C71
,
I par. 27. It is interesting to remark that the validity of (iv) contains for instance 171 I11 cor. p. 164 , including the exact value of the constant
C~ '
It is reasonable to guess that the validity of Langland's conjecture (cf. c-131) will allow us to extend these results to the case where G
is reductive, but
not necessarily in the Harish-Chandra class (as we have done for compact groups in example 3).
3. Non aZgebraic groups :
The results ofparagraphs 1 and 2 b are easy to generalize to non algebraic algebras. It is known how to generalize those of paragraphe 3 c to connected solvable groups (cf. C31 ). Consider first the case of a Lie group G with a finite number of connected components, and locally isomorphic to an algebraic group. Then one has to make two modifications in the formula (1).
First, it may happen that the
compact part of the connected center of GO is not a direct factor. In this case, one decomposes the regular representation of G using the Fourier
transform on t h i s subgroup, and f o r each c h a r a c t e r of this.subgroup one w r i t e s a "projective" P l a n c h e r e l formula analoguous t o ( 1 ) . happen t h a t
x
~
Secondly, i t may
~ i~s n(o t ~f i n )i t e . Never t h e l e s s it i s provided w i t h a
canonical Plancherel formula (because
G(g)/G(g)O
i s d i s c r e t e ) which r e p l a c e s
t h e sum i n formula ( 1 ) . Remark t h a t ( i i i ) and ( i v ) , modified a s above, imply f o r l o c a l l y a l g e b r a i c groups with a f i n i t e number of connected components p r e c i s e c o n j e c t u r e s f o r t h e e x i s t e n c e , p a r a m e t r i z a t i o n , and t h e formal degree of square i n t e g r a b l e r e p r e s e n t a t i o n s . They a r e coherent with t h e r e s u l t s of N. H. Anh [ I ] When
G
.
i s not l o c a l l y a l g e b r a i c w i t h a f i n i t e number of connected components,
t h e r e a r e s e r i o u s problems due i n p a r t i c u l a r t o t h e f a c t t h a t some groups a r e not n e c e s s a r i l y of type I , and o r b i t s not l o c a l l y closed. See C31 t o unders t a n d what happens i n a concrete
example (connected s o l v a b l e groups).
1. Le lenmre 1 (i) et son corollaire sont un cas particulier d'un resultat de
V. Kac ("Infinite root systems, representations of graphes and invariant theory", Invent. Math. 56 (1980) 57-92), valable pour toutes les representations lineaires des groupes algebriques. 2. Examinons ce qui a Bt6 fait sur la "conjecture" du paragraphe 2. c. ( 0 ) C o m e toutes les formes lineaires fortement regulilres sont bien
polarisables, ces notes fournissent les representations T g ST (i) On peut demontrer que ces representations verifient (i). C'est mtme vrai en supposant seulement g bien polarisable. (ii) I1 n'est pas vrai $ue toutes les orbites
$2 E
g * / ~ soient a
tempLr6es. I1 faut se restreindre P un ouvert de Zariski de g* que rencontre
.
tous les R Avec cette restriction elle est vraie dans le cas resoluble j (come on le ddduit de resultats de Pukanszky et de Charbonnel-Dixmier). ag * / ~ est tempGree, la formule pour la trace de Tg ST est vCrifiee par M.S. Khalgui dans le cas des groupes connexes P radical co-compact,
Lorsque !2
t:
et est bien probablement vraie en general compte-tenu des rdsultats de
W. Rossmann [I21
.
(iii) La demonstration du theoreme 2 chapitre VII peut ttre modifiee
2
avec g c et r c g ST 2 suffisent B decomposer L (G), ce qui est un premier pas vers la formule (1).
pour montrer que les representations T
(iv) Cette partie de la conjecture peut
ttre gSndralis6e de la manilre
suivante aux T non fondamentaux : la formule j pj(gn.~) = ( (~(g~)/~(g~)~))-' dim T
est asymptotiquement vraie quand
gn
tend vers
m
. Sous cette forme, elle
donne une interprdtation intdressante des constantes apparaissant dans les cas connus de la formule de Plancherel
-
en particulier pour les groupes de
.Lie semi-simples et la formure dlHarish-Chandra.Dans le cas des series principales sphdriques des groupes de Lie semi-simples reels, R. Mneimnei a vdrifi6 que cette interprdtation est correcte.
REFERENCES
N. H. Anh. Lie groups with square integrable representations. Annals of Math. 104 (1976) 431-458 P. Bernat and al. Dunod, Paris 1972.
. Representations
des groupes de Lie resolubles.
J. Y. Charbonnel. La formule de Plancherel pour un groupe r6soluble connexe 11. Math. Annalen. 250 (1980) 1-34. Duflo. Sur les extensions des reprgsentations irreductibles des groupes de Lie nilpotents. Ann. Sci. Ec. Norm. Sup. 5 (1972) 71-120.
Y-
M. Duflo. Operateurs differentiels bi-invariants sur un groupe de Lie. Ann. Sci. Ec. Norm. Sup. 10 (1977) 265-288. M. Duflo and M. Vergne. Une propri6te de la representation cotadjointe d'une algsbre de Lie. C.R. Acad. Sci. Paris. 268 (1969) 583-585. ~arish-~handra. Harmonic analysis on real reductive groups I. J. of Functional Analysis 19 (1975) 104-204. 111. Annals of Math. 104 (1976) 117-201. A. A. Kirillov. Representations unitaires des groupes de Lie nilpotents. Uspekhi Mat. Nauk 17 (1962) 57-110. A. A. Kirillov. The characters of unitary representations of Lie groups. Functional Analysis and its applications 2 (1968) 40-55.
A. Kleppner and R. L. Lipsman. The Plancherel formula for group extensions 11. Ann. Sci. Ec. Norm. Sup. 6 (1973) 103-132. R. W. Richardson. Deformations of Lie subgroups and the variation of the isotropy subgroups. Acta Math. 129 (1972) 35-73. Rossmann. Kirillov's character formula for reductive Lie groups. Inv. Math. 48 (1976) 207-220.
W.
W. Schmid. ~ ~ - c o h o m o land o~~ the discrete series. Ann. of Math. ( 1976) 375-394.
103
CENTRO INTERNAZIONALE MATEMATICO E S T I V O (c.I.M.E.)
ON A NOTION
OF RANK FOR UNITARY REPRESENTATIONS OF THE
CLASSICAL GROUPS
ROGER HOWE Yale University
R e s e a r c h supported p a r t i a l l y by NSF grant M C S ~ ~ - 0 5 0 1 8
1. Review of the Heisenberg group and the oscillator representation. The goal of these lectures is to introduce some general concepts concerning unitary representations of locally compact groups, and to apply these concepts to the study of representations of semisimple Lie groups, especially the symplectic group. Historically, what we may call "general representation theory", on one hand, and the representation theory of semisimple Lie groups on the other, have, with some notable exceptions, tended to go their separate ways; general concepts have not proven very powerful in semisimple harmonic analysis, which has needed its own special methods.
The key phenomenon permitting at least a partial
merger here is the oscillator representation of the symplectic group. Although this representation has received increasing attention in recent years, it may be still unfamiliar to some. Because of that, and because it will play such a pervasive role in the present study, I will begin by reviewing the basic definitions and salient properties of the oscillator representation. General references for the following material are [Cr], In1 I , twill. Let F be a local field assume F
- k,
a, or non-Archimedean. We will
is not of characteristic 2.
We let W be a vector space of
dimension 2m over F, on which is defined a symplectic form < choose a basis
where
f ei,fi)y=l
6ij is Kronecker's
, >.
We may
for W, such that
6. We will suppose such a basis chosen once
and for all. We will refer to it as our standard symplectic basis.
Taking
coordinates with respect to the standard basis identifies W with F'~,
and we will make this identification when convenient. The group of linear isometries of the form < symplectic -of SP(W, c
, > 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<
... < k8 5 n
P
5 Sp(W)
denote t h e unipotent r a d i c a l of
N
P.
Nm-rank.
We begin by d e s c r i b i n g t h e p o s s i b l e p a r a b o l i c s
0 < k
Let
ei
for
be a sequence of i n t e g e r s .
P.
A s i n $1, l e t
i 5 k.
Let
We c a l l t h e
n e s t e d sequence of subspaces
a s t a n d a r d flag.
By
P(F,W) = P(F), we mean t h e subgroup of
preserves a l l t h e subspaces of to
P(F)
N(F)
of
P(F)
and on
5Ci1t-,
P(F)
We a b b r e v i a t e k < m.
that
Each p a r a b o l i c subgroup i s conjugate F
.
The unipotent r a d i c a l
c o n s i s t s of t r a n s f o r m a t i o n s which a c t a s t h e i d e n t i t y on
obviously those
f o r which
GC/xkC. The maximal p a r a b o l i c subgroups a r e F
P({Xk)) = Pk.
=
(Xk)
is a singleton.
We review t h e s t r u c t u r e of
Pk
We have t h e decomposition
Here as above Wk
i s t h e span of
orthogonal complement i n W.
N({%},
.
f o r a unique standard f l a g
each q u o t i e n t
for
F
Sp
W) = Nk(W) = Nk.
{ei,fi)
for
A.
Wk
is its
We have a l s o abbreviated
The u n i p o t e n t r a d i c a l
It f i t s i n a n e x a c t sequence
i 5 k, and
Nk
i s two-step n i l p o t e n t .
Here
Nk.
ZNk denotes t h e center of
It i s n a t u r a l l y contained i n
Sp(Wk), and i n f a c t we have
The quotient
The f i n a l isomorphism i s described by formula (1.7). homomorphism from
Nk(W)
to
Hom(w;,
defines a map from W;
x € Nk(W), then x-1
To f i n i s h t h e d e s c r i p t i o n of
consider i t s a d j o i n t
Given
we describe t h e 2 S (Xk)-
T € H ~ ~ ( W ; , X ~ )we , may
* , (Wk) I * ).
, > , the
r e s t r i c t i o n of t h e form <
Then
.
Hom(Wk, Xk)
T* € Horn((%)
\.
This w i l l e s s e n t i a l l y be an
L
valued b i l i n e a r form on
to
Nk(W) = Nk
commutator of two of i t s elements.
* (Wk) .
i s r e a l i z e d by observing t h a t i f
Xk)
space W;
But by v i r t u e of t h e
i s isomorphic t o i t s dual
Explicitly, define
a
-1
0
T
*
maps
T, S € Hom(wt,xk)
<
I
to
Wk.
Hence given two maps
we may form T
a
o
-1
o
*
S
*
€ Hom(Xk,Xk)
We compute
-1
(Tea
since (3.5)
a
* = - a.
* * = S ~ ~ - ' * O* T=
0 s )
- S e a
Therefore ~
o
-1
c
r
+,s*
*
- ~ o a - l o ~
-1
.T
*
i s s e l f - a d j o i n t , and s o may be regarded a s an element of
s2(X,).
b i l i n e a r form (3.5) is t h e commutator form on Hom(~;, Xk)
.
The
J u s t a s i n t h e formula (2.1) we may i d e n t i f y t h e P o n t r j a g i n dual
(q) " with
2* S (Xk).
t h e n a t u r a l a c t i o n of Pk
centralizes
* Ad Pk
Z Nk
orbits i n
The a c t i o n of on
GL(X2
on
2*
(\I.
S
denote t h e q u o t i e n t of
.
On
Horn(~t, Xk),
Let
Providing
complement t o
S2
and c of
N(X,)
B # 0,
(5)
,
W in
subspace
B
in
t h e group
HO~(W;,\),
zg i s non-degenerate
NB
%
$ and consider
ZNk.
Let
N
B
on
SO
W
N
ZNk
obtained by d i v i d i n g
t h e commutator form of
R
of
(Z Nk) ".) Thus t h e
Choose such a form
Xk.
8
by t h e k e r n e l
w i l l s t i l l be two-step n i l p o t e n t . N
denote t h e r a d i c a l of t h e form <
I$
s~(w;)
( 2 N ~A ) correspond t o t h e isomorphism c l a s s e s of
i t i n i t s r o l e a s l i n e a r f u n c t i o n a l on
$
then becomes
(The subgroup
and s o a c t s t r i v i a l l y orl
symmetric b i l i n e a r forms on
of
(Z Nk)"
B
is clearly
, >B , and
W denote a
let
that
.
Observe t h a t t h e i n v e r s e image
i s isomorphic t o t h e Heisenberg group
RB may be l i f t e d t o a subgroup, a l s o denoted
H(W), while t h e
R
B
, of
N
8'
Thus we have
Let (2.1).
b e t h e c h a r a c t e r of ZNk attached t o $ a s i n formula 8 Then t h e r e i s a unique r e p r e s e n t a t i o n p of H(W) w i t h c e n t r a l
character
X
B
X
B '
It then follows e a s i l y t h a t a general r e p r e s e n t a t i o n of
which i s a m u l t i p l e of
N~
X
on
B
Z
Here we have made t h e convention t h a t
Rg , and,
i t be t r i v i a l on i t be t r i v i a l on
has t h e form
Nk
similarly,
by l e t t i n g by l e t t i n g
Np
H(W). Rp
.
Let
%
considered a s a symmetric b i l i n e a r form on Xk.
p
Np
is extended t o
$
It remains t o determine t h e space
of
is extended t o
PB
To s e e t h i s , it is convenient t o regard
be t h e r a d i c a l Then we have
a s a s e l f - a d j o i n t map
p
by t h e r u l e
Then map '1
Rp = ker L -1 T o a -1 a o S
*
0
B'
*
Given maps
S, T €
*
0
S
goes from Xk
o
LB
takes
Xk
to
H O ~ ( W$1~ ,
%.
to itself.
,
t h e composite
Thus t h e f u r t h e r composite Wemay r e w r i t e t h e formula
(3.6) a s
where tc
tr
i s t h e u s u a l t r a c e f u n c t i o n a l on
Then
H~(W;. RB).
a n n i h i l a t o r of image
I
Rg
.
R
P
in
a-1
xi.
Hence t h e map
0
S*
Since t h e map a-1
o
End(%).
* 6 Hom(Xk, w;) S*
C.
La
LB
Suppose
annihilates
S belongs
R;,
is s e l f - a d j o i n t ,
i s zero, w h i l e t h e map
the it has
s
o
a-1
o
T*
LB
0
has kernel containing
and must t h e r e f o r e have t r a c e zero.
i s zero independent of
cS,T>p
hand s i d e of e q u a t i o n (3.9)
R
B
and image contained i n
Thus we s e e f o r such t h i s i s t o say
T;
S E
R
R
B
S , t h e number
.
B
The l e f t
t h e r e f o r e c o n t a i n s t h e r i g h t hand s i d e .
The
v e r i f i c a t i o n of t h e r e v e r s e i n c l u s i o n is l e f t t o t h e reader.
p
For us that for
fl
R
of rank k , which of c o u r s e i s t y p i c a l , formula (3.9) t e l l s = 10)
B
of rank
k
, so
that
N
i s i t s e l f a Heisenberg group.
B
t h e r e w i l l be a unique r e p r e s e n t a t i o n
Hence of
p
B
Nk
such t h a t
p
on
Z Nk.
We know a l s o [MW] t h a t
these
a r e square i n t e g r a b l e modulo
ZNk.
The r e g u l a r r e p r e s e n t a t i o n
of
p
B
B
i s a m u l t i p l e of X
8
o r any s u b r e p r e s e n t a t i o n of i t , w i l l a s s i g n zero s p e c t r a l
ZNk,
measure t o t h e s u b v a r i e t y of
( z N ~ ) ' c o n s i s t i n g of forms of rank l e s s than
k. Together, t h e s e f a c t s g i v e u s one d i r e c t i o n of t h e following r e s u l t . Lemma 3.1: o n l y i f i t is Proof: implies
A representation
of
.-.
is
Sp(W)
Nk-regular i f and
ZNk-regular. A s noted, t h e remarks above show t h a t
Nk-regularity.
G 2H1 1 H 2
(T
ZNk-regularity
The o t h e r i m p l i c a t i o n i s q u i t e g e n e r a l .
be any l o c a l l y compact group and two subgroups.
representation
a
of
G
it is also
is H1-regular,
because t h e r e g u l a r r e p r e s e n t a t i o n of
H1,
m u l t i p l e of t h e r e g u l a r r e p r e s e n t a t i o n of Since
Z Nk(W)
= Nk(Wk)
is t o
Then i f a
Hz-regular.
restricted to
Let
H2,
This is
is a
Hz.
Sp(Wk)
as
Nm(W)
is t o
Sp(W) ,
we have a second r e s u l t a s a consequence of l e m a 3.1, lemma 2.9, and p r o p o s i t i o n 2.14. P r o p o s i t i o n 3.2:
is
Nk-regular
A representation
i f and only i f
C 2 k.
a
of
.-.
Sp
of pure Nm-rank
8
Remark: of
*
Sp
Since Corollary 2.11
allows u s t o decompose a r e p r e s e n t a t i o n
i n t o a d i r e c t sum of r e p r e s e n t a t i o n s of pure rank, p r o p o s i t i o n 3.2
i s e f f e c t i v e l y a c r i t e r i o n f o r a general r e p r e s e n t a t i o n of
t o be
gp
Nk-regular. Now consider a general p a r a b o l i c subgroup P ( n standard f l a g
F
.
Let
defined by some
.
F
Xk be t h e l a r g e s t subspace of
N(F)-regularity
r e s u l t of t h i s s e c t i o n g i v e s a c r i t e r i o n f o r
The main i n terms of
rank. P r o p o s i t i o n 3.3:
is
N(D-regular Proof:
A representation
i f and only i f
We n o t e t h a t
N(F).
", Sp
of pure Nm-rank
8
8 2 k.
P(n
c P(Xk),
and
i s a normal, though n o t c e n t r a l , subgroup of in
of
0
Nk
c N(n.
N(F), and
Also
i s a l s o normal
Nk
P r o p o s i t i o n 3.3 w i l l follow from
Z Nk
3.2 and t h e
following lemma. Lemma 3.4: only i f i t i s Proof: between N(F)
N(F)
which is
A representation
$
and
Nk.
up
It could be reformulated:
a r e p r e s e n t a t i o n of
Nk-regular i s equivalent t o a s u b r e p r e s e n t a t i o n of t h e
p
We w i l l prove t h i s .
2* 6 S (Xk). Nk
Let
p
have rank k.
with c e n t r a l character
Let X
B
p
B
.
be t h e I claim t h a t
2* S (Xk), t h e induced r e p r e s e n t a t i o n
o u t s i d e a proper s u b v a r i e t y of
i s irreducible.
N(F)-regular i f and
This r e s u l t has e v i d e n t l y o n l y t o do with t h e r e l a t i o n
unique r e p r e s e n t a t i o n of for
is
Nk-regular.
regular representation. Select
Sp
of
By Mackey's theory [My], t o e s t a b l i s h i r r e d u c i b i l i t y of
it i s enough t o show t h a t
~ d * r n ()~# p
B
B
for
m t N(F)
but not i n
Nk.
The decomposition (3.1) of
where
nGL(V.
N1 = N(F)
t o a n a c t i o n of
N
1
.
Pk
gives us
*
The a c t i o n of
Ad N(F)
ZN;
on
It i s c l e a r from t h e c o n s t r u c t i o n of
pe
factors t h a t we
have
where m($) of
GL(Xk)
i n d i c a t e s t h e transform of on
s2*(%).
simply show t h a t s u b v a r i e t y of of
$
N
I
t o e x h i b i t one
i n t h e standard a c t i o n
we must
a c t s f r e e l y on t h e complement of a proper c l o s e d Since t h e c o n d i t i o n t h a t t h e i s o t r o p y group i n
b e non-trivial $
by . m
Hence t o prove i r r e d u c i b i l i t y of
s2*(xk).
c' S2* (Xk)
p
N'
i s a n a l g e b r a i c c o n d i t i o n , it w i l l s u f f i c e
f o r which s a i d i s o t r o p y group is t r i v i a l . k {ei)i=l
I n t h e standard b a s i s
of
t h e elements of
Xk
a s upper t r i a n g u l a r m a t r i c e s with 1's on t h e diagonal. t o check t h a t f o r any i n n e r product
p
on
a n orthogonal b a s i s , t h e isotropy group i n
N
of
1
appear
It i s then easy
Xk f o r which t h e I
N
ei
form
p is t r i v i a l .
This
e s t a b l i s h e s t h e claim of g e n e r i c i r r e d u c i b i l i t y of '
Now c o n s i d e r a r e p r e s e n t a t i o n N
k-
regular.
Then
representations
CJI P~
'
o. of
N(F).
Suppose
0
is
w i l l decompose a s a d i r e c t i n t e g r a l over t h e
Nk
for
p
of rank k
in
2* S (Xk), and t h e s p e c t r a l
measure of t h i s decomposition w i l l be a b s o l u t e l y continuous w i t h r e s p e c t t o Eaar measure on
s2*(%).
The s p e c t r a l measure moreover allows
automorphism group, s o t h a t it is of c o n s t a n t m u l t i p l i c i t y along It follows t h a t
alNk
i s quasi-equivalent t o
N
N' I
as orbits.
But since u
p
is irreducible for almost all
regular representation o of N(F) Nk.
Therefore
a
, we
see that any
Nkis determined by its restriction to
is quasi-equivalent to
and this representation is obviously N(F)-regular.
Indeed, if G
2H
is a group and a subgroup, then it is easy to see that
It follows that any subrepresentation of the regular representation of H induces a G-regular representation of G. Remark:
In fact, the above analysis, in conjunction with lemma
2.9 allows us to determine the of
.Sp
This concludes lemma 3.4.
of pure rank
?
k
N(F)
spectrum of any representation
from its Nm-spectrum.
2 63
4:
Description of
(&I)'
8 '
Fix a symmetric b i l i n e a r form of
and
( i p ) l be t h e subset
Let
'
B '
Let t h e p a r a b o l i c
Pk 5 Sp be a s defined i n (3.1).
Theorem 4.1:
a be a r e p r e s e n t a t i o n of
Then i f
8 c m.
a
(zp)
on X.
The business of t h i s s e c t i o n i s t o
defined a t t h e end of 52.
provide a d e s c r i p t i o n of
cr(ik)
$
Let
k 2 4
, t h e (weakly
-
Sp
closed) algebra generated by
i s equal t o t h e a l g e b r a generated by a l l of
a r e equivalent i f and only i f Proof:
C
and I r r e d u c i b l e , then
,.
ol lpk
As in theorem 2.10,
cr(gp).
In particular
a ) i k is i r r e d u c i b l e .
is i r r e d u c i b l e i f and only i f
a2 a r e of pure Nm-rank
of pure Nn-rank
'"
and
u2 Ipk
And i f
al
a2
and
a r e equivalent.
t h e proof i s by induction.
t h e theorem i s t r u e , s i n c e from [EM] we know t h a t
ol
"
For
C = 0,
A
( S P ) ~ c o n s i s t s of t h e
t r i v i a l r e p r e s e n t a t i o n alone. C P 1,
Assume t h e r e f o r e that r e s t r i c t i o n of
o to
",
P1.
i n which case m 1 2 .
Consider t h e
According t o t h e argument of theorem 2.10,
a / i l t h e decomposition (2.8), which we reproduce:
we have f o r
Xere t h e
vi
scalars.
(This follows from the d e s c r i p t i o n of t h e
t
formula (2.8),
a r e r e p r e s e n t a t i o n s of
and c o r o l l a r y 2.13,
r e p r e s e n t a t i o n s of
Gp(q)
the
Consider f i r s t t h e c a s e
iXa c t s
since the
following
i a r e of pure rank.) As vt
i vt a r e of pure C = 1 and
v:
by
~*~(w>-rank
k = 1. Then t h e
C
- 1.
i vt a r e
of rank 0, s o t h a t they a r e simply lrmltiplee of t h e t r i v i a l representation. Hence we a c t u a l l y have
i nt a r e non-negative i n t e g e r s .
where t h e
w
c a s e amounts t o showing t h a t p
.
Consider
n
Sp(W1)
i p (w)
We n o t e t h a t *
Pl(W) = Pl(Wl).
algebra a s
u
o(?,(w)
g e n e r a t e s t h e same a l g e b r a a s
o(P1)
5 gP(w).
iP(wl)
To prove t h e theorem i n t h i s
Suppose t h a t
C Y ( ~ ~ ( W Then ~)). Sp(w1)).
But
o(P (W ) ) 1 1
o(? (W)) 1
F1(w)
generates t h e same
g e n e r a t e s t h e same a l g e b r a a s g e n e r a t e s t h e whole group
U iP(w1)
. Thus t h i s c a s e of t h e theorem w i l l f o l l o w i f we show
o(:l(~l))
= o(SP(w1)).
From formula ( 2 . 7 ) ,
Since
i s 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 . gp(w)
t o a subgroup of
iP(w>,
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
<.
and
On t h e o t h e r hand,
a non-square have d i s t i n c t we s e e t h a t t h e
<
and
t
Y
{ % .N1(W1) 'I} 5 F1(wl)
on
a s given by formula
L (Y)
The r e p r e s e n t a t i o n s
t o b e i n e q u i v a l e n t on
f o
2
can be taken a s t h e even f u n c t i o n s :
u+
2t
4 decomposes i n t o 2 i r r e d u c i b l e components
N1(W1)-spectrum
a;
is
of m u l t i p l i c i t y
2 supported on
Ops
1 s = -($t,
where
each of which c o n t a i n s
and t h i s i s divided among t h e
w i t h m u l t i p l i c i t y 1. With
Ops
spectrum of m u l t i p l i c i t y one on one
A ~ * ? ~ ( wo~r b) i t ,
N1(W1)-
i r r e d u c i b i l i t y of
s
on P1(W1)
i s immediate.
Remark:
The arguments j u s t r e c i t e d may be regarded a s t h e r e v e r s e
of t h e arguments i n t h e proof of theorem 2.10 leading t o t h e c o n s t r u c t i o n i -ct.
of t h e
Also, they j u s t i f y t h e comments made t h e r e about t h e
connection between t h e
We now know t h a t t h e a s r e p r e s e n t a t i o n s of
i
zt.
and t h e
' a t
co;
gl(wl).
a r e i r r e d u c i b l e and p a i r w i s e i n e q u i v a l e n t Since o n l y they occur i n
i n t h e c a s e a t hand, t h e d e s i r e d f a c t , t h a t
,.
same a l g e b r a a s
generates the
a /(Fl(wl))
(namely t h e sum of t h e f u l l m a t r i x a l g e b r a s on
a(Sp(W1))
t h e spaces of t h e
U ~ S ~ ( W ~ )
ui which occur i n
a).
This proves t h e theorem when
C=l, k = l . I n a l l o t h e r c a s e s we w i l l have as
C -1.
k-1,
C -1 and
as
i
-
.
i
1 @ zt(g)
gX -
, where
N1(W).
g
-cl t
Hence f o r
FX . N1(W)
?k-1 (
o(ibl(<))
and
I
g F Sp(W1), t h e o p e r a t o r s
o(FX
. N1(w;)).
.
1~ 3 Hence t h e o p e r a t o r s
o(jx
. Nl(W)).
we conclude t h a t t h e o p e r a t o r s
simply by
il(w)
a r e a l r e a d y i r r e d u c i b l e and pairwise
h e r e i s t h e i d e n t i t y on t h e space of by
of
1 h e r e i s t h e i d e n t i t y on t h e space of
l i e i n t h e a l g e b r a generated by
is true for
k-1,
-
In particular, the
d i s t i n c t on
k-1 2 1 a s w e l l
i v ~ ( P ~ - ~ ( w ; )w) i l l generate t h e same a l g e b r a .. i But t h e r e p r e s e n t a t i o n s T~ of P1(W) were obtained
That i s ,
from r e p r e s e n t a t i o n s of t h e subgroup extension.
1, and t h u s
By induction we may assume t h e theorem i s t r u e f o r
rn-1.
v t (SP(w;))
k
-ct,
i vt,
already
I n particular, t h i s i vt (g) @ 1. where
1
a r e i n t h e a l g e b r a generated
By t h e previous c a s e of t h e theorem,
i vt(g) 8 1 f o r a l l
g 6 Sp(w;)
are i n the
. FX . til(W)).
U(;~-~(W;)
algebra generated by
Hence f i n a l l y t h e
operators
for a l l and
g
c
&(q) a r e i n t h i s algebra. ZX
a($k-l(~;)
N 1(W))
-
Thus we have shown
generate t h e same algebra.
u(P1)
But now
observe t h a t
Hence
o(it(w)
Hence
u(Gk(W))
n il(w))
generates t h e same algebra a s
generates the same a l g e b r a a s
Since k z 1, t h e union
il(w) U
gk(W)
o(il(w)).
o(G1(w)u
ik(w)).
gp (w) .
generates a l l of
The
theorem follows. Theorem 4.1
immediately implies t h a t r e s t r i c t i o n induces an
i n j e c t ion
k 2 t
when
.
I n f a c t , a much sharper r e s u l t holds, a s we s h a l l now
show. We focus a t t e n t i o n on rank 0
8 c m.
Consider
0 (
i s concentrated on t h e
(ip)"
8
(iP);
*
f o r a symmetric form
.
8
of
By d e f i n i t i o n , t h e Nm-spectm of
Ad Pm-orbit
a
Op 5 Nm.
W e may assume t h a t
the form
i e non-degenerate on X
t and so d e f i n e s a n inner product on XC
W e b o w from theorem 4.1 t h a t t h e r e s t r i c t i o n of
u
.
t o t h e maximal
parabolic subgroup P E (W) structure of NE(W)
is irreducible. We want to investigate the
a l ~ ~ Let .
of PE(W).
Z NE be the center of the unipotent radical
Z Nc-spectrum of
lemma 2.9 tells us that the
* Ad PL(W)
0 c (2~~)'.
orbit
denotes the
Z NE = NE(W) fl Sp(WE) = NE(WL).
We know that
with Y
Y-
u
Hence
is concentrated on the
as in equation (4.1).
If
\
Z N E-spectrum of u , then nu satisfies a transformation
-
But equation (2.2) says n
law analogous to equation (2.2).
system of imprimitivity in the sense of Mackey, for P8(W)
is a u and a
.
,
*
based on 0 Let J 5 (W) be the isotropy group of Y under Ad P Y E 8' Then Mackey's Inprinitivity Theorem says that crljL(w) is realizable as a representation induced from some representation
7
of J.
In symbols
We will make (4.2) more precise by giving sharper descriptions of J and 7.
We have the decomposition (3.1) of P (W). It lifts to a 8 decomposition of iE(w). The subgroup ip(~t ) - NE(W) of iC centralizes N8
, SO it will belong to J. Clearly the subgroup of GL(XE)
leaves the inner product Y fixed is just 0
Y'
that
the isometry group of y
.
Hence we have
Consider the representation
7
of J.
Mackey's theory implies that
712 Nt is a multiple of the character Since y is non-degenerate on XE tell us that
TIN
,
as defined in formula (2.1). Y' the arguments leading to formula (3.8) X
E (W) is a multiple of the unique representation
NE with central character
X
Y
.
p~
of
Recall that p actually factors to the group Y
the Heisenberg group attached to Hom(WE, X8), with symplectic structure given by formula (3.6). The action by conjugation of Sp(W8)
on NE factors to an Y action on N It is evident that this action maps Sp(W8) 0 into Y' Y Sp(Hom(W8, XE), c , > ) as a dual pair. Let us abbreviate Y Sp (Hom(W C, XE), c ,> ) = Sp We know that the representation p of Y Y Y N extends to a representation, still to be denoted py , of the semi-direct '
0
.
Y
,.
.
.
product Sp N Obviously we want to convert p into a representation Y Y Y of J. It is fairly clear how to do so, but we encounter the same technicalities as we did in theorem 2.10.
We summarize the details.
,.
The formula (1.18) says that Sp(W8)
may be mapped into
,.
by taking an appropriate E -fold tensor product of oscillator Y Y ", ,. representations. Pulling back by p maps Sp(WE) into Sp Since p (Sp )
-
Sp(WE)
Y
Y
.
is its own commutator subgroup, this mapping may be characterized
.
as the unique lift of the natural embedding of Sp(W8)
into Sp Y is contained in Sp(W ), and the action by conjugation of
Similarly, 0 Y E 0 on NE/Z N8 may be regarded as the (m-8) fold direct sum of the Y natural action of 0 on Wt. Thus formula (1.15) tells us 0 maps Y Y into py(Spy) by an (n-!,)-fold tensor product of the restriction of
,.
-
,.
the oscillator representation of Sp(W8). Thus we can map both Sp(wE) the representation p
Y
semi-direct product
and
5Y
into
gpY , so we can extend
to a representation, still denoted p
Y'
of the
to factor to a representation of J, we must have
In order for py
Let Z2 denote the common
a compatibility condition as in theorem 2.10. kernel of the projection maps
Z2 is mapped into
ipy
Sp(wi)
+
SP(W;)
and
iy
0
+
Y'
This group
&(%)and as a subgroup
both as a subgroup of
order for p to factor to J, these two mappings of Z2 Y must coincide. Let E be the non-identity element of D2. By the tensor of " O n Y'
.,
product descriptions of the mappings to Spy, we know that via the ;p(w>
pY(c)
mapping we get
= ( -I)',
while via the ' Oy mapping we
Therefore p will factor to J Y even, and will not factor if m is odd. pY(r)
get
=
character of
is
.,
*
-
When m
if m
(-l)m-e.
is odd, we may alter the mapping of 0 to Sp by a Y Y Oy of order 2, non-trivial on Z2. This modification will
(Jp(wt)
then produce a representation of
x
5y).Ay
satisfying the
compatibility condition. Hence in any case we can produce an extension, now denoted
p ;
, to
indicate the possibility that some modification
of the "natural extension'' may have been necessary, of py J.
The representation
from N to Y is def'ined up to modification by a character
p;
of order 2 of 0
Y'
Having constructed the representation
p ;
, we
continue according
to Mackey's Theory. It tells us that any representation is a multiple of X
where
Y
on
ZN'
of J which
has the form
rl is a representation of
will be irreducible if and only if applies to the representation
7
7
iP(wt) .5y Tl
.
The representation
r
is. The factorization (4.5)
of equation (4.2).
But for that
7
",
the representation
Tl
must be trivial on SP(W>.
This may be seen
,
i n s e v e r a l ways.
For example, we n o t e t h a t
NmcJ. I n
fact
An a n a l y s i s p r e c i s e l y analogous t o t h a t l e a d i n g t o formula (2.12) y i e l d s
t h e conclusion t h a t t h e
zl.
Nm-C(~>-rank of of
Nm-rank of
Hence f o r our
z1 must be zero, whence Hence, f o r our
z
T
i n (4.5) is 8
z from (4.2), t h e Nmm8 ( ~ 2 - r a n k
z11~p(w;)
, the
plus the
is t r i v i a l , a s stated.
representation
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 then r e q u i r e s t h a t
z1 f a c t o r s t o ol
-Oy.
But
factor further t o
O1'
Thus we have a sequence of mappings
Combining a l l t h e s e g i v e s us a map
Theorem 4.2: (S~(W)); into Proof:
6Y
The map
A
of (4.7) d e f i n e s an i n j e c t i o n from
.
This follows d i r e c t l y from theorem 4 . 1 and Mackey's
general results. To a c t u a l l y compute
A(o)
for
u € (ap);
according t o t h e above
r e c i p e would involve going through a l l t h e mappings of sequence (4.7). O f t h e s e , t h e f a c t o r i z a t i o n (4.5)
i s t h e most d i f f i c u l t step.
It is i n
general no very s t r a i g h t f o r w a r d matter t o f a c t o r a r e p r e s e n t a t i o n i n t o a
t e n s o r product.
However, t h e s p e c i a l s t r u c t u r e of t h e Heisenberg group
allows u s t o compute
' c ~= A(u)
i n a r e l a t i v e l y e f f i c i e n t manner.
R e c a l l t h e d e s c r i p t i o n (4.4) of
The summands w i l l be t h e spans of 8
+
1 5 i 5 m, r e s p e c t i v e l y .
~ o m ( c ,XE)
h.
N8/ZN,
of
Hom(Xm ll fact
Nm
Nm
n
abelian).
Hom(Xm n
NE
, so
Finally note that
Nm
n N,
C a l l t h i s complement
WE
X,)
@
Hom(Ym fl
w,;
X,)
and
Z
Xm
NE
m and
fi
for
n W;
w,;
Ny
i t s image i n
.
of N Y Ny
.
is normalized by in
Xe)
i s t h e image i n
(Note t h a t i n
is
a fortiori
0 and by Y
s ~ ( w > . Of course
i s a l s o normalized by t h e s e groups.
N,
a unique complement t o
with both
+15iS
This decomposition induces another:
abelian i n
n
8
i s a maximal a b e l i a n subgroup of
P ~ - , ( w ~ ) ,t h e s t a b i l i z e r of
Z N 4 -c Nm
for
Also, t h e i n v e r s e image i n
NE.
w;, X8) n NE i s
ei
Hom(Xm n ,;w
It i s n o t hard t o check t h a t
We may w r i t e
N Y'
In
Nm fl NE
there is
which is a l s o normalized by t h e s e groups.
WE.
Then we have
Z N8
normalized by
and by
0
The following method f o r computing
Y
Tl
Pm-8(~t).
could be formulated purely
i n terms of H i l b e r t space c o n s i d e r a t i o n s by f u r t h e r use of Mackey's theory, but t h e a l t e r n a t e formulation chosen seems s l i g h t l y more convenient i n t h e present c o n t e x t . p
Let
Y be t h e H i l b e r t space on which t h e r e p r e s e n t a t i o n
of equation (4.5) i s defined, and l e t
factor
't1
on t h e r i g h t hand s i d e of (4.5).
ffl
be t h e space of t h e o t h e r Then t h e space of t h e t e n s o r
product
is of course ff
T
3
ffl
Y.
@
but it is already irreducible under the action of quotient N
Let
Y'
N
Y-
Then
Y'
taken to be
Y°o
then
Y- 5 Y
Y
The space
is a module for J, acting through its
NC
denote the subspace of smooth vectors for
has a natural nuclear locally convex topology.
L 2 (NC/(NC
fl Nm))
If
Y
is
with action analogous to formulas (1.41,
is the Schwartz-Bruhat space
in ff
the space of smooth vectors for N
Y
.
S(N~/(N~fl N ,)
In any case
is clearly
The tensor product is well-defined as a topological vector space since
Y
-
is nuclear. Let
to NC fl Nm
X'
Y
from Z NC Y of equation (4.9).
denote the extension of the character X
which is trivial on the complement
[Hl],[Cr] says there is a
A basic fact about the Heisenberg group unique linear functional A
ym
on
UC
such that
*
-
#"
The character X ' is clearly invariant under the action Ad (Pm-t(WC)'Oy) Y Therefore, since it is unique, the functional X of on (NC fl N ~)'. ",
#"
equation (4.11) is an eigenfunctional for
Pm-80Jt)'Oy
.
Let
$ be the
#"
associated (quasi-) character of of
ffM/(ffl @ ker A)
" D n
Y
0
H1
Y'
Then we see easily that the action
is just
p1 8 4
We may further observe that, since Nm+(WC) subgroup of
4-&(~;),
Nm,
is in the comnutator
it will actually leave the functional A
From equation (4.6) we therefore see that of
.
and we may write
invariant.
X is an eigenfunctional for all
where X
is the character of Nm which extends
B
and is trivial on
X' Y
(w?. (Recall y is related to the original form $ by equation Nm-~E (4.1).) We may retrospectively assume that the radical of p is exactly Xm
v8. Then our use of L
X
B
in (4.12) is consistent with our earlier
definition (2.1) .) Let us put
Then we may take the following point of toward the representation that it is the extension to J'
of NmVNE
.
of the representation
The space of smooth vectors for N ; N 8
of smooth vectors as for N
E
above.
is just the same space
In summary, we have the following
result. Proposition 4.3:
The representation of 0
Y
H-/ (ff where A (4.11),
on the space
o ker XI ,
is the Nm-eigenfunctional on the space Y-
is equivalent to the representation
is as in equation (4.5) and
T
O
defined by equation where
JI is the eigencharacter of 5
Y
T~ =
~(u)
defined by
A. Remarks: a) The analysis leading to proposition 4.3 can be extended to conclude that
T~ Q\lr
is equivalent, as admissible representation.
to the action of O1 on the space of functionals on subrepresentation of o character
X
f"
a
, which are elgenfunctionals for
, the smooth Nm with eigen-
However, the technicalities leading to this result would
l e a d us too f a r a s t r a y .
b)
It i s obviously of i n t e r e s t t o know t h e image of t h e map
of formula (4.8).
A f a i r l y s t r a i g h t f o r w a r d argument based on t h e study of
t h e Well r e p r e s e n t a t i o n of spectrum of evidence t h a t
,.,
Sp(W)
attached t o
i s c e r t a i n l y in t h e image of
Oy
A
A
i s surjective.
0
Y
A
.
shows t h a t t h e tempered However, t h e r e i s
See f o r example t h e Onofri example
discussed i n $5. w
c)
Both
A
(SP(W))~ and
A
0
Y
have t h e s t r u c t u r e of t o p o l o g i c a l
spaces, defined by t h e F e l l topology [Fl]. some d e t a i l i n $6
and $7).
r e s p e c t t o t h e s e topologies.
It i s obvious t h a t
A
is continuous w i t h
The techniques t h a t show t h a t
whole tempered spectrum a l s o show t h a t homeomorphism onto i t s image.
(This w i l l be d i s c u s s e d i n
A
A
h i t s the
has a closed image and i s a
5:
Examples We offer some examples illustrating theorem 4.2.
Our discussion
will not include all details. The simplest situation is offered by the rank 1
representations.
dim W = 2m 2 4, theorem 4.2 implies that the only N -rank 1 m ,. + representations of Sp(W) are the two components at and at of
For
-
the oscillator representations themselves. to the trivial and the signum characters of
These correspond respectively
O1
N
{+I).
Altogether,
then, taking the different oscillator representations into account, we have
In particular, this is a finite set. Next consider rank 2 representations. representations of
Sp rather than of
representations of
Sp
Here we are dealing with
zp, and they are t)-iesmallest rank
(except for.the trivial representation).
According
to theorem 4.2 they are classified by representations of 2-dimensional orthogonal groups for dim W 5 6. of
Sp(W)
via the Weil representations associated to 2-dimensional
orthogonal groups for dim W 2 4. 02's
But theorem 1.1 constructs representations
All unitary representations of the
are matched to unitary representations of Sp; and for
dim W E 6
one can see by using proposition 4.3 and developing formulas (1.8) that r t
8,
is matched to
A-'(T)
c ip.
Thus a11 rank 2 representations of
Sp occur in this way. Among binary quadratic forms, all but the hyperbolic plane are
anisotropic, so that their isometry groups are compact. Thus the representations of
Sp coming from tnese forms make up a discrete set.
The Weil representations of Sp corresponding to these 0 !s have been 2 studied by Asmuth [Am].
For dim W = 4 and for the signum character of
02, one gets when F, the base field, is non-archimedean a supercuspidal representation of Sp, analogous to Srinivasan's representation for Sp over finite fields [Sn].
e10
For dim W 2 6, the range of validity
of theorem 4.1 for rank 2 representations, one gets no supercuspidal representations. Indeed, proposition 2.18 prwents representations of Nm-rank less than m from even being tempered, let alone cuspidal. We note that Asmuth identifies the components of rank 2 Weil representations for dim W 3 6 as being very small constituents of cerfain induced representations. For the hyperbolic plane (the split 2-dimensional form), whose "sometry group we label 01,1, we get a continuous series of representations instead of a discrete set. We will analyze this case in somewhat more detail. The group OlS1 has the semidirect product structure
-1 where Z2 acts on FX by sending t to t $
of 'F
.
Thus for each character
that is not of order 2, there is a unique 2-dimensional
representation
of O,,,.
w ~ . For
The restriction of p
$
$ €
iX
to FX is
$ 6B i l , and this characterizes
of order 2, there are extensions $+ and
9- of $ to
linear characters of
O1,l.
Here one has
ind Consider the Weil representation of Sp(W) associated to O1,l. 2 This may be realized on L (Y @ Y) with action given by formulas (1.8) on one factor and the complex conjugate formulas on the other factor. 2
Alternatively it can be realized on L (W) with Sp(W) linear action on W. A
(SP(W))(~,~)
.
acting via its
It is a direct integral of the representations in
The individual representations may be described as follows.
The action of SO =t # on L~(w) 131
is
Fix a character $ of 'F and consider the space
functions f on W
-
{O)
such that
o(t)f
= $(t)f,
YO.($)
of smooth
or in other words
Consider the evaluation map
defined by
where el is as before the first standard basis vector for W.
On the
parabolic P1 of Sp stabilizing X1, the line through el, define a rational character
al by
It is easy to compute that
Since the projective space @(W) to conclude that q
is isomorphic to
P1\Sp,
Y-(+)
defines an isomorphism between
it is easy and the smooth
vectors in the induced representation
Here we are using the convention of normalized induction, which sends unitary representations to unitary representations. 2 The action of the Z2-factor of 0 on L (W) is by a "symplectic 191 Fourier transform". We will not write down the exact formula. It is evident, however, that o(Z 2) must interchange ym($) and y m ( $ ' ) . -1 Therefore, when + # + the action of 0 x Sp on la1 -1 However, if \Jr = JI , ye(+) @J ym(+-l) is equivalent to 8 oy then o(Z2) provides an extra endomorphism of yM(\Jr), which therefore
.
?(++)
breaks up further into eigenspaces
and
ym(+-).
Since Sp
must respect this decomposition, we see a+ must in this case be reducible into 2 inequivalent parts. Theorem 4.2 prevents any further decomposition of the a
JI
Proposition 5.1:
'
In summary:
All irreducible representations of Sp(W),
dim W g 6, which have split rank 2 Nm-spectrum are constituents of indSP $ 0 al for some P1 irreducible, and A ( )~ = ~1
0=
JI
+
+
(
$
.
If
$
1 $-I, then
-1
If + = $
, then
+*
+
+
0
is
decomposes into 2
inequivalent constituents, and these correspond in some order to the extensions
$+
and
+-
of
+
to
OlS1.
For a more direct approach to the
a+
, see
[Fr] (Gs].
As a third example, we mention holomorphic representations of For our purposes, a holomorphic representation may be defined
SpZm(R).
as a representation whose Nm-spectrum is supported on positive semidefinite classes of bilinear forms. Theorems 1.1 and 4.2 yield the following result. Proposition 5.2:
The holomorphic unitary representations of
",
a
SP~~(R) of rank 8 < m are in natural bijection with 08. They are *
realized as summands of the Weil representation of Sp2,(R) to 08.
corresponding
These representations account for all holomorphic representations
.
of s ~ ~ ~ ( Rwhich ) are not Nm-regular
Our final example is rather different. a representation of OnY2(R)
instead of Sp.
rather than rank less than 2.
In the first place, it is Secondly, it has rank 2
(The ranks of representations of orthogonal
groups are always even integers, and the only unitary representation of rank 0 is the trivial representation.)
Therefore it may not be apparent
from our current vantage point that this example really illustrates the phenomenon under discussion. However it does, and very nicely. On that account, and because of its intrinsic interest, we present it. The example is constructed by Onofri [On]. It is a unitary 2 2m-1 representation p of S02m,,(!R) on L (S ), where sk is the k-sphere in Euclidean (k+l)-space.
Actually Onofri constructed a
representation for OnY2 for all n, not only even n, but it seems that for n odd, his representations must be representations of a covering group of SO
n,2
'
The maximal connected compact subgroup of
SOgm x SO2. The action of
p(S0
)
2m
is
is the natural action, and by the
classical theory of spherical harmonics breaks up into a direct sum
of irreducible representations, where the harmonic polynomials on spaces for
p(S02),
Hj
is the action of SO2,
of degree j. The
which acts on
Hj
by
, ' * ' x
on
ff. are the eigenJ where X is the
basic character of SO2. The subgroup SO~m,l acts irreducibly via and
p defines a principal series representation for S02m,l.
sphere S2m-1
is a homogeneous space for
S02m,1
p
,
(The
acting by conformal
transformations, and the principal series may be realized as "multiplier representations", i.e., actions on homogeneous vector bundles (in our case, line bundles) on S2m-1 ) It is clear from this description that p is a very small representation of S02m,2.
It has only a one-parameter family of K
types, as opposed to the typical 4-parameter family. It has functional dimension 2m-1, as opposed to the typical 4m-3. Moreover, it is the only known example of such a small representation of this group. How are we to understand this quantization of
p? Onofri constructed it as a
* the Kepler problem , inspired by
"geometric quantization".
the philosophy of
Here we point out its relation to the theory of
reductive dual pairs and the oscillator representation. We remark generally that geometric quantization and the oscillator representation are pretty clearly compatible, indeed deeply connected, but the relation is not well understood.
*It has been suggested therefore
[Zl] to call it the planetary representation.
Consider the dual pair
(OZm, (R)
, SL2(R))
.
This is a stable pair
in the sense of [H2], and so we get a pairing, as in Theorem 1.1, between the tempered dual of SL2(R)
and certain representations of O2m,z(~)'
These representations of 02m,2 have a 2-parameter K-spectrum and functional dimension 2n, so they are not Onofri's
p
.
However, if one takes the
distributional approach, one can look for other representations of OZmp2 corresponding to non-tempered representations of SL2. The least-tempered unitary representation of SL2, and the only one which is smaller than average is the trivial representation. And indeed, when we look at the representation of
O2rn,2 which is paired in the sense of [H3] with the
trivial representation we find it is the representation of 0 induced 2% 2 from Onofri's p on SOzm,z. 1t's restriction to SO is just 2m,2
P@
*
P
This is an encouraging fact. One thing it suggests is that the map
A of Theorem 4.2 may be surjective. This would be very interesting to know. However, at the moment while one can see that the representation a
of 02m,2
corresponding to 1 E SL2 will be uniformly bounded, no
a priori proof of unitarity now exists. Unitarity now comes only from Onofri's construction.
6:
kymptotics of matrix coefficients, generalities. In this section we switch our attention to a rather different
topic, namely, the asymptotic behavior of matrix coefficients of representations. The topic is also somewhat technical. But it has been of basic importance in the work of Harish-Chandra, and has had a number of surprising and important applications elsewhere [~n], [BW], 1221. We have already had occasion to use the weak results of [HM] in Theorem 2.10. We will see in $8 that these seemingly general and unrelated considerations are in fact closely connected with the theory of Nm-rank for representations of
Sp. We begin very generally. Let G be a locally compact group, and
let p be a unitary representation of G on a Hilbert space ff x,y
where
.
Given
€ ff, the function
(
,)
here indicates the inner product on
(left) matrix coefficient of
H , is
p with respect to x
called the
and y.
Let L and
R denote the left and right action of G on functions on G:
It is trivial to verify that
Hence for fixed y the map x - + ( P ~intertwines ,~ p with the (not unitary) left regular action L.
Similarly, for fixed x, the map y - C ( P ~ , ~
intertwines p with the right action R. bounded: one has the obvious estimate
The functions qxSy are trivially
where
11 11
indicates the norm in ff
.
lluch of this section will be
devoted to improving estimate (6.4) in certain cases. Here let us simply note that (6.4) implies we may convolve
with LI functions. Ipx,y Then it follows directly from
Suppose G is unimodular and f E L1(G). and x,y E
(6.3) that, for f € L ' ( G )
Here
where
g+
-
*
H ,
denotes convolution on G, and
- denotes complex conjugation. We are interested in the asymptotic behavior of in G.
If G
(Px,y(g)
as
is a simple algebraic group such as Sp and
not contain the trivial representation, then we know from matrix coefficients of
p all vanish at
closely at the rate of decay of
(Px,~'
-.
p does
[athat the
Here we shall look more
In this connection we entertain
several definitions. (6.7)
a)
p is said to be strongly mixing if for all x,y E ff
-
vanishes at on G. %,Y p is absolutely continuous if p is quasi-equivalent to a
coefficient b)
, the matrix
subrepresentation of the regular representation on L~(G). c)
p is strongly
if there is a dense subspace of ff
such
6 L~(G). (Px,~ For our last definition we need additional structure. Let K
that for x,y
in this subspace,
5G
A
be a compact subgroup, and let K be the unitary dual of K. We may decompose the space ff
of
p into an orthogonal direct sum
such t h a t
H
CL
equivalent t o of
in
p
of
p
n p
.
Let
function on such t h a t
is
K
The subspace ff
5G
and t h e a c t i o n of
f o r some integer
.
H
(6.8)
p
i s p(K)-invariant,
Cr
n
+ - , called
or
is t h e
K on
p-isotypic
be a compact subgroup.
+
Let
ff
P
the multiplicity
component of
or
K, and
I
+ (g-l) , and
let
Y
i f given
p, v
E K, and x f
(+,I)-bounded,
p
be a p o s i t i v e
G, i n v a r i a n t under l e f t and r i g h t t r a n s l a t i o n s by @ (g) =
is
be a function on K.
.
Ir
,y E
W e say ffV,
one
has t h e estimate
We l i s t below some straightforward observations concerning t h e s e concepts. a)
Because of estimate (6.4), a representation
mixing i f there i s a dense subspace of when
x
and
b) For i f
L (G)
Q
X¶Y
vanishes a t
y belong t o t h i s dense subspace. p
i s absolutely continuous, then
p
is strongly mixing.
u,v € L (G), then q ~ , , ~ ( g ' ) = fC, u(g) v(g1-'p)dg
-
denotes complex conjugation, and
An easy argument shows 2
such t h a t
w i l l be s t r o n g l y
2
(6.9) where
If
H
p
u r v
is strongly mixing.
*
vanishes a t
= u
v
-
*
* v*(g)
is a s i n (6.6).
2 f o r u,v E L (G).
Hence
Suppose
p = pl
and
ff = tll @ H2
x f
ff
,
@
is a d i r e c t sum of two s u b r e p r e s e n t a t i o n s ,
p2
is t h e corresponding decomposition of
write
x
=
+ x2 w i t h xi c Hi.
xl
Then f o r
fi
.
For
ff
x,y 6
Therefore a d i r e c t sum of s t r o n g l y mixing r e p r e s e n t a t i o n s i s s t r o n g l y Every a b s o l u t e l y continuous r e p r e s e n t a t i o n embeds i n t o a d i r e c t
mixing.
sum of copies of
L'(G),
Hence such r e p r e s e n t a t i o n s a r e
by d e f i n i t i o n .
a l s o s t r o n g l y mixing. c) mixing.
If
is strongly
For suppose
t h e function i.e.,
p
f
LP
for
f L' (Px,~
-
p c
for
, then
c ff.
x,y
1f
p
f f Cc(G), i . e . ,
i s continuous of compact support, then
vanishes a t
o..
Letting
f
is strongly
f
* cpxPy
c
Co(G)9
-a
r u n through a "Dirac sequence"
sequence of non-negative f u n c t i o n s each w i t h t o t a l i n t e g r a l equal t o 1 and
- and
whose s u p p o r t s s h r i n k t o t h e i d e n t i t y t C (G) f o r a dense s e t of (px9~ 0 p is s t r o n g l y mixing.
d)
If
p
This i s shown i n
is s t r o n g l y
[m]and
f o r a dense subspace of
H
u n i t a r y embedding of
.
Then
ff
in
L2, t h e n
i n [~a].
i n t e r t w i n i n g o p e r a t o r from have a c l o s a b l e graph.
x,y
Let
using formula (6.5) shows
p
By remark a ) , then, we s e e
i s a b s o l u t e l y continuous.
y f ff
cpy:x + cp
2
.
be such t h a t
ff i n e)
strongly
-
L'(G),
Y
is e a s i l y verified t o
Hence t h e g e n e r a l Schur's lemma [Ra] produces a A 2 (ker cpy) i n t o L (G). We may s p l i c e t o g e t h e r a
p
y's
e s t a b l i s h i n g t h e a b s o l u t e c o n t i n u i t y of
Since m a t r i x c o e f f i c i e n t s a r e bot nded, we s e e t h a t i f LP, t h e n
L'(G)
i s a densely defined
c o l l e c t i o n of such.embeddings coming from a dense c o l l e c t i o n of embed
(
2Y
X9Y
H t o L (G), and cp
cpx
i s strongly
Lq
for a l l
q 2 p.
p
p
We w i l l s a y
to
. is p is
s t r o n g l y L+' ' f)
if
If
is
p
is s t r o n g l y Lq
p
for
q > 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 <
such that
- (6)'
consists of the trivial
representation alone, providing a strong and quantitative version of Kazhdan's Theorem [ ~ n , ]
and further emphasizing the distinctive
nature of semisimple harmonic analysis. b)
A serious weakness of these corollaries is that they provide
sharp estimates only for even integral p.
In particular they provide
no distinctions between L ~ +and ~ L ~ .a very serious lack of resolution. It is possible by various ad hoc tricks to improve the situation for symplectic groups. However, it is natural to wonder whether, for any p 2 2, if a representation of G is strongly LP, is it then (E2Ip, (dim p) 2)-bounded ? We conclude by filling in the proof of proposition 2.18.
If
A
p
E G is tempered, then the K-finite matrix coefficients of
p
are
bounded by some constant times
%
, according to theorem 7.1, or the
estimates of Harish-Chandra. But Harish-Chandra has shown [HCL], [Sl] that the restriction of %
to the maximal unipotent subgroup N of
G is in L~+'(G).
PIN
Thus
, hence in
is strongly
absolutely continuous; or in other words, p
particular
is N-regular.
Remark: After this was written, I realized the argument for theorem 7.1 could be simplified, and the result improved to
(3,dimp)-boundedness.
Furthermore, Cowling showed me that the estimate for left K-invariant functions follows directly from elementary considerations, and does not need the Plancherel formula. Thus theorem 7.1 can be strengthened and given a much simpler, essentially elementary proof.
8:
Asymptotics of m a t r i x c o e f f i c i e n t s and rank f o r
Sp.
We f i n a l l y apply t h e r e s u l t s of t h e foregoing s e c t i o n s t o symplectic groups. {ei,fi)
Let
be our symplectic v e c t o r space and l e t
W
be t h e standard symplectic b a s i s of formula (1.1).
i s t h e span of t h e
ei
for
f l a g c o n s i s t i n g of a l l t h e The diagonal subgroup rational characters
A
of
B
of
B
X
i s a s p l i t Cartan subgroup.
A
j
preserving t h e maximal
i s a minimal p a r a b o l i c subgroup of
Xi
ai
The group
i I j.
Recall
Sp(W).
Define
by
These c h a r a c t e r s form a b a s i s f o r t h e l a t t i c e of a l l r a t i o n a l c h a r a c t e r s of
A.
The s e t
Z+
o f , p o s i t i v e r o o t s of
A
with r e s p e c t t o
B
are
the characters
6B of
Thus t h e modular f u n c t i o n
Here a s b e f o r e
2m
=
dim W.
I f t h e base f i e l d
B
is
We a l s o n o t e
F
is R o r
(C,
let
K
be t h e maximal compact
subgroup p r e s e r v i n g t h e (Hermitian) i n n e r product f o r which t h e fi
a r e orthonormal.
For
F
non-Archimedean,
let
K
ei
and
be t h e maximal
compact subgroup p r e s e r v i n g t h e l a t t i c e generated by t h e
ei
and
fi'
These choices f o r
s a t i s f y t h e c o n d i t i o n s (7.1).
K
We begin by studying t h e asymptotics of t h e matrix c o e f f i c i e n t s of
q.
the oscillator representations
Then we e s t a b l i s h a r e l a t i o n between
asymptotics and rank.
at.
Consider 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 2
L (Y), according t o formulas (1.8). Schwartz space of
Take
We can i d e n t i f y
Y.
Y
u
and
with
coordinates with respect t o the basis vectors for
+,
It is r e a l i z e d on v
Fn fi.
in
S(Y), t h e
by i n t r o d u c i n g Then we compute,
T € A1
I
= I7 l q ( ~ > 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.
References [Ar] J. Arthur, Harmonic analysis of the Schwartz space on a reductive Lie group, I and 11, mimeographed notes. [Am] C. Asmuth, Weil Representations of Symplectic p-adic groups, Am. J. Math., 101(1979), 885-908. [Bn] I. N. Bernstein, All reductive p-adic groups are tame, Fun. Anal. and App. 8 (1974), 91-93. [BW] A. Bore1 and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Annals of Math. Studies 94, Princeton University Press, 1980, Princeton, New Jersey. [Cr] P. Cartier, Quantum mechanical commutation relations and theta functions, Proc. Sym. Pure Math. IX, A.M.S., 1966, Providence, R.I. [Cg] M. Cowling, Sur l'algGbre de Fourier-Stieltjes d'un group semisimple, to appear.
*
[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.
v.94(1960),
[FZ] J. Fell, Weak containment and induced representations of groups, Can. J. Math., v. 14(1962), 237-268. [F3] J. Fell, Non-unitary dual spaces of groups, Acta Math., v. 114 (1965), 267-310. [Gs] K. Gross, The dual of a parabolic subgroup and a degenerate principal series of Sp(n,C), Am. J. Math. 93 (1971), 398-428. [HCl] Harish-Chandra, Discrete series for semisimple Lie groups 11, Acta Math., v. 116 (19661, 1-111. [HCZ] Harish-Chandra, Harmonic analysis on semisimple Lie groups, B.A.M.S., v. 76 (1970), 529-551. [HC3] Harish-Chandra, Harmonic analysis on reductive p-adic groups, Proc. Symp. Pure Math. XXVI, A.M.S. 1973, Providence, R.I.
[Hn] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, 1962, Nev York. [HI] R. Howe, Reductive dual pairs and the oscillator representation, in preparation. [HZ] R. Howe, 'L
duality for stable reductive dual pairs, preprint.
[H3] R. Howe, &series and invariant theory, Proc. Symp. Pure Math. XXXIII, Part I, 275-286.
[HM] R. Howe and C. Moore, Asymptotic properties of unitary representations, J. Fun. Anal. 32 (1979), 72-96. [Kn] D. Kazhdan, Connection of the dual space of a group with the structure of its closed subgroups, Fun. Anal. and App., v. 1 (1967) 63-65. [Ly] J. Lepowsky, Algebraic results on representations of semisimple Lie groups, T.A.M.S. 176 (1973). 1-44. [My] G. Mackey, Unitary of group extensions, Acta Math., v. 99 (19581, 265-301. [McD] I. MacDonald, Spherical functions on groups of p-adic types, Publ.Ramanujan Inst. 2, Univ. Madras, 1971, Madras, India [MW] C. Moore and J. Wolf, Square integrable representations of nilpotent groups, T.A.M.S. 185 (1973). [Nk] M. Naimal-k, Normed Rings, P. Noordhoff 1964, Groningen, Netherlands. [On] E. Onofri, Dynamical quantization of the Kepler manifold, J. Math. Phys. 17 (1976), 401-408. [Ra] J. Repka, Tensor products of unitary representations of SL2(R), Am. J. Math. 100 (1978), 747-774. [Rg] J. Rosenberg, A quick proof of Harish-Chandra's Plancherel Theorem for spherical functions on a semisimple Lie groups, P.A.M.S. 63 (1971), 143-149. [Sh] D. Shale, Linear symmetries of free boson fields, T.A.M.S. (1962), 149-167.
103
[Sl] A. Silberger, Introduction to Harmonic Analysis on Reductive p-adic Groups, Princeton University Press, 1979, Princeton, New Jersey. [Sn] B. Srinivasan, The characters of the finite symplectic group Sp(4,q) T.A.M.S. 131 (1968). 488-525. [TV] P. Trombi and V. Varadarajan, Asymptotic behavior of eigenfunctions on a semisimple Lie group, the discrete spectrum, Acta Math. 129(1972), 237-280.
[V] [Wr]
V. V a r a d a r a j a n ,
C h a r a c t e r s and d i s c r e t e s e r i e s f o r L i e groups, Proc. Symp. P u r e Math. XXVI, A.M.S. 1973, Providence, R.I. G. Warner, Harmonic A n a l y s i s on Semi-simple L i e Groups I, 11. Grundlehren d e r Math. W i s s . 188, 189, Springer-Verlag, 1972, H e i d e l b e r g , New York.
[Will A. W e i l , S u r c e r t a i n s groupes d l o p e / r a t e u r s u n i t a i r e s , Acta Math. 111, (1964), 143-211. [Wi2] A. Weil, B a s i c Number Theory, Grund. d e r Math. Wiss. 144, Second E d i t i o n , Springer-Verlag 1973, H e i d e l b e r g , New York.
-
[Zl]
G. Zuckerman
o r a l communication.
[22]
G. Zuckerman, Continuous cohomology and u n i t a r y r e p r e s e n t a t i o n s of r e a l r e d u c t i v e groups, Ann. Math. 107 (1978), 495-516.
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 <_ 4
r
2
1
.
( I t i s here
r e q u i r e some modifica-
With a n obvious n o t a t i o n we can then w r i t e
H
, since
the operator
V
comutes with the action of
G
, we
clearly
have
with
This space c a r r i e s t h e r e p r e s e n t a t i o n Sn-l
Dr'
of harmonic polynomials of degree
Go: qa: Q
O
r
and c o n s i s t s of r e s t r i c t i o n s t o
-
1 ; i t is contained i n
b u t i s i t s e l f n e i t h e r normal nor t a n g e n t i a l .
more e x p l i c i t information by noting t h a t
One proceeds t o g e t
and t h a t
r ' th summand by
a c t s on t h e
G
type formula says t h a t t h i s i s
D
~ @' D l~y o
.
A Clebsch-Gordan
@
.
Comparing represen-
D
~ @' Dr+"O ~
t a t i o n s and degrees t h i s shows t h a t
M ~ - ~ i s' a~ subspace of type
where
D~-'"
; i t can a l s o be d e s c r i b e d more
p r e c i s e l y i n terms of harmonic polymmials (cf. Using some c l a s s i c a l f a c t s about a c t i o n of the Riesz transform
L~" Tan
with the m u l t i p l i e r
(1
R
Xr
, it
[12]).
i s now easy t o d e s c r i b e t h e
on t h e r e p r e s e n t a t i o n s :
n-2 + F)
*
compact p e r t u r b a t i o n of a n isometry.
, which
It maps
Go:
onto
shows i n p a r t i c u l a r t h a t i t is a
Similarly, P
has a n obvious d e s c r i p t i o n
i n terms of the r e p r e s e n t a t i o n s . One can use these methods t o s o l v e some f u r t h e r problems of a n a l y s i s on Sn- 1
.
For example one may ask, i n analogy w i t h the case of holomorphic n f u n c t i o n s on the u n i t b a l l i n C , whether the subspace H of L can be
c h a r a c t e r i z e d by d i f f e r e n t i a l equations. f = (fl,.
..,f
)
on
n A
As w e l l known,
the Riesz systems
a r e c h a r a c t e r i z e d by the analogues of t h e Cauchy-
Rienann equations :
It i s easy t o g i v e a meaning t o the r e s t r i c t i o n of these equations t o
sn-'
.
They a u t o m a t i c a l l y a n n i h i l a t e every normal v e c t o r f i e l d , and the theorem one can prove i s the following [12]. The vector f i e l d s on element i n
H
S
n-1
t h a t a r e t h e t a n g e n t i a l p r o j e c t i o n s of some
a r e c h a r a c t e r i z e d by the r e s t r i c t e d Cauchy-Riemann e q u a t i o n s
(taken i n the sense of d i s t r i b u t i o n s ) . For the proof one shows t h a t the r e s t r i c t e d ~ a u c h yRiemannequations amount
t o a c e r t a i n s i n g l e G-equivariant d i f f e r e n t i a l o p e r a t o r a n n i h i l a t i n g t h e vector f i e l d i n question ( i t is r e a l l y j u s t the operator of e x t e r i o r differenn-1 t i a t i o n , i f t h e v e c t o r f i e l d i s r e i n t e r p r e t e d a s a d i f f e r e n t i a l form on S ). By e q u i v a r i a n c e and by t h e decomposition of
i n t o i r r e d u c i b l e subspaces
LTan
i t s u f f i c e s now t o show t h a t t h i s o p e r a t o r does n o t a n n i h i l a t e any o f t h e
I;::
; t h i s can be done by e x h i b i t i n g a c o n c r e t e element i n t h e space t h a t i s
n o t mapped t o zero.
L
Another v e r y i n t e r e s t i n g a p p l i c a t i o n o f t h e harmonic a n a l y s i s o f g i v e n r e c e n t l y by M. Reimann [14]. [ I ] defined an operator
S
mapping
R~
d e f i n e d on t h e u n i t b a l l of
was
To d e s c r i b e t h i s , we r e c a l l t h a t Ahlfnrs $-valued
functions
u = (u
t o matrix-valued f u n c t i o n s
;
,..., u n )
1
the
(i,k)-th
e n t r y i s g i v e n by
-
& o p e r a t o r and i s of i n t e r e s t f o r t h e t h e o r y o f
i s a n analogue o f the
S
quasiconformal mappings. equation
S*Su = 0
Of obvious importance i n connection w i t h
, where
S*
d e n o t e s t h e a d j o i n t of
systems a r e p a r t i c u l a r s o l u t i o n s of t h i s e q u a t i o n . )
L
i r r e d u c i b l e p i e c e s of f
, i. e.,
.
S
is the
(The Riesz
By working w i t h t h e
it i s shown i n [ 1 4 ] t h a t , g i v e n a n a r b i t r a r y
t h e r e i s a unique s o l u t i o n with
S
u
of
f
E
L,
S*Su = 0 whose boundary v a l u e s c o i n c i d e
L~
t h e " D i r i c h l e t problem" w i t h
boundary d a t a h a s a unique
solution.
62.
Let
G
=
K
.
KAN
G
B i r a d i a l f u n c t i o n s on n i l p o t e n t groups.
be a connected r e a l semisimple Lie group of r e a l r a n k one.
be a n Iwasawa decomposition, and l e t
We a r e going . t o consider t h e groups
N
M
be the c e n t r a l i z e r of
which occur i n t h i s manner.
Let A
a r e of i n t e r e s t s i n c e they can be i d e n t i f i e d w i t h open dense s u b s e t s of t h e boundaries of t h e r a n k one symnetric s p a c e s G = SU(n, 1 )
G/K ; t h e
N
g i v e n by
is, by t h e way, t h e well-known Heisenberg group.
The Lie a l g e b r a o f
N
i s of t h e form
-n =
1 2 g +g
in
They
where p, q
g1,g2 -
a r e the
A-root spaces corresponding t o the r o o t s
g1,g2
denote the dimensions of
y,2y
.
Let
respectively; i t is possible t h a t
p=0.
(This convention may seem strange, b u t i n t h i s way t h e case where t h e r e i s only one r o o t f i t s much more smoothly i n t o the g e n e r a l case. )
A
commutes with follows t h a t If
M
, the
a d j o i n t a c t i o n of
normalizes
N
, so
and
Since 2
g
gi
.
N
f u n c t i o n s on
2
q = 1
(When
1x1 , Y
biradial,
.
, the
.
M
I t follows t h a t t h e
f (exp(X
+ Y)) , X
1
E g
,
exactly the functions t h a t
As i n [7], we c a l l t h e s e f u n c t i o n s
M-invariant f u n c t i o n s a r e the ones t h a t depend
In t h i s c a s e we can s t i l l c a l l t h i s s l i g h t l y l a r g e r c l a s s
s i n c e a l l t h a t follows a p p l i e s , w i t h only minimal modifications.
The s p e c i a l s t a t u s of the case M
.
> 1
( X I , IYI , a r e
a r e i n v a r i a n t under t h e a c t i o n of on
q
which, when w r i t t e n i n t h e form
depend only on the norms
biradial.
M
By ~ o s t a n t ' sdouble t r a n s i t i v i t y theorem
[15] t h i s a c t i o n i s t r a n s i t i v e when
(131,
Y E
S1 X S2
It
(with r e s p e c t t o t h e u s u a l inner
product given by .the Cartan i n v o l u t i o n and t h e K i l l i n g form), then obviously o p e r a t e s on
M
.
i s a s e m i d i r e c t product.
MN
denotes t h e u n i t sphere i n
S.
preserves
M
q = 1 would disappear e n t i r e l y i f i n s t e a d of
we used i t s analogue i n t h e e n t i r e , n o t n e c e s s a r i l y connected isometry
group of the symmetric space. ) Under the map
n +MnM
with the left-and-right
the
M-invariant f u n c t i o n s on
M-invariant f u n c t i o n s on
o f t h i s s e c t i o n i s now t h e following.
(MN,M) f u ~ c t i o n son
i s a Gelfand p a i r .
MN
1
Hence t h e
L -algebra
.
N
a r e identified
The main o b s e r v a t i o n
A
of biradial
i s commutative.
N
The proof i s immediate:
writing
n
=
exp(X
+ Y)
a s before,
- 1 = exp(-X - Y) , and n exists
m E M
by the double t r a n s i t i v i t y theorem ( i f q -1 -1 such t h a t mnm = n I t follows t h a t MnM = The c a s e
>
1 ) there
Mn-b
.
i m p l i e s the d e s i r e d commutativity.
one h a s
and t h i s
q = 1 can a l s o be d i s c u s s e d
w i t h o u t using the c l a s s i f i c a t i o n of symmetric spaces, along the l i n e s o f [15]. I f one i s w i l l i n g t o use t h e c l a s s i f i c a t i o n , occurs only when
N
then one knows t h a t
q = 1
i s t h e Heisenberg group; t h e statement (even a s t r o n g e r
statercant) follows then from t h e remarks below. In g e n e r a l we may a s k i f t h e r e e x i s t s a l a r g e r c l a s s than the b i r a d i a l f u n c t i o n s t h a t form a commutative a l g e b r a under convolution. case of the Heisenberg group w i t h the product
H2n+l
parametrized a s
I n the special
(z, t ) ; z E C
,t
E R
a function
f(z,t)
.,z
,t .
1z
. .
i s c a l l e d m u l t i r a d i a l [ 4 ] i f i t depends only on
It i s a n important f a c t [ 4 ] t h a t these form a commutative
algebra.
There i s a Gelfand p a i r i n t h e background h e r e too:
elements
e = (el,.
autmrphisms
..,en)
-
(z, t )
of the
(ez, t )
i s a s e m i d i r e c t product and map
-
a: (z, t )
+
(z, - t )
.
Since
a
where
T~
.i s
a(g)
H2n+l
a c t on
..,c n z n ) .
ez = (elzl,.
h n (T H2n+lJT )
by t h e Then
i s a Gelfand p a i r .
i s a n automorphism of
t h a t the double c o s e t of g = (z,t)
n-torus
l e t the
T
~
H
~
I n f a c t , the
and i t i s easy t o see
H2n+l
t h e same a s t h a t of
g
-1 , for
every
induces a n automorphism of the convolution a l g e b r a of
b i - i n v a r i a n t f u n c t i o n s while
g
-
g
-1
induces a n antiautomorphism, commuta-
t i v i t y follows.
A s i m i l a r l a r g e r commutative a l g e b r a was e x h i b i t e d by Cygan [ 3 J i n the case of t h e q u a t e r n i o n i c analogue of the Heisenberg group. Let us r e t u r n t o b i r a d i a l f u n c t i o n s i n t h e g e n e r a l c a s e and d e s c r i b e t h e i r Fourier a n a l y s i s .
Since we a r e d e a l i n g w i t h a Gelfand p a i r ,
Ch. X I t h a t the maximal i d e a l space of
known [5,
A
it is well-
i s given by t h e bounded
s p h e r i c a l f u n c t i o n s ; the s p h e r i c a l f u n c t i o n s , i n turn, a r e t h e j o i n t eigenfunctions of t h e a l g e b r a c o n s t a n t term on
{xi]
Let
N
IY. \
v i t y the a l g e b r a o f and
2
Mi
.
.
g of MN.-invariant d i f f e r e n t i a l o p e r a t o r s without
be orthonormal bases of
51
, g2
M-invariant symmetric t e n s o r s on
.
By double t r a n s i t i -
1 i s generated by
Using t h e "symmetrization map" [5, Ch. X I one s e e s e a s i l y t h a t
Mi2
2
i s generated by t h e o p e r a t o r s
where,
%
=
Yi
t h i s time,
t h e square is taken i n t h e enveloping algebra.
i n t h i s case. ) If
(If
q = 1
,
; we a r e n o t going t o keep t r a c k of t h e t r i v i a l m o d i f i c a t i o n s r e q u i r e d
u
is biradial, l e t
u0
be d e f i n e d by
~
+
~
A:
and
by
(Aiu) Since
A:
0
0 0
=
Aiu
g2
involves only t h e c e n t e r
o f t h e Lie a l g e b r a
the r a d i a l Laplacian of t h e Euclidean space
4"
O O
=
2 2 ( 4 1 ~ 1 D2
+
2
g
g
, it
i s simply
,
2q D ~ ) u O .
A simple computation, c a r r i e d o u t i n [ 7 ] g i v e s 0 0
Alu
= (41x1
2
2 Dl
+
2p Dl
+ b 2 1x12
0 0 A2)u
where
(of course, the length o f the r o o t
P
and
1 y(
can e a s i l y be expressed i n terms of
q ).
Now suppose t h a t
u
i s s p h e r i c a l , i. e.,
that i t i s a biradial function
such t h a t
w i t h some numbers variable, on
y
where
u0
x1,x2,
and
u(e) = 1
.
Since
4
involves only the second
i s a product of functions of one v a r i a b l e ; t h e f a c t o r depending
i s a r a d i a l e i g e n f u n c t i o n of the Euclidean Laplacian.
So
(do denotes the normalized s u r f a c e measure ; j (') a Bessel f u n c t i o n ) . t i a l equation f o r cian.
I
.
If
A = 0
Alu = xlu
, we
a g a i n g e t t h e r a d i a l Euclidean Laplab , the equation becomes
t h e c o n f l u e n t hypergeometric equation, with
Every s o l u t i o n which g i v e s a smooth f u n c t i o n on at
is, of course, e s s e n t i a l l y
now g i v e s an o r d i n a r y d i f f e r e n -
h f 0 and we s e t @ ( t ) = exp(-ykt)v(bht)
If
i. e.,
The equation
x = 0)
N
which i s r e g u l a r
( i . e.,
g i v e s a s p h e r i c a l function.
However, we a r e r e a l l y i n t e r e s t e d only i n the bounded s p h e r i c a l
It i s n o t hard t o s e e t h a t we g e t these e x a c t l y when
functions.
(then we may take number.
k 20
The s o l u t i o n s
v
, since
j (')
i s even) and i f
n
i s real
i s a natural
of our d i f f e r e n t i a l equation a r e then the Laguerre
So f i n a l l y we f i n d the following bounded s p h e r i c a l f u n c t i o n s :
where
u
%,n(e)
2 0
,a
>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. ).
<.
on
I,
{q,- ..,Ar
Let
More precisely, we have t h e following
Corollary t o Theorem
] be a complete s e t of representatives i n
t h e various conjugacy classes of CSG's. and write
ElJi = MiAi,R
*
ai, I
(15 i _< r )
We assume t h a t each Ai
is
B-stable
for t h e Levi cmponent of t h e psgrps
A? has a natural map onto a l a t t i c e in 191 :A it is c l e a r with f i b e r s of cardinality at mast A
associated with ( - 1 )
19, p. 545, [ 5 I,
Ai.
Since
t h a t t h e notion of a function on
n
(ldild X aR ,:
of a t most polyncanial s k h
makes sense. meorem. (15 i
h
W e give
( M ~ the ) ~ d i s c r e t e topology.
m e r e are unique continuous functions
on
Ci
X
*
"i,R
5 r ) with t h e following properties Z
(i) (ii) (iii)
each
i s of a t most p o l y n d a l growth
Ci
c i ( s ~:SV) = ci(u :V ) (a f o r a l l functions
m e f'unctions
ci(w : v)
f
are
h
a:,R7
v
F
s
i n the Schwartz space
F w(G,A~~))
C(G)
2 0 and t h e s e r i e s on t h e r i g h t converges
absolutely. m e s e r i e s of representations
r
W,"
h
(w
1 _< i 4
V F
F
axe known as the principal s e r i e s of representations of
V)
G.
W e look a t s m e examples. 1)
G = S L ( ~ R,).
group of rotations
There a2e two Cartan subgroups,
ue =
kt
(0 E
diagonal matrices : L = A U YA wheFe
all matrices a
=
#
0),
-1 0
\
t ) .
L. B is t h e
I, i s the group of
R)
')' = ( O
B and
and A
i s t h e Goup of
/
The characters of the d i s c r e t e classes are t h e
which are the distributions defined by the invariant
en (n E
Z, n
locally
L,l functions on G (with respect t o standardHaar measure) given by
The formal degree of t h e class corresponding t o we find t h a t the character
en
is
Inl.
mning to
L
A
M = { l , y ] and s o M = LI consists of t h e t r i v i a l character and E
t h a t sends
y to
-1. We combine these with t h e character
a u eivt
(v
t
E
and use t h e construction v i a induced r e p e s e n t a t i o n s t o
R)
T
get the characters
T,"
(U
E
Mh).
A simple calculation gives
The Plancherel f o r m u l a then becomes (cf. Lang
2)
G i s cmplex and simply connected.
G regarded a s a Lie group over
~ l g e b r sand of
only one upto
G-conjugacy.
(H. H'
t
I*).
for i;hich
a
in
Any element
element of the complex dual of
aC
g be the Lie .algebra of
174; Harish
in
X
gC.
+
I
L i s a CSG of E
*
by s e t t i n g
IR
is a ~ ~ A o f
G and is t h e
can be regarded as an
IR
X(H+ JR ) = X(H)+ i k ( R )
This allows us t o introduce the l a t t i c e E c I*
exp H H e
G induces a
is a empact f o r m
I(
dC=I
IR=JII, G,
Chandra [ l o ] )
gC far t h i s cauplex Lie
(-I)112
and
1
L i s the centralizer of
If
gC.
i s sCSAof
Let
We w r i t e
g.
J for multiplication by l I c t
If
G.
p.
The canplex s t r u c t u r e of
R.
cauplex Lie algebra s t r u c t u r e on
111,
(H E I = ) is a character of
and so the irreducible unitary representations of
of all k
I=. ~ u now t H =
MA = L =
%
E
1;
4
a r e of t h e
form exp H exp HI
HeE
( H ) ~ ~ v ( H ' ( H E I*,
R
E
IR, 5
E
IL, v
E
1):
The corresponding induced representations a r e a l l irreducible (cf. Wallach [I]).
Let
T*,, be t h e i r characters.
Then the Plancherel farmula takes t h e rollow-
ing form (cf. Harish Chandra [ 9 1). such t h a t for all
f
There is a positive constant
i n the Schwaxtz spare of
G,
c
,0
p(l : v)
where
is t h e function on IL x I
gC and pC i s a p o s i t i v e system of r o o t s of
t h e Cartan-Killing form of
2.
.
(-, ) i s
defined as follows : i f
Eigenfunction expansions on
G.
The case of s p h e r i c a l functions.
1. We s h a l l now introduce the c e n t r a l theme of these l e c t u r e s , namely, t h e L ~ ( G ) and i t s connection with t h e
theory of eigenfinction expansions on Plancherel formula.
kt ~ ( 9 ~be )t h e universal enveloping algebra of
We s h a l l regard i t s elements as l e f t invaxiant d i f f e r e n t i a l operators on if
f
E
c~(G), a
Elements of
E
u(%)
u(gc),
we write
f(a;x)
for
fa(x);
where t h e s u f f i x
t
= t =
---
0
if
a
fa.
a =X X 1 2
We a l s o write
-.
Xj
5.9
=t
instead of
E
on
f.
f(x;a)
for
(af )(x)
g,
r = 0. The adjoint representation
~ d ( x ) ( a ) , for
Ad
again) of
a E u(gc),
x
Ad
of
G on
g extends
G on ~ ( 9 ~ )we; o f t e n w r i t e E
G.
m e connection between
t h e lef't and r i g h t d i f f e r e n t i a l actions of elements of by t h e f a m u l a
a
G;
indicates t h a t t h e derivatives axe evaluated f a
t o a representation (denoted by X
f o r t h e a c t i o n of
.
a l s o a c t as r i g h t invariant d i f f e r e n t i a l operators, the
corresponding a c t i o n being w r i t t e n and
af
g
u(gc)
i s then given
A s i n $ 1 we write
it i s obvious that
8
c E T ~ , ... ,Td 1
'J
fur the centralizer of
3
8
i s t h e center of
u(gc).
G
(via A&)
It i s well known t h a t
1 = rk(G).
where
Let us now consider the r i g h t regular representation L2(G).
Denote by
Elements of tation a, f t+ af
of
of
G in
of
u ( ~ ~on) 8;
it is none other than t h e action
admits a canonical anti-linear ant iautamorphism
- id
on
are all closeable and t h a t
r.
cm(G). By d i f f e r e n t i a t i n g we obtain a represen-
We now r e c a l l t h a t
U(CJ~ ) on cW(G) introduced above.
adjoints, which i s
r
&I the subspace of vectors which are d i f f e r e n t i a b l e f o r
&I are all i n
a t+ r ( a )
i n u(~,);
u(%)
t, the operation of formal
3. It i s well known t h a t the operators r ( a ) ~ l ( r (ta)) = ~ l ( r ( a ) )where ~ t h e t on t h e r i g h t In p a r t i c u l a r , i f
r e f e r s t o Hilbert space adjoints. adjoint (m normal),
r(a)
Hilbert space sense.
The operators
a
i s formally s e l f
i s essentially self-adjoint (or nmmdl) i n t h e ~ l ( r ( a ) ) (a
E
9)
are thus narmal am3
mutually conmuting and it makes sense t o ask for t h e i r e x p l i c i t s p e c t r a l expansion.
I f we now observe t h a t the matrix coefficients of irreducible
unitary r e p ~ e s e n t a t i o n sof
G
the spectral decanposition of
are eigenfunctions f o r r(8)
8, it
is c l e a r t h a t
i s intimately r e l a t e d t o t h e Plancherel
expansion on
G. However, i n view of the i n f i n i t e m u l t i p l i c i t i e s t h a t will 2 a r i s e on L (G), it w i l l be convenient t o consider t h e s p e c t r a l theory of
r(8)
on subspaces of
by elements of sentations of
K K.
2
L (G) t h a t transform umler r i g h t and l e f t t r a n s l a t i o n s
according t o fixed but a r b i t r a r y f i n i t e dimensional repre-
2 Since t h e union of such subspaces i s dense i n L (G),
t h i s i s not a serious r e s t r i c t i o n . 2.
Gfi
The simplest of such
K-finite subspaces i s L2(G//IC)
f a r the double co-set space
IC\G/K.
The elements of
where we w r i t e 2
L (G//K)
are
those t h a t are b i i n v a r i a n t under We
najr
regard them as t h e
K
and a r e known as s p h e r i c a l f u ~ c t i o n s .
K-invariant functions on
G/K
t h a t are i n
L ~ ( G / K ) I n t h i s case t h e r e i s an algebra t h a t i s s l i g h t l y bigger than which i s t h e one t o b e considered f o r e i g e n f u m t i o n expansions.
g be t h e c e n t r a l i z r of a
K
i n u ( ~ ~ )For . each element
G-invariant d i f f e r e n t i a l operator
G-invariant d i f f e r e n t i a l operators on manner.
(
a
on
t h e rank of t h e Riemannian symmetric space
rGIK(d and
E
In fact, l e t
a, r ( a ) we
and a l l t h e
cm(G/K),
RI c[!P1,.
G/K.
. .,Td 1
Furthermore
where
d
is
cm(G//K) i s
one can therefore ask for t h e s p e c t r a l theory of 2
t h e cammuting algebra of (unbounded) narmal operators i n L (G//K) by
have
C-(G/K) may be obtained i n t h i s
r G/K(D) i s abelian and i s
Moreover
s t a b l e under
)
a
8
determined
rm(d.
I n t h i s s p e c i a l case one can see very c l e a r l y t h e close r e l a t i o n ship between the eigenfunction expansion problem for s p h e r i c a l functions and t h e r e p r e s e n t a t i o n t h e o r e t i c problem of d e c q o s i n g t h e n a t u r a l representation of
G
in
of c l a s s
2
L (G/K). 1, i.e.,
We a r e concerned here with unitary representations of admitting nonzero
K-invariant vectors.
irreducible u n i t a r y representation o f c l a s s space
E ( B ) ~ i s of dimension
function
cpT
on
The function cpT eigenfunction f o r
If
T
i s an
1 i n a Hilbert space
H(T),
G
the
1 and defines a well determined s p h e r i c a l
G by
i s i n c~(G//K), i s normalized by
(~~(= 1 )1, and is an
r G I K ( d . Of course not d l representations of c l a s s
are r e l e v a n t f o r t h e decanposition of exceptional s e r i e s of representations).
L2(G/K).
( c f . remarks i n $ 1 on t h e
Specializing the s i t u a t i o n discussed
i n $ 1 we consider t h e p s m
P =MAN,
decanposition of
i s the c e n t r a l i z e r of
G
and M
1
where
G = KAN A
i3
in
now an Iwasawa
K (so that P i s
a m i n i m a l psgrp).
-(a
map inverting
a)
manwe
a
If
+A),
G
a d i s denoted by
of
a*,
t h e representation
It i s irreducible f a r all v
*,,T-I
E
a*
is of c l a s s
1
( c f . Kostant [l];
The basic r e s u l t in this hamewark i s due t o
It a s s e r t s t h a t t h e
are inequivalent and
a*)
E
obtained by unitary induction h a n P
s l s o Parthasarathy e t al [ l ] ) . Harish Chandra.
v
is the
P is one dimensional and unitary; t h e corresponding
of
representation of
then, for any
log(^ + a)
A and
i s the Lie algebra of
for
T ,,
v
E
(the p o s i t i w chaniber
a*+
n is a d i r e c t i n t e g r a l of t h e
T,
with
respect t o a measure c l a s s t h a t i s mutually absolutely continuous with respect t o t h e Lebesgue measure on
a*
( c f . Harish Chandra [ 2 I ) .
It i s not d i f f i c u l t t o s e t up an intertwini*
operator t o go frm
c ~ ( G / K ) t o the space of smooth sections of a Hilbert space buadle whose f i b e r s are the spaces of t h e ( ~ f ) ( x:Y ) Since f o r
a'
E
n'
A,
E
Tv. For
f
E
c~(G/K) l e t
=Jm f(xan)e(wp)(lOg
a)daan
N,
it is clear t h a t for fixed v,
( J f ) ( - : V) i s an element of t h e space on
which
7: acts.
of
and s o gives us t h e intertwining operator we axe seeking.
G
spherical, For
The operator
(Jf)(- :
J
canmutes with l e f t t r a n s l a t i o n s by elements
i s c m p l e t e l y determined by
( ~ f ) (:lV ) we have the expression
where Af
i s the Abel transform of
f
defined by
( ~ f ) (: l v),
If as
f
is
G = KAN.
The question of proving t h a t of
J
extends t o an intertwining u n i t a r y isomorphism
L ~ ( G / K )with a s u i t a b l e d i r e c t i n t e p s l of t h e
question of proving t h a t t h e map morphism of
L ~ ( G & ) with
on
a*.'
2.
Let us w r i t e
cp(v :x )
f
H
.
(Jf) ( l : )
\
then reduces t o t h e
extends t o a unitary i s o -
2
L (a*+,~dv) f o r a s u i t a b l e density f'unction
for t h e matrix c o e f f i c i e n t
cp Tv
.
B
A simple calcula-
t i o n shows t h a t
-
therefore as a transform of
f.
Formally, f o r any
spherical function f ,
t h e Harish Chandra transform
#f
of
We think of
function on
(Jf)(l : )
f
is t h e
&+ given by
Since t h e representations
T,
and
vsV ( s
E
w(G,A)) a r e equivalent, we have
t h e functional equation
Finally, as
G = KAN,
t h e representation
such a manner t h a t t h e constant function
i'rv 1 is
we then obtain an i n t e g r a l representation over
Here
dk is t h e Haax measure on K with
can be r e a l i z e d on
IK
K-invariant. K
For
2 L (Kh)
cp(v :x )
given by
dk = 1. The function H
is
in
from
G to
~ ( x E)
D
a
and i s defined by t h e requirement t h a t f o r q x
i s t h e unique element for which
x
E
G,
E
K exp ~ ( x N. )
The fundatxental theorem of Harish Chandra's theory a s s e r t s t h a t t h e map 2 f I+ #f ( f E c ~ ( G / / K)) extends uniquely t o a unitary isomorphism of L (G//K)
3.
with
2 L (a*,$dv)'
where
f3
i s a smooth nonnegative function on
invariant under t h e Weyl group W(G,A)= i.e.,
a*
which is
m and which has moderate growth,
i t s e l f and each of i t s derivatives has atmost polynomial growth; t h e
superfix
m indicates t h e subspace of functions invariant under m.
over (with
More-
s u i t a b l y normalized) we have an inversion formula
dv
With these r e s u l t s established, it i s n a t u r a l t o ask f o r an e x p l i c i t determination of
f3.
Harish Chandra's theory shows t h a t
determined from t h e asymptotic behaviour of
on
a*',
such t h a t f o r any
v
E
a*',
H
E
+ a ,
can b e completely
I n f a c t it is possible
cp(v : x).
t o prove the existence of a meromorphic function on
$
,a:
say
s,
analytic
we have t h e asymptotic
r e l a t i o n (with exponentially decaying e r r o r terms)
.. c
etp(H)cp(v :exp t ~ )
c(sv)e
i t s V(H)
SED
Furthermore
c
.w
has no zeros on
a*,
and ~ ( v )= I ; ( v ) ~ - * = s ( ~ ) - l s ( - ~ ) - l Furthermore, t h e h c t i o n
2
( Y E
8)
has a remarkable i n t e g r a l representation
v
This i n t e g r a l converges i f
E a:
and
'd roots a > 0
Im(v,a) < 0
(of
(g,a))
It was evaluated e x p l i c i t l y by Gindikin and ~ a r p e l e v i ; [ 11.
This - l i c i t
evaluation implies i n particular the product formula
where
5
y >0
is t h e
is a constant,
i s t h e s e t of short positive roots, and
A*
c-function associated with t h e rank one subgroup of
only short p o s i t i v e r o o t i s a.
G whose
In particular 2
n
B = Y
ad* The
c
wla!
and hence the
Pa
can be determined by e x p l i c i t calculation i n the
rank one case (more on t h i s l a t e r ) . These r e s u l t s i n the spherical s e t up are the prototypes of the general case.
Therefore it may be worthwhile t o take a closer look a t them as a
preview f o r t h e general theory.
I s h a l l do t h i s i n t h e next lecture.
3. Asymptotic behaviour of spherical eig;enfunctions 1. We begin t h e deeper study of the spherical theory by looking a t the eigen-
.
functions
cp( v : )
and the eigenvalues t o which they belong.
The
cp( v : )
are given by t h e formula
f o r any
V
i n t h e complex vector space
a homomorphism of
Q
into
C.
a$.
The corresponding eigenvdue i s
This homomorphism car- be e x p l i c i t l y determined
by using t h e representati.cn (1); i n particular one finds t h a t f o r a given
u
element
i t s value a t
0
E
v depends polynomially on
Indeed, we h a w
V.
a homomorphism
such t h a t
Since
cp(v : ,)
it follows t h a t of
n) as
i s invariant under t h e Weyl group y(v)
is
.
ro-invariant
a function of
It i s a b a s i c f a c t t h a t every element
~ ( a , ) can ~ be thus obtained:
It i s also t r u e t h a t
v
For fixed
I(D).
C-(G//K) which i s
.
cp( v : )
,a:
E
yG,K(d with
f a c t o r s through t o an isomorphism of
y
i s t h e unique eigenfunction f o r
c ~ ( G / / K )(normzlized t o be only i f
v
v'
and
cp(V
1 at
1).
l i e i n t h e same
since t h e homomorphisms of \D\~:. The
.
Finally,
cp(
V
projection
c
+ U(%)n),
( tu($
= t @ a @ n
w
G.
a
=
We note
and hence it makes sense t o speak of t h e
U(a,)mod( T U ( % )+ u(%)n);
<+
y.
implies t h a t
then one has
be t h e Casimir element i n U( Sc).
For instance, l e t 0:
3
if and
a r e naturally i n b i j e c t i o n with
a r e c a l l e d t h e elementary s p h e r i c a l functions of
: )
Ea : u ( % )
Laplacian on
-
.
m-orbit; t h i s i s immediate from (4)
~ ( m ) into
f i r s t t h a t t h e Iwasawa decomposition @
Q in
: ) = cp( V 1 : )
It i s not d i f f i c u l t t o give an e x p l i c i t description of
= U(ac)
0 in
1 a t t h e i d e n t i t y element corresponding t o t h e eigen-
homomorphism v tt y(v)(iv) ; there are no other eigenfunctions of
u(,)
v,
. + d-r
where
Denote by
w
a
the
( H ~ ) is ~ an < orthonormal ~ ~
--
b a s i s of
a
r e l a t i v e t o t h e K i l l i n g form (which is p o s i t i v e d e f i n i t e on
a).
Then
The basic method of studying the d i f f e r e n t i a l equations
2.
polar coordinates.
The map
(kl,h,k2)
H
klh%
(3) i s t o use
from K X A+ X K
submersive everywhere and so we obtain a (unique) homomorphism i n t o t h e algebra of d i f f e r e n t i a l operators on A+
into
G
of
D
6+
is
with analytic coefficients
such t h a t
+
It is n a t u r a l t o c a l l 6 ( v )
t h e r a d i a l component of
v
+.
on A
By a simple
c a l c u l a t i o n one can show t h a t
where t h e
v' i
a r e i n ~ ( )a and have degree
c
functions which a r e i n t h e r i n g
m
t h e functions
(e
- 1)'l
+ %(A )
(a: E A').
< deg(v),
while t h e
gi
are
(without u n i t element) generated by Thus, f o r t h e Casimir element
w,
an
e x p l i c i t c a l c u l a t i o n gives
where, f o r ant
(H
E
X
E a*,
3
i s t h e element of
a), (-,.) being t h e K i l l i n g form;
space corresponding t o t h e root
n(a)
a
defined by
&,H)
= >-(H)
i s t h e dimension of t h e root
a.
The c r u c i a l consequences of t h e formula (8) are as follows: (i)
if we consider only t h e highest degree terms,
6+(v)
i s a constant
c o e f f i c i e n t operator (ii)
If
h
E
+
A
goes t o i n f i n i t y i n such a way t h a t
log h ) ++m
for
dl roots a c A+,
t h e coefficients
tend t o m o (exponentially), and
6+(v) may b e thought of as perturbative f o r
hence the lower order terms i n
goes t o i n f i n i t y i n t h e sense
the question of studying asymptotics when h described above (iii) Y(v)
since t h e leading term of
by
(cf.
P
(5))
and since
6+(v)
is
E ( v ) which i s a t r a n s l a t e of a
is an a r b i t r a r y element of
y(v)
one
I(IU),
would expect t h a t cp(v:h) =
E
cse (isv-p)(log
h,
+
t e r m s decaying exponentially.
SEm
I n a c e r t a i n sense the e n t i r e analytical theory of t h e d i f f e r e n t i a l equations
(3)
comes out of an attempt t o make t h e above remarks more precise.
+
and l e t 6: the
m
.
6: be the l a t t i c e generated by t h e simple roots a1,. .,a r
Let
be the subset of elements of t h e form mlOll
are all integers
j
2 0 with
m
f
t h a t each element of t h e r i n g R ( A )
1
+
+ m > 0. r
+
-
+ mrOl2
in
a+
where
It i s then c l e a r
has an expansion of t h e form
c eq, c E C. A v w i a t i o n of the c l a s s i c a l Frobenius method of finding q ~ q9 9 power s e r i e s solutions of ordinary d i f f e r e n t i a l equations applied t o t h e
Z .
system (7) of d i f f e r e n t i a l equations now shows t h a t
(q v :h)
has an i n f i n i t e
s e r i e s representation of the form
cp(v :h)ed"g
h, =
c
c,(v)e i s v ( l o g h)
where most
v d,
E
I, ( v : h)e <(lo@;h )
r
(v :h ) depend polynomially i n l o g h of degree a t 9 d 2 0 being a fixed integer. Taking v = 0 and writing a*
and t h e
c
q.Pu{ol
SEW
we find for the spherical function
E
t h e estimate
From t h e formula (1) we get, f o r all
For t h e derivatives of
cp(v : .)
V E
we have similar estimates; t h e constant
of these estimates i n general depends on sowth i n
a*,
C
but is of a t most polynomid
V
The estimates (11) and ( E ) , together with the corresponding
v.
-
estimates of t h e derivatives of
cp( v : ),
are the fundamental i n i t i a l
estimates of t h e theory. Write
f o r R(A+). For aqy compact L c a+
B
The funct.ions of form A+[LI. the
dv)
a
let
go t o zero exponentially f a s t on t h e conic s e t s of t h e
+
6+(v)
Hence on A [L], exhaust only
i s a nice perturbation of
~ ( m ) and not
i s not of t h e f i r s t order.
u(ac)
t h a t the unperturbed system th As i n t h e c l a s s i c a l cases where we go frcnn n SO
n components, we can
order equations t o f i r s t order systems of vectors with get a f i r s t order system as follows. module over
We observe t h a t
is a f i n i t e
u(ac)
~ ( m ) . It is i n f a c t a f r e e module of rank w =
ul = l,u2,
...,uw
representation of
u(ac)
basis
~ ~ ( v )But .
of homogeneous elements.
I rnl ,
having a
Using t h i s basis we get a
by w X w matrices of elements of
I(B),
such t h a t
We now replace jth
(14)
cp(v: - ) by the w
component is
~ ( v h; : u
j
.
eP).
X
1 vector function
cD
on
Y
aC
The equations (7) now becane
UQ(V:- ) = r ( ~ ~: v ) o ( v :.)
+n
( ~ v: : . ) c ~ ( v :.)
X
A whose
.
S(u : : - ) i s a w X w matrix of elements from R[x].
where
The system
(14) i s obviously a perturbation of a f i r s t order l i n e a r system since we can take
u =H iJ (Hi)l
being a b a s i s of
The eigemralues of t h e matrix perturbation problem (14).
u = Hi
If we take
itnd
514) i s thus s t a b l e f o r
a.
r ( u :i v )
a r e c l e a r l y important i n t h e
One knows t h a t these are t h e numbers
*,
V E
a
these a r e all purely imwinary.
v
a*;
t h i s i s t h e c e n t r a l a n a l y t i c a l f a c t of t h e
E
The system
theory. The s p e c t r a l theory of t h e commuting family of matrices ( u e U( ac), v
E
a:
f i x e d ) i s &o
is t h e polynomial f u ~ c t i o non r o o t s it can be proved t h a t -
important i n t h e study of
r ( u :i v )
(14). If n
*
ac which i s t h e product of all s h o r t p o s i t i v e
~ r ( v ) ~ ~ ( (vs) E W) depends polynomially on
V
i s t h e s p e c t r a l projection on t h e one dimensional space where
has eigenvslue
u(ismlv)
for a l l u
E
I(B )
.
Simple arguments from t h e theory of d i f f e r e n t i a l equations now allow us
+
t o conclude t h a t on every conic region of t h e form A [L] we have
f o r all h uniform in Moreover,
E
A+[L], v v
5
E
(= subset of r e g u l a r elements in a );
and h, $(H) =
mi&*++(^)
(H
E
i s t h e Harish Chandra c-function.
and i f b = m, then b 6 w
*
a*'
0
is a constant.
It i s meromorphic on
a*
C
i s holomorphic and everywhere nonvanishing on a
,-'
n*
s u i t a b l e tube domain i n
a); m 2 0
the
:a
containing
a*,
with both
2
and
L-l
having
a t most polynomial growth t h e r e i n For estimates on c o n i c a l s e t s A'[L]
all of
where
c ~ ( A + ) t h i s is not e n o u a ; we must consider L
i s s t i l l compact but i s -wed
t o be i n
is
Cl(a+).
The above considerations can be generalized suitably t o handle t h i s The r e s u l t obtained may be described a s follOWs.
more general s i t u a t i o n .
Fix Ho The elements
Let Mo
cl(a+), Ho f 0.
E
v
E
a:
s t e b l i z e r of
So t h a t
Let
f o r all v
E
a*'
and
approximation for
Here
Eo
>
0, p
Mo
Let
a;L1
h
po be the Bnalogue of n(C%)or. Write
mo
p
for
for the
sl = 1, s2,..., sm be a ccmplete s e t of
&
be t h e counterpart of
Then we can f i n d a compact neighborhood
Mo.
4.
and l e t
robo.
representatives for
8(v : ).
Po = 1/2 Za9,a(Ho)=o
m
in
Ho
i n G.
parametrize the elementary spherical functions of
also; we s h a l l denote these by the P W Mo,
Ho
be t h e centralizer of
L of
Ho
5
f o r t h e Poup
i n cl(a+)
such t h a t
i n t h e conic s e t A+[L], we have t h e following
V( v :h) :
2 0 are constants and t h e 0 i s uniform i n h and
V.
Wave packets i n spherical Schwartz space
1. The estimates (15) and (16) of
goes t o i n f i n i t y on
$ 3 allow us t o study the behaviour, as x
of t h e "wave packets"
G,
The approximation (15) of $3 suggests t h a t we take Schwartz space
do*)
of t h e r e a l vector space
5
a*;
a t o be i n the usual however, a s t h e function
= m w is holpmorphic, we need t o replace a with c has poles and only rrr Ta. The approximation (16) of $3 coupled with induction on dim(^) now shows
t h a t the wave packet
i s a well4efined element of following sense:
cm(G//K)
for any integer
and i s rapidly decreasing i n t h e
rn 2 0,
elements
ul,%
u ( % ~ ) ,there i s
E
a constant C = ~ ( m :ul :u2) > 0 such t h a t
I
IT;(ul;h;u2)
j Ce-'(log
h)(l
+
Illog hll)'
for all h e c ~ ( A + ) . Motivated by t h e above remark l e t us w r i t e f
E
@ ( G ) f o r the space of all
cm(G) which are rapidly decreasing i n t h e above sense.
obvious family of seminorms it becomes a Frechet space. Chandra Schwartz space of
G,
2
G.
map
*g
I+ f
It i s s t a b l e under
G which actually define represen-
@(G) i s furthermore closed under convolution and i n f a c t t h e
t a t i o n s of f,g
It i s t h e Harish
and i s contained i n L (G).
l e f t and r i g h t t r a n s l a t i o n s by elements of
Equipped with the
@ ( GX) @ ( G-+ ) @ ( G ) is continuous, making @ ( G ) a
from
topological algebra.
c(G//K) = @(G) n cw(G//~).
Let us put
I f we now observe t h a t f o r
v
E
a*
we can prove the following r e s u l t which I c a l l t h e f i r s t wave packet theorem
( f o r spherical functions ) Theorem.
For any
i s an element of
.
a E
t h e wave packet
C(G//K); and t h e map
The next s t e p i s obviously t o form
pa
defined by
a w (P, is continuous.
and attempt t o show t h a t it is
dependent of
a.
a(v) where y
y
It i s not d i f f i c u l t t o show t h a t there is a unique
-
m-invariant continuous function Y( ) on a
E
i s a nonzero constant in-
a*
such that f o r all b i n v a r i a n t
@(a*),
JT&x)d-v:x)dx=~~)a(v)
(2)
ere we must note t h a t for using
f = O(N)
v
instead of N
one finds that f o r any
f E
E
( v c a*)
", 'm= rp(-v : -)).
a
O n the other hand,
for computing the Harish Chandra transform,
@(G//K)
So, using Fourier inversion on A,
with the measure dh being dual t o dv,
the formula (2) reduces t o ep(log where we r e s t r i c t
h)L
qa(&)d;i = a*'
dv)a(v)eiv("g
h)av
a t o be i n C;(O*)~ for obvious convergence reasons.
We s h a l l now evaluate the l e f t side of (3) directly, with a r e s t r i c t e d t o be i n C:(a*'),
and obtain for such
replaced by a constant.
a the same formula (3) with
This w i l l then prove that
')'
Y i s a constant. Our
calculation w i l l e x p l i c i t l y evaluate the constant too for a very specific normalization of
dz.
I s h a l l now describe i n a very heuristic way Harish Chandra's beautiful
method f o r evaluating the i n t e g a l
W e f i r s t remark t h a t the study of the asymptotic behaviour of
cp(v : .)
leads
t o t h e following approximation. x
(4) Then, i f
a
E
a compact i n
=e A+
:x
=x
E
G, v
E
$I,
.E c(sv)eisv('(x)) sctu "
a+ ( r a t h e r than i n (%(a+)), we have
- cp,(v :xa) -+ o
cp(v :xa)
c ~ ( G ) topology, u n i f o r m i n
subsets of
2'.A s
a E c:(a**),
V
a s long as
v varies over compact
a f k s t step we form t h e "truncated wave packet" vm),(x)
(6) where
x v
x
and goes t o i n f i n i t y on a conic region A+[L] where L i s
(5) i n the
Write, f o r
=l*
cpm(v: x ) a ( v ) ~ v ) l - ~ d v
and e v a h a t e
We f id, proceeding formally,
To evaluate the inner i n t e g r a l we use the i n t e g r a l formula f o r
i n t e g r a l coming f r o m t h e element
s
E
tu
is then $(-sv)
s; t h e
( c f . 92).
So,
remembering t h a t
we get
The rigorous proof of (8) is a l i t t l e more d e l i c a t e since' t h e i n t e g r a l s over
g
do not converge when
where they converge
v
E
a*
and so one has t o go into the complex dmain
ge.
The formula (8) i s exactly of t h e same form as (3) with (P,,,
i n place of
$ v)
=w
and
To complete t h e proof of (3) one i s l e f t with proving
pa.
t h e remarkable f a c t t h a t
for all h
.
The idea for proving t h i s i s t o study
. - qm( : .)
q( v : )
t h r o u a t h e system of d i f f e r e n t i a l equations s a t i s f i e d by
V
cp(v : ).
on
fh
It turns
out t h a t the difference
can be expressed as a linear combination (with coefficients depending on h) q( v : ),
of derivatives of over
the derivatives being from
of derivatives with respect t o
-n
F.
are zero, we obtain ( 9 ) .
the actual carrying out of t h i s argment i s very delicate. ge of
Since integrals However,
Since an interchan-
integrations i s involved, one needs absolute conver@;ence;but t h i s i s
not available, and it becomes necessary t o smooth things by convolutions with elements of
~06)( (causing 9
no problems since these convolution operators
ccnmnute with integration over
8.
The actual r e s u l t , which I c a l l the second
wave packet theorem ( f o r spherical functions), i s then as follows. Theorem.
Let
Here the measure
3
a
d ;
E
C ( a*)m.
men
i s the one t h a t occurs i n the integral representation of
and m a y be specified by the condition From t h i s theorem and ( 2 ) we then obtain
e - ~ P ( H ( C ) ) ~=; I.
For obtaining completeness and hence t h e Plancherel formula it only remains t o prove t h a t the map
a i+pa
is p &
c(G//K).
A simple argument reduces
f I+ @ is i n j e c t i v e on
t h i s t o showing t h a t t h e Harish Chandra transform C(G//K); i n f a c t , l e t
- cpa)
shows t h a t g = pa.
g E C(G//K) and l e t = 0,
a =
-1
W
and the i n j e c t i v i t y of
Then t h e formula (10)
#g.
#
on
@(GI K) would
The i n j e c t i v i t y is proved by an argument using induction on
which reduces the assumption t h a t character
@ = 0
8 of t h e discrete series,
give
dim(^)
t o t h e conclusion t h a t f o r some
O(f)
f 0.
This is a contradiction
since the discrete s e r i e s representations axe never of class .1. This axgument is a special case of a more general one which w i l l occur again.
The
main theorem of the spherical theory m a y now be formulated as follows. Theorem.
The map
f l-3
Hf, ( # f ) ( v ) =
JG
f(x)(~(Y:X)dx
is an isomorphism of the convolution algebra a l ~ b r aC( a*)m,
@(G//K) onto the multiplication
both being regarded as Frechet algebras.
For t h e inverse
map we have
Here
w =
Iml,
measure t h a t goes over v i a
.fz e 2 p ( H ( S ) ~ = we have,
dh dn where
dx = e
1; and
8
t o t h e measure
sKdk = 1, d g on
ff
and
dn
is t h e
f o r which
dh and dv are d u d t o each other.
Furthermore,
5. Asymptotic behaviour of tempered eigenfunctions f o r fl.
The constant term
1. I n order t o t r e a t t h e problem of eigenfunction expansions on
G
in f u l l
generality it is c l e a r l y necessary t o go beyond t h e spherical case discussed
in §§3 and 4.
To t h i s end Harish Chandra introduced the very elegant notion
of eigenfunctions f o r
8
on
transforming c w a r i a n t l y .
K
i.e., T2
a pair
X
2
in U
a representation of
(on the vectors of
U)
V
K
*om
in U.
We say
T
i s unitary i f
and
T
in U
is meant a
K
T2(k2) T~
are.
(u
T2(k)
E
and
r1(k)
act
a c t from t h e r i g h t :
U, kl,%
Afbction
K,
i n U,
It w i l l be convenient t o l e t
t h e l e f t and t o l e t
7(5.'%)u = Tl(kl)u
K
i s a representation of
T1
and
I ?i s the. group opposite t o
where
( 7 7 ) = T where 1' 2
K
U be a f i n i t e dimensional
More precisely, l e t
By a double representation of
complex vector space. representation of
G, with values i n a bimodule of
E
K)
f :G+U
iscalled
7-spherical i f
cm(G : T)
w i l l be t h e space of smooth
=-spherical
f;
C( G
:T)
and
C:(G : 7)
its obviously defined subspaces. To see the naturalness of t h i s concept l e t sentation of
in a Hilbert space H,
G
~ ~+ (c). 8 We
assume t h a t
t h e isotypical subspace of A
$
a be an irreducible repre-
possessing an infinitesimal character A
= nl i s unitary. For any b E K l e t H be K b H of vectors transforming l i k e b; and for any T
= %EF Hb.
finite set F c K
let
dimensional.
I$ be t h e orthogvnal projection
U
Let
We know t h a t each H
F
H -+ $
.
is finite W e then define
t o be t h e algebra of endomorphisms of t h e Hilbert space H., regarded as a
Hilbert space i t s e l f under the scalar product
(u,v) = tr(uvt).
Write
rF(k) = %n(k)EE, so t h a t
T
then
E
$
(k
E
K),
'Cl(k)u = $(k)u,
u ~ ~ (= k urF(k) )
i s a unitary double representation of
C ~ ( G : T ) and
?rF
K i n U.
9
i s an eigenfunction of
(k
E
K, u
E
u),
I f we put
f o r t h e eigenhomo-
morphism XB:
The fundamental problem i n doing harmonic analysis is t o determine t h e behaviour of G = KAJTO
f
if
7-spherical eigenfunctions a t i n f i n i t y on the group.
i s an Iwasawa decomposition of
i s any
7-spherical eigenfunction on
t h e behaviour of h
m, i
-de
f ( h ) when h
E
behaviour of
f ( h ) when
f o r all roots
Or
of
it is a question of studying
G,
and goes t o i n f i n i t y .
C~(A;)
it i s not necessary t h a t a ( l o g h )
f i x a psgrp P = MAN
Then
G.
Bow, when
for all positive roots
-t m
containing Po = M$IONO h
Suppose
+ G = K c I ( A ~ ) K and so,
r e l a t i v e t o P,
a. So
("standard") and study t h e
i.e.,
when a ( l o g h)
+m
Po.
The central observation of the theory is t h e following.
Since
G = K(MA)K, one can reduce t h e d i f f e r e n t i a l equations s a t i s f i e d by
a
E
A
on G
o r a t l e a s t on a suitable open
t o a system of d i f f e r e n t i a l equations on MA, subset of it; these equations, when
f
and a
4 rn
r e l a t i v e t o P,
may
be regarded a s perturbations of the eigenfunction equations on t h e group MA. Consequently it i s reasonable t o expect t h a t
f
may be approximated asymptoti-
c a l l y by a suitable eigenfunction or l i n e a r combination of eigenfunctions from t h e coup MA, which
m%
the approximation being v a l i d on conic subsets of
a root of P a ( l o g h)
+
A.
i s uf the same order of magnitude as
on lllog hll.
Basically t h i s i s t h e same s i t u a t i o n t h a t presents i t s e l f i n t h e study of elementary spherical f'unct ions.
There is however a c r u c i a l additional com-
plication.
On converting t h e perturbation problem t o one involving only f i r s t
order equations we f i n d (unlike i n the spherical case) t h a t t h e unperturbed l i n e a r system admits eigenvalues which do not all have r e a l p a r t s
so
5 0,
Fortunately it turns out t h a t the
t h a t the problem i s not stable.
assumption of temperedness acts as a s u b s t i t u t e for t h e eigemralue property mentioned above. 2.
We s h a l l now make these remarks precise.
We begin by introducing t h e
Given any CSA
so-called Harish Chandra homomorphism.
9 c g,
t h i s is a
homomorphism
which i s actually an isomorphism of
with t h e algebra of elements of
8
u($)
invariant under t h e Weyl coup W(%, 9,) :
If
91c g
group
Gc
is another CSA of such t h a t
Suppose ml c g be the center of also.
g and y
y( 9 ) = C
91c'
i s an element of the complex adjoint
we have the commutative diagram
i s a reductive subalgebra with U(mlC).
If
9 c ml
So, as t h e inclusion w(&,bc)
we have a unique injection
rk(?)
is a CSA of 3
~
(
~
5,
= rk( 9).
Let
it i s a CSA of
!$%) g
~ gives , 9 t~h e )inclusion
making the following diagram c m t a t i v e :
P 94.
It is also clear from t h e previous diagram t h a t choice of
P is independent of t h e 915 Finally, using well lnaown r e s u l t s from the theory of f i n i t e
b.
reflexion groups we have Proposition A.
%g(ml)
i s a free module over
~
~
~ o f( r a n8k =)
[w(yc,!lc) : w(gc,bc)l. Next, we consider a psgt-p P = MAN. of p s p p s
Q = MAN
Q
homomorphism of
MA
with t h e sane A
We w r i t e P(A) for t h e ( f i n i t e ) s e t
(and hence M).
We write d
Q
for the
a
inverts
into the positive r e a l s given by d (ma) = e
pQ(lo@; a)
(m
Q
E
M, a
E
A)
where
(a,
nQ are the respective Lie algebras of
exp : a + A).
Let algebra of
5
Given P,
be the centralizer of
MA
crucial:
E
A
and log.: A
-t
P(A) by
(or
a)
and i s a reductive subalgebra of
g
Q'
in
g.
men
m.,
is the Lie
g of t h e same rank a s
although it is independent of t h e 9/12 P within H A ) . For our purposes t h e following camputation is
We often write choice of
we define
A, N
instead of
P
g.
Proposition B.
3
unique
zl
Let
u(?~)
E
and l e t notation be as above.
P = MAN
Zl
(ii)
z
(iii)
=q
If
For
ut
3t
then E
o
ap
is the antiantomorphism of
I
any z
Suppose f
zl:
mod 8(n)u(gc)n $2)
u
gc,
3,
g?)
c
z1 =
(iv)
zc
such t h a t
m a e w e r , we have t h e following properties f o r the map (i)
Given
=
9
end
~ ( 9 ~ which ) is
~ $ 2 )= ~ p ( z ) f~ o r all
z E
-id
on
9.
l e t us write
8
is a smooth function on
Then, f o r
G.
ml
E
MA,
We s h a l l now make an estimate of t h e term
Since rp(z) E 8(n)u(gc),
b
U( gc )
E
.
E
sP(m1) = Ib(ml)e(nlll
= -(x,Bx')(x,x'
~ ( n ) . Let
fur
Cj
Let
u(?~)
X c n,
where t h e s u f f i x denotes r e s t r i c t i o n ,
being defined r e l a t i v e t o t h e Hilbert space structure on
(x,x')
where
We have t h e following lemma.
Lemma C.
11 11
it is enough t o estimate f(ml;8(x)b)
q1,q2
E
%). Let
E
u(?~),
depending on I! and
smooth function
g
and any ml
6
g given by
<
@(x.) ( 1 5 j p ) be an orthonormal basis J b E u ( % ) and X E n. Then 3 elements
v2 MA,
with the following property.
For rtny
For, since
ad
5
stabilizes
8(n),
q2e(x) = for suitable
3. J
E
u(%).
we can w r i t e
C
15s
e(xj)Cj
Furthermore, as MA s t a b i l i z e s
8(n)
and
I au(ml) I
5 %(ml)-
The lemma follows now from t h e following calculation:, dql;ml;q20(x)b)
= E dql;ml;e(xj
)Cjb)
j
dq1e(xj)
=
m 1 ;ml;Cjb)
3
= Remark.
2s aij(ml)drll~(~i);ml;~jb)i j
Given a vaxiable element
a
€
A,
l e t us w r i t e
P a + m (a goes t o i n f i n i t y along P)
if a ( l o g a ) + m f o r each r o o t a of
(P,A).
I f we put
min % l o g a )
pp(log a ) =
a
E
s e t of r o o t s of
(P,A)
t h i s means
pp(log a)
For
-1
( ~ d ( a ) ) ~ (the ~ ) eigenvalues are
e "(log
a)
and hence we have the estimate
This estimate, i n conjunction with Lemma C, r e v e d s that the second term on the r i g h t of (5) may be thought of as a perturbation term r e l a t i v e t o the first. Let
be a homanorphism and l e t X,
f
be an eigenfunction for
8
with eigenhomorphism
i-e.,,
By Proposition A we can find elements
such t h a t the v
i We now go over *om
ccmponent 0
j
form a basis for f
$ml)
t o the vector
@
8(ml)
E
Clearly
8
components, the
j
may now be identified with an r X r
matrix representation with elements i n ij
with r
+(8). th
being given by
The regular representation of
z (v)
regarded as a module over
such that
8; indeed, for v
E
~ ( y )3 ,
unique
i s the representation fiescribed above.
using ( 5 ) .
W e now have, fin-
5E
MA
and
Let
(7) Then
where vi = d-'0
P
vi 0 d P
E
$5) once
again.
The equations (8), taken together w i t h Lemma C a d the remark follawing
it, constitute the perturbation problem on MA that I *oh Since A
is abelian and
in (8) where the
Hi
a
cdml), w e m w t a b e
form a basis of
a;
v=Hi
about e a r l i e r .
(1 f i z d j d a ) )
the resulting equations
are of the f k s t order in the unperturbed p a r t -
It i s well known t h a t i f write X
in the form
9
is a
CSA
of
5
contabhg a,
we can
f o r same A
E
* gC.
The ~ ( g ~ , ~ ~ ) - oof r b iAt i s uniquely determined by X,
and the matrix
r(v)
has eigenvalues
In particular,
r(H. ) has the eigenvalues
Since it i s the complex Weyl group t h a t governs here, it is i n general not possible t o guarantee s t a b i l i t y
3.
- by eigenvalue considerations.
To overcome the & i f ficulty suggested above Harish Chandra works only with
tempered eigenfunctions.
Let us r e c a l l the Iwasawa deccmposition
Analogous t o the estimates f o r the spherical functions
a function
fm all
f
$,q
on
G
c K, h
s a t i s f i e s the
E
c~(A:).
If
weak
f
cp(v: .)
derivative
af%
i s a K-finite
& f i n i t e function,
f
f
K
defines a
i s tempered; i n t h i s case, each
(a,b c u ( ~ ~ )s )a t i s f i e s the weak inequality.
double representation of
we say t h a t
inequalitx i f for some m 2 0, C > 0,
satisfying the weak inequality i s equivalent t o saying t h a t tempered distribution, m , briefly, t h a t
G = KA$IO.
i n a f i n i t e dimensional space U.
Let
T be a
W e then denote
by
t h e subspace of
cm(G :'G)
consisting of all & f i n i t e tempered elements.
For
functions i n A(G :T) one can develop a powerful asynptot i c theory using the
perturbative method outlined i n the preceding paragraph.
I s h a l l describe
the basic r e s u l t s of t h i s theory now. Let me s e t up some notation.
Q =MAN
i s apsgrp, we write
If
=TI
T
usual the maximal compact subgroup of
.
where
54
$
fixed by
M
= K
n~
8.
We write
=K
n~ 2
i s as for
Since (see Varadarajan [l], Part 11, $8, Proposition 17)
cp(0 : ).
we can replace let
is a double representation and
7
a(x)
e
-0
by
3 i n the d e f i n i t i o n of the weak inequality.
be the distance of
Riemannian space with a
xK
G-invariant metric.
Since we often vasy t h e group Theorem A.
Let
f
fYom K
E
we write
G,
/A(G :z)
G/K
considered as a
Then the weak inequality becomes
EG f o r 8 t o be more precise.
and Q = MAN a psgrp.
f Q E IA(MA :T ~ )such t h a t , for each m a ~ ( m a )(ma) f
in
E
Then
3 a
unique
MA
- fQ(ma) -+
fQ i s called t h e constant term of
Also,
f
0 as
Q
a +m
along Q.
Let me now describe t h e most important of the properties of the msp f
H
for
fQ. F i r s t we have
m
E
MA, k
same notation.
E
K.
Next we have the t r a n s i t i v i t y .
Fix a psgrp
---
& =MAN.
To formulate it we need
Then there i s a canonical bijection
between the s e t of psgrps
Q C
Q
*Q
and psgrps
E;
of
the correspondence
is defined by
and the Langlands decompositions
Q = MAN, *Q = YA%
(15) are related by
% - be a p s g q of
Theorem B.
Let
&
correspondence.
Fix
f s A(G :T).
=
G
and l e t
; E z ,
For any
let
Q
2 *Q
be the above
f ~$3 , = f$m3
(m 3. Then E
and
Fina3J.y we have the following approximation r e s u l t justifying our terminology of
f
Theorem C. p s m Q = MAN.
f
E
&(G:
h
E
c~(A:)
7).
Q Let
a s the constant term. QO =
M$i$lo
Write
= infa a root of (Q,A)a
Then 3 constants
In p m t i c u l m , for any t > 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) <
-.
For any regular
k
E
(-l)1/2g:
let
v[X]
be t h e subspace
of functions defined by
For any discrete c l a s s
A
o
the matrix coefficients of
E
let M d' w;
L ~ ( M be ) ~ the Hilbert space spanned by
and l e t
1,
where we a r e n a t u r a l l y identifying This i d e n t i f i c a t i o n , since space s t r u c t u r e on
@(M:
T
M
) as a subspace of L2( M ) 8 U.
i s a l s o a H i l b e r t space gives a n a t u r a l Hilbert
U
We have the orthogonal deccrmpositions
V.
v =!I vrw1 =II v[XI.
(8)
X
W
These a r e a l l f i n i t e since
i s f i n i t e dimensional.
V
The theory of $6 now leads t o t h e following theorem introducing the c-functions. Theorem B.
Let
t h e s e t of p o i n t s (P,%),
P
E
v
Ply P2 of
be psgrps i n P(bR),
5 such t h a t
v
Q(QR). Then, f o r any
E
(v,a)
#
m
= b(bRlqR).
0 f o r each root
5' and s
E
m,
Let
5'
be
a of
we have uniquely
defined endomorphisms
of
V
such t h a t
V $ E V , ~ E M , ~ E % .
a r e much nicer.
1
-
.
( s : ) a r e c e r t a i n l y cm on '3' Actually they 1 The point i s t h a t t h e estimates furnished by Proposition A
The functions
cp
2
f o r t h e E i s e n s t e i n i n t e g r a l when
v
varies in 5
perturbation theory of $5 not only f o r small tubular domain 5c( 8 )
Let
P E P(QR)
and l e t
allows us t o do the
v E 5 but f o r
containing 5
where
v
i n a sufficiently
4,...,aq
where
p l i c i t y of
(changing
cyi
(P,L ), mi R
a r e a l l the d i s t i n c t roots of P changes
s only by 2 1 ) .
being t h e m u l t i -
Then we g e t t h e
following consequences of doing t h e perturbation theory in ~ ~ ( 5f o)r some 8
> 0. There i s some
Theorem C.
8
>
such t h a t
0
extends t o a holomorphic function on zc(6),
f o r all P1,P2
E
and
P(bR)
S E ID.
For any we write exp(m
n
P
E
P(gR), P = ML It, R
~ ( x ) ,~ ( x ) ,exp H&x),
PI),LR and N
Theorem D. such t h a t
Fix P
we have and
n(x)
G = K exp(m
n @)LRN.
For
x
x
in
f o r t h e components of
G,
E
K,
respectively.
E
P(QR) and l e t zc(p)
(vI,a) > 0 V roots
a of (P,$).
extend t o holomorphic fbnctions on Zc(P);
be t h e s e t of a l l
Then
V E
zc
c ( 1 : v ) and c- ( 1 : - v ) p p there, they a r e given by the
PI
PI
f o l l a s i n g (convergent) integrals :
Here
Jr
E
V,
V E
;Fc(P), m
E
N,
and
d :
i s normalized by
Jz-2pp(Hp(;))
-
dn= 1.
The proof of Theorem D resembles closely t h e proof of the analogous r e s u l t f o r the
c-functions t h a t occurin the spherical theory.
eigenfunction
$ of type II(X)
given on some Zc(ZiO) x G,
on 5 x G
holomorphic i n
the kind introduced i n Proposition A .
Let
We s t a r t with an
but assume t h a t it i s a c t u a l l y and s a t i s f y i n g estimates of
v, P
E
63(gR) and l e t
Ho
E
QR
be
such t h a t
a(Ho)
7
E =
~ ( 6> ) 0
such t h a t
as
t
E
5;(5,X)
= m exp t Ho
where
m
t
-t +m,
v
a of ( P , L R ) Then we can f i n d
0 'v' roots
v ~ ( s ~< H vI(sHO) ~ ) Vs
( = s e t of m
+
E
6
E W,
#
v
with m(X + i v )
sc(6)
E
In particular, i f
so
roo
E
$
$
=
E( P : Jr : v)
.
(13) e
v (H (m)) +
m
~ ( m - I )=
Choose m = -.a'1' 0 (14)
lim e
-P
h his $,,(v
N.
i n t o an i n t e g r a l over
t h e formula, v a l i d f o r
E
where
P
E
P(gR), Jr e V.
K/%
To
that
%:
m e M, a 0
E
K(?))-'e
$,
v-( lip(
$(moa) =
-
P m. a -t
JiT( K(;)
needs some d e l i c a t e analysis on
f).
31-v+( ~p(aO) -
dn.
We g e t
-( iv+pp)( l o g a )
)v( p(;)m0)e
v-(~~(")dn.
By ( l l ) , the l e f t s i d e is
: v)Jr)(mo).
:mo) = (cpl
Corollary 1. Fix
w
,.
e Md.
Then
V[w]
i s s t a b l e under
c p , p ( l : v). Corollary 2.
i s such t h a t
This gives, with
JN ~(&-'P(P)-')T(
where
O),
$p, s o( v : m ) .
compute t h e l i m i t (11) we f i r s t transform t h e i n t e g r a l over represents
#
and
so,
giving a method o f e x p l i c i t determination f o r a p a r t i c u l a r This technique is applied t o
S -> 0
d e t cpl p ( l : v)
is not i d e n t i c a l l y zero.
c-
PI
( 1 : v) p
and
A s a much deeper consequence we mention t h e f a c t t h a t the second wave
packet theorem now y i e l d s an e x p l i c i t formula. Theorem E.
For
a
$ E V,
6; on
N
C;(S' ), PI, P E P(qR)
and l e t
=L
Pa Then, with
E
O ( V ) E ( P:~9 : V : -)dl'.
being normalized a s above, we have f o r
I f we had more information on the
c-functions
-
m
E
M, a E LR,
such a s f u n c t i o n a l
equations s a t i s f i e d by them, the above formula would be a very s u b s t a n t i a l s t e p i n the e x p l i c i t determination of t h e Plancherelmeasure. deeper study of t h e E i s e n s t e i n i n t e g r a l .
This needs a
I n the next lecCure I s h a l l sketch
t h e o u t l i n e s of t h i s study and indicate how one can obtain an e x p l i c i t Plancherel formula from it. To sketch a t l e a s t formally the argument f o r deriving Theorem E we proceed a s follows.
We have, f o r
$a( (m)
m
E
MLR, a s Pp(~p(:))
2 0,
= lim J- e - ( ~ + E ) P $ H $ ~ ) ) &+O+ N $p,dG)dz
while
4 V)T(K ( ~ ) ) EP( P1:
=
For f i x e d
E
: V : p(K)m exp H ~ ( ; ) ) ~ V
> 0, we observe t h a t f o r any p
SC(P). Hence, f o r
m
E
M, a E
L~'
E
5, pE = p
+ imp
i s in
Letting
E +
(
O+,
3(ma)
we g e t
r a ( v ) ( c p I p ( l : SV)CPI JS
=
( S : v)).)(rn)e isv(1og a )
pl
which is e q u i v a l e n t t o Theorem E.
8. The Plancherel measure and t h e P l a n c h e r e l formula 1. Our first o b j e c t i v e i s t o s t a t e p r e c i s e l y t h e main theorems o f Harish
Chandrals a r t i c l e [ 5 ] on t h e P l a n c h e r e l formula. To begin w i t h we work with a f i x e d so t h a t 3 = a*,
CSA
g
We p u t
C g.
a =
qR
and use t h e n o t a t i o n o f §§6 and 7 without f u r t h e r comment;
i n p a r t i c u l a r we r e c a l l t h a t 5' f o r a l l roots
+stable
a of
(P,A)
Let
P = MAN
i s t h e s e t of
v
E
5 f o r which
f o r some (hence every) psgrp
(v,a)
#
0
P E P ( A ) . A s usual
m =m(ala). Theorem A.
*
A
E
P ( A ) , w E Md, v
class of the unitary representation not change i f V E
5',
lr
p,w,v
TheoremB.
(w,~)
is replaced by
is irreducible Let
lr
p,w, v (sw,sv)
E
5. Then t h e equivalence
i s independent o f for
s
E
w.
P and does
Moreover, i f
.
P , P EP(A), 1 2
A
S E W ,
Y E ~ ' ,u
f i n i t e dimensional u n i t a r y double r e p r e s e n t a t i o n of
c Ma. K
Let and
T
bea
V = OC(M: T ~ ) .
Then
c
( s : V ) d e f i n e s a d i j e c t i o n of p.- l p, t h e r e is a ( u n i q u e ) f u n c t i o n 11 > 0 on
V[d]
$X
h
Md,
,: E
V E
such t h a t f o r a l l
T,P~,P~,
3' cp l P - ( s : v) 2 I
on
3'
V[sw] . Moreover,
onto
~ [ w ] , where
Theorem C.
t
cP 2
lP
- idw
( S : v) = p(w: v ) - l 1
idw denotes t h e i d e n t i t y o p e r a t o r on
Let
T
be a s i n t h e previous theorem.
~ [ w ] ; and
We t h e n have t h e
following .
(i)
For
P1, P2
F(A), s
E
E
b, cp
2
f u n c t i o n on Jc. (ii)
V[w]
extends t o a meromorphic
1
( s :v) = cp 1 2
I
( 1 : sv)-' 2
CP2]
( s :v).
( s : V ) is holomorphic and u n i t a r y everywhere on 5, and maps p2Ip1 onto V[sw]. Moreover, i f P , P 1 , P 2 ~ P(A), s , t E w, OC
The E i s e n s t e i n i n t e g r a l s s a t i s f y t h e f u n c t i o n a l e q u a t i o n s
(iii)
for
(s :v )
1
Put Ocp 2
Then
1
P,Q
E
P(A), s E m, Jr E V.
The next theorem d e s c r i b e s t h e a n a l y t i c p r o p e r t i e s of t h e f u n c t i o n Theorem D .
For any
h
w E Md,
meromorphic f u n c t i o n on Zc
p(w : . )
( denoted by
such t h a t
(i)
~ ( w : .)
i s holomorphic on Zc(6)
is t h e restriction t o p(w :
-)
also).
5
Moreover,
p.
of a
3
6 > 0
(ii)
Let
f o r constants
rl,...
)rr
property. K.
Fut
A i = (ri)R
There a r e constants
Let
we have an estimate
be a complete s e t of
which are conjugate. Theorem E .
b > 0, r 2 0,
@-stable CSGfs of
and l e t C ( A )> 0
S
.
( A ~ ,. . ,Ar).
be the s e t
( A E S)
no two of
G,
with t h e following
be any f i n i t e dimensional unitary double representation of
T
Then, f o r any
f
@(G : T ) ,
E
n
Here, f o r given A
d(u)
E
MA
S,
i s the c e n t r a l i z e r of
denotes the formal degree o f t h e d i s c r e t e c l a s s
i z a t i o n of Haar measure on M);
A w
in
for
G;
A
w E Ma,
(using some nomal-
and
If one chooses s p e c i f i c Haar measures, say standard ones, then t h e constants
C(A) can be e x p l i c i t l y evaluated.
From t h i s , the usual
Flancherel formula i s q u i t e easy t o derive. show t h a t t h i s formula is t r u e f o r a l l
2
L -version of t h e
A more c a r e f u l treatment would
f E @(G) ( n o t only f o r
K-finite
f).
It i s c l e a r l y important and useful t o i n v e s t i g a t e whether t h e r e a r e e x p l i c i t formulae f o r
k.
Actually t h e r e i s a product formula f o r
~ ( w :
.)
which i s a f a r reaching generalization of t h e product formula f o r the spherical Y
Plancherel measure t h a t Gindikin and Karpelevic established.
To formulate t h i s
we need some notation. A root
a
0< r < 1 (r
of
R).
E
reduced roots of z(FP)n~(pl)
(g,a)
i s c a l l e d reduced i f
For any psgrp
(P,A).
If
(i;,=o ( P ~ ) ) .
P1,P2
P E $(A ') E
m
i s not a root f o r
we write
P ( A ) , we write
d(p1,p2)
Z(P)
f o r the s e t of
c(P~P I1)
f o r the s e t
= [ I ~ ( P ~ / P (~[ ) I I i s the
cardinality),
d
Suppose of zeros of
P
E
-
P ( A ) and
a; Za,
aka.
= q zbl
i s a metric on P ( A ) .
a
C(P).
E
Let
t h e centralizer of
Then Z
a
= M&la
a(a)
a(a)
be the hyperplane i n
in
*pa = M A
and
A
G;
a
= exp p Ha;
i s a psgrp of
N
a a
a
N,
=
(here
Ma
Ma = OZa a s usual; see Varadarajan [ l 1, Part 11, p. 20 f o r the d e f i n i t i o n of t h e function L let
wr)
see also Harish Chandra [
H OL;
be t h e function defined as above on
9he product formula f o r Theorem F.
p(w :
a constant
3
.)
3 1, $2).
For
A
w E
M d'
a*
a and l e t
i s then given by t h e following theorem.
c >0 1
such t h a t
h
for a l l
P E P(A), w E Ma.
In view of t h i s theorem, the problem of e x p l i c i t calculation of t h e Plancherel measure reduces t o t h e case when maximal.
We distinguish two cases according t o whether
rk(G) = rk(K); CSA
9
dim(^) = 1, i.e.,
r k ( ~ )> r k ( ~ ) o r
t h e f i r s t a l t e r n a t i v e is equivalent (when
being fundamental i n
g.
P is
dim A = 1) t o t h e
We r e c a l l t h a t f o r a fundamental CSA there
a r e no r e a l roots. Theorem G1. Let
R+
Suppose
1)
i s fundamental i n
be a positive system of roots of
s e t of
R+
choose
R
+
(but
g
( g ,gc),
dim A
and l e t :R
consisting of complex (= nonimaginary) roots. so t h a t
chosen thus, l e t
R:
+ [Re]
and l e t
w+ = k R + Ha; C
( -llP u+(A)
i s r e a l and
2 0
be t h e sub-
We can always
i s stable under complex conjugation. = 2p,
arbitrary).
then
With
R+
for a l l
A
i g*.
E
3loreover, there i s a constant
c
2
> 0 such t h a t f o r a l l
h
w E Ma,
V E
5, p(o : V ) = (-1lpc
where
(w)
(
F
-
1
corresponds t o
2
+TB (X(w)
w
+
+
iv)
i n the usual parametrization of
h
Md. Remark. (P
E
F(A))
When
tj
is fundamental, the s e r i e s of representations
i s c a l l e d the fundamental s e r i e s of
p,w, v These representations
G.
were proved by Harish Chandra t o be i r r e d u c i b l e f o r
n
all v
E
5
in [ 5
1.
For
complex groups t h i s had been established much e a r l i e r bywallach [ l ] and Zhelebenko
ill,
t h e case of
S L ( ~ , C ) going back t o Gel'fand and Neumark [ I ] .
Recent work on representations from t h e infinitesimal point of view has l e d t o an a l t e r n a t i v e approach t o t h e fundamental s e r i e s ( c f . Varadarajan [ 3 1, Enright [ l ] and Enright-Wallach [ I ] ) . t r e a t t h e case of complex
v
This approach makes it possible t o
and t o determine completely t h e
k-multiplicities
of these representations. To complete t h e e x p l i c i t determination of remains t o consider t h e case of roots of
(gc,tjc)
rk(G) = rk(K).
such t h a t the s e t
R:
p(w : .)
E
dim(^) = 1 it
We s e l e c t a p o s i t i v e system of complex ( = nonreal, non-
imaginary) roots i s s t a b l e under complex conjugation.
v
when
We put, f o r
w E
M^d'
3, w+(w : V ) = u+(x(w) + i v )
where w+ =
GR;Ha
to
[R:]
w.
If
= 2p,
and
X(w)
F
(-1)Y2q;
i s the parameter corresponding
we have
For any i r r e d u c i b l e character
a* E :L
(L
i s the CSG defined by
9)
let
R+
T ~ ,
be a representation of
L with character I
a*
and l e t
ma sinh ma
p0(aX : v) = d(af)-I t r
Pa
where
va =
-*, 2 v a
Theorem G2. 7 0
Let
- %Ta*(?)
i s t h e unique r e a l root i n
a
element of order atmost
C3
ma
2
i n LI;
d(a*)
dim(^) = 1 but
such that f o r a l l w
n
E
M
d'
V E
+
R+,
rag?-11)
and
y i s a certain
i s t h e dimension of
rk(G) = rk(K).
Then
3
oa*.
Put
a constant
5,
with notation as above. 2.
It i s c l e a r l y not possible t o discuss except i n outline how these theorems
a r e proved. The s t a r t i n g point is t o identify Eisenstein i n t e g r a l s with matrix coefficients of the
Kp , y v .
To do t h i s we introduce C(K x K) = C ~ ( Kx K)
with the pre Hilbertian s t r u c t u r e determined by
It c a r r i e s the double representation
For any f i n i t e s e t
n
FC K
T
defined by
i s t h e f i n i t e dimensional subspace of a l l
where %(k) = ZbEF dim(b)chb(k), c\ s t a b l e under
TI
such t h a t
being the character of
b.
UF
is
.
By varying F we obtain a family of U~ double representations such t h a t every f i n i t e dimensional double representation of
K
I.%(~) V
7
=
i s contained a s a d i r e c t summand of a d i r e c t sum of these.
Fix now w space
z
and we put
v
A
E
choose a representative
Md,
~ ( o ) and l e t a s i n $6.1.
a s usual by
H
o
in
a c t i n g i n a Hilbert
w
be t h e H i l b e r t space of the representation (of h
Fix a f i n i t e F C K,
take
(uniquely) represented by
( ~ n d O ( ~ ( o ) means )
cm kernels
a s above, and define
U = UF
$
V = O@(M: T ~ ) . The endomorphisms T of
K)
a r e then
K~
End(~(o))~ f o r a f i n i t e subset
&)
RC
so that f o r a l l
h e I$'
a p a r t from t h e s e smoothness and f i n i t e n e s s conditions, symmetry conditions
We can then form t h e function
defined by
K
T
must s a t i s f y t h e
414
Then the basic r e s u l t i s t h a t
is a linear bijection
and has the property
f o r a l l x E G, kl,k2 E K. A l l t h e representations
a c t on H. This allows one t o define P,W," intertwining operators between them a s operators on H. I f P1,P2 E P(A), v E Sc,
the operator
J
T
(v) p2Ip1
( s e e $6.1 f o r t h e spaces
J,2 I p 1( v )
@(v))
: 0 (v) + 0 (v) p1 p2
i s defined formally by
The i n t e g r a l s converge only when
v
i s therefore t h e problem of a n a l y t i c continuation; the constants > 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.
RFFERENCES
M. Xguchi [l]
Asymptotic exapnsions o f E i s e n s t e i n i n t e g r a l s and Fourier transform on symmetric spaces, Jour. of Functional Analysis
2 (1979),
167-216.
L. Ehrenpreis and F. I. Mautner [l]
Some p r o p e r t i e s of t h e Fourier transform on semisimple Lie groups, I, Ann. Math.
61 (1955),
406-439; 11, Trans. Amer. Math. Soc.
(1957), 1-55; 111, Trans. Amer. Math. Soc.
(1959), 431-484.
T. J. Enright [I]
On t h e fundamental s e r i e s of a r e a l semisimple Lie algebra:
their
i r r e d u c i b i l i t y , resolutions, and m u l t i p l i c i t y formulae, Ann. Math. 110 (1979), -
1-82.
T. J. Enright and N. R. Wallach [I]
The fundamental s e r i e s of semisimple Lie algebras and semisimple Lie groups ( p r e p r i n t ) . Y
I. M. Gellfand and M. A. Naimark [l]
Unitgre Darstellungen der Klassischen Gruppen, Akademic-Verlag, B e r l i n 1957. Y
S. G. Gindikin and F. I. Karpelevic
111
Plancherel measure f o r symmetric Riemannian spaces of non p o s i t i v e curvature, Dok. Akad. Nauk. SSSR. l & (1962), 252-255.
Harish Chandra
111
Spherical functions on a semisimple Lie group, I. Amer. Jour. Math.
& (1958), [2]
Spherical functions on a semisimple Lie group 11, Amer. Jour. Math.
80 (1958), [3]
241-310.
553-613.
Harmonic Analysis on r e a l reductive groups I.
The theory of the
constant term. Jour. of Functional Analysis
(1975), 104-204.
2
[4]
Harmonic Analysis on r e a l reductive groups 11. Wave packets i n t h e Schwartz space, Inv. Math.
[5]
(1976), 1-55.
Harmonic Analysis on r e a l reductive groups 111. The Maass-Selberg r e l a t i o n s and the Flancherel formula, Ann. Math.
104 (1976)~ 117-201.
[61 Discrete s e r i e s f o r semisimple Lie groups, I, Acta Math.
113 (1965),
251-318. [71 Discrete s e r i e s f o r semisimple Lie groups, 11, Acta Math.
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
<
i f u i s a l i m i t ordinal,
iv)
Xu = X.
Xo = llm{Xg,
o}
we have an exact p i c t u r e o f t h e s t r u c t u r e o f
I f 6 a c t s c o n t i n u o u s l y on a compact m e t r i c space X, I; i s
c a l l e d d i s t a l on X i f x,y s X,
x
#
y,
implies
C l e a r l y any i s o m e t r i c a c t i o n i s d i s t a l . admits an i n v a r i a n t m e t r i c .
For example,
i n f d(xg,yg) g c 6
>
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
<
We would now l i k e t o see which a c t i o n s a r i s e i n t h i s fashion.
D e f i n i t i o n 4.8.
C N
o f X such t h a t
a.
iii)
I n l i g h t o f Theroem 4.5,
r
Xu
i s an extension w i t h r e l a t i v e l y d i s c r e t e
Xu
spectrum, f o r
such actions.
More p r e c i s e l y , t h e r e i s a
N/r
i s distal.
T h i s was f i r s t
111.
D e f i n i t i o n 4.9
( P a r r y [381)
If
(S,v)
i s an e r g o d i c G-space, c a l l t h e
a c t i o n measure-distal i f t h e r e i s a decreasing sequence o f s e t s o f p o s i t i v e measure
{Ail
with
v(Ai)
+
sequence g i c G, then x = y.
0,
such t h a t i f x,y
c S,
xgi ,ygi
c
Ai
f o r some
(We have ignored some measure t h e o r e t i c i s s u e s
i n t h i s d e f i n i t i o n , which a r i s e i f G i s n o t d i s c r e t e .
See C481 f o r a more
c a r e f u l formulation.) Any d i s t a l a c t i o n w i t h an i n v a r i a n t measure t h a t i s p o s i t i v e on open s e t s i s c l e a r l y measure d i s t a l .
Theorem 4.10
[481
A f i n i t e measure p r e s e r v i n g ergodic a c t i o n (on a non-
atomic measure space) i s measure d i s t a l i f and o n l y i f i t has g e n e r a l i z e d d i s c r e t e spectrum.
T h i s i s an analogue f o r measure t h e o r e t i c a c t i o n s o f t h e F u r s t e n b e r g s t r u c t u r e theorem f o r ~nimimal ( i .e.
e v e r y o r b i t dense) d i s t a l a c t i o n s on
compact m e t r i c spaces r167. Another s i t u a t i o n i n which a c t i o n s w i t h g e n e r a l i z e d d i s c r e t e spectrum arise i s the following. Theorem 4.11.
L e t N be a n i l p o t e n t group.
N-space f o r w h i c h L'(s)
Suppose S i s an e r g o d i c
i s a d i r e c t sum o f 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 ( n o t
necessarily f i n i t e dimensional).
Then S has g e n e r a l i z e d d i s c r e t e spectrum
(and t h e o r d i n a l i n n e f i n i t i o n 4.7 can be t a k e n t o be f i n i t e . ) T h i s theorem i s f a l s e f o r s o l v a b l e groups.
L e t N be t h e Heisenberg group,
such a p r o p e r l y e r g o d i c !-space. i n t e g e r p o i n t s , so t h a t
r
+
r
i s a lattice.
where K i s compact.
K
Let
X = ind:
(Y).
Then Y i s a
r -
space
w i t h d i s c r e t e spectrum.
t o N decomposes i n t o a d i r e c t sum o f i r r e d u c i b l e s , and where
the
K = IIIZ,pZ , where t h e p r o d u c t
9 i n c e a f i n i t e dimensional r e p r e s e n t a t i o n o f
1 expressed as n = i n d r ( o )
r
YZ =
There i s an i n j e c t i v e homomorphism
F o r example, l e t
i s t a k e n o v e r a l l primes.
L e t us g i v e an example o f
II
induced
on L ~ ( x )can be
o i s the representation o f
r
on L ~ ( K ) , i t
f o l l o w s t h a t L ~ ( x ) i s a d i r e c t sum o f i r r e d u c i b l e s . I t i s n a t u r a l t o ask which groups have a c t i o n s o f t h e s o r t we have been
d i s c u s s i n g i n a n o n - t r i v i a l way, say an e f f e c t i v e o r e s s e n t i a l l y f r e e action.
A group w i l l have an e f f e c t i v e o r f r e e a c t i o n w i t h d i s c r e t e spectrum
i f and o n l y i f t h e r e a r e enough f i n i t e dimensional u n i t a r y r e p r e s e n t a t i o n s t o
separate points.
I n t h e connected case, such groups a r e i d e n t i f i e d by a
c l a s s i c a l theorem o f i r e u d e n t h a l Theorem 4.12
.
( F r e u d e n t h a l 7141).
A connected group has a f r e e ( o r
e f f e c t i v e ) a c t i o n w i t h d i s c r e t e spectrum i f an o n l y i f i t i s isomorphic t o R~
x
K
where K is compact.
To describe t h e analagous r e s u l t f o r g e n e r a l i z e d d i s c r e t e spectrum, we r e c a l l t h a t a connected group i s s a i d t o be o f polynomial growth i f f o r any compact neighborhood o f t h e i d e n t i t y , W, t h e Haar measure m(wn) grows no
( I f t h i s i s t r u e f o r one compact neighborhood,
f a s t e r than a polynomial i n n.
i t i s t r u e f o r a l l such neighborhoods.)
e q u i v a l e n t t o t h e group b e i n g of t y p e (8)
F o r L i e groups, t h i s c o n d i t i o n i s
[19! 1231.
14e r e c a l l t h a t t h i s
means t h a t every eigenvalue o f Ad(g) l i e s on t h e u n i t c i r c l e f o r a l l g c 6. For example, n i l p o t e n t groups and euclidean motion groups a r e t y p e (R), semisimple groups and t h e ax + b group are not. w i t h C.C.
while
The f o l l o w i n g i s j o i n t work
Moore.
Theorem 4.13.
[341
A connected group has a f r e e ( o r e f f e c t i v e ) e r g o d i c
a c t i o n w i t h generalized d i s c r e t e spectrum i f and o n l y i f i t i s o f polynomial growth. Proof.
We i n d i c a t e t h e p r o o f f o r t h e ax + b group.
The general p r o o f i s
based on t h i s argument and some s t r u c t u r e t h e o r y f o r L i e groups, p a r t i c u l a r l y t h a t o f solvable L i e groups. L e t G = AB be a s e m i d i r e c t product where R = R i s normal and A = R+ a c t s on R by m u l t i p l i c a t i o n .
I f X i s a G-space w i t h g e n e r a l i z e d d i s c r e t e spectrum,
and X1 i s t h e f a c t o r o f X w i t h d i s c r e t e spectrum, then B must a c t t r i v i a l l y on
Y1 s i n c e a l l f i n i t e dimensional u n i t a r y r e p r e s e n t a t i o n s o f G are one dimens i o n a l and t h u s f a c t o r through f o l l o w i n g : suppose v:X
+
Y
B
= [G,G!.
I t t h e r e f o r e s u f f i c e s t o show t h e
i s an extension o f 6-spaces w i t h r e l a t i v e l y
d i s c r e t e spectrum and suppose B acts t r i v i a1 l y on Y; t h e n R a c t s t r i v i a l l y on X.
To prove t h i s a s s e r t i o n , l e t
L-(Y)-module
which i s G - i n v a r i a n t .
Y r e s p e c t i v e l y , and decompose
Thus, we w r i t e
2 L (X) =
p
Let
rQq , p,v
Wi
a f i n i t e l y generated
be t h e given measures on X and
w i t h respect t o
jp
v
over t h e f i b e r s o f
dv(y) where py i s supported on Y 11s a d i r e c t i n t e g r a l decomposition L?(x) = ~ ' ( v - l ( y ) ,py)dv. p =
~7
cp.
This gives For each
~ , gwe have [a(y,g)f](z)
a(y,g)
:
= f(zg)
generated over
2
L ( r - l ( y g ) .uyg)
z 6 cp-l(y).
for
L-(Y)
+
For
= Vy.
given by
and G - i n v a r i a n t m a n s t h a t t h e r e i s
g
Wi
R, yg = y , so
E
Say dim Vy = n.
t i o n o f B on Vy.
(Y) .uy)
F i x i. Saying t h a t Wi
a f i n i t e dimensional subspace, such t h a t a(y,g)Vyg
-1
2 L (
=
$ VYd v ( y )
a l{y}
x
B
i s finitely Vy
c L'( ) - l ( y ) ) ,
and
i s a u n i t a r y representa-
Then f o r each y, we have n elements i n
A
B = c h a r a c t e r group o f B. t h e cocycle i d e n t i t y f o r
(*
(al{y}
x
B)
Furthermore, G a c t s on a
g
(5, i s t h e in//,,
Let
B and one can check t h a t
implies a l{yg} x
E
R,
where n means u n i t a r y equivalence.
s y m e t r i c group on n l e t t e r s ) be t h e set o f
i.
unordered n - t u p l e s o f elements o f
We have a map r: Y
* in/sn,
and (*)
A
r
implies that
i s a 6-map.
The a c t i o n o f G on
namely t h e o r i g i n and t h e 2 h a l f l i n e s . every G
-
orbit i n
o n l y compact G
-
in (and
hence i n
I
R has t h r e e o r b i t s ,
From t h i s i t i s easy t o see t h a t in/sn)
o r b i t i s t h e i d e n t i t y ( i .e.
f i n i t e i n v a r i a n t ergodic measure on
R
in/sn,.
i s l o c a l l y closed, and t h a t the the origin).
But
@,(u)
is a
By smoothness, t h i s must be
suported on an o r b i t and by f i n i t e n e s s and invariance, t h i s must c l e a r l y be t h e zero o r b i t .
Thus,
t r i v i a l l y on each Wi we1 1
.
.
al{y}
i s t h e i d e n t i t y f o r a l l y, so B acts
B
x
Therefore, B i s t r i v i a l on L ~ ( x )and hence on X as
F i n a l l y , we remark t h a t t h e n o t i o n o f generalized d i s c r e t e spectrum y i e l d s a t y p e o f s t r u c t u r e theorem f o r general a c t i o n s w i t h f i n i t e i n v a r i a n t measure t h a t i s sometimes u s e f u l . always t r u e t h a t
S
x
S
I f S i s an ergodic G
-
space, i t i s not
i s a l s o ergodic, where G acts by ( s , t ) g = (sg,tg).
I f t h i s a d d i t i o n a l e r g o d i c i t y p r o p e r t y holds, t h e a c t i o n i s c a l l e d weakly mixing.
More g e n e r a l l y , i f
product
X
x
yX
X
+
Y
has a n a t u r a l G
-
a c t i o n no l o n g e r need be ergodic.
i s an ergodic extension o f Y, t h e f i b e r e d i n v a r i a n t measure on i t [471, b u t t h i s Once again, i f t h i s e x t r a e r g o d i c i t y holds,
t h e extension X i s c a l l e d r e l a t i v e l y weakly m i x i n g over Y.
Given any ergodic
G-space X, t h e r e i s a unique maximal f a c t o r Z o f X such t h a t Z has generalized d i s c r e t e spectrum and X i s r e l a t i v e l y weakly m i x i n g over 7.
Thus we break X
up i n t o a f a c t o r whose s t r u c t u r e we know e x p l i c i t l y , and an e x t e n s i o n w i t h extra ergodicity properties.
O f course, simply by knowing t h a t an a c t i o n o r
e x t e n s i o n i s weak m i x i n g does n o t say very much about i t s d e t a i l e d strucure, so f o r most questions, t h i s i s not a s a t i s f a c t o r y s t r u c t u r e theorem aside from t h e f a c t o r Z.
Nevertheless, weak m i x i n g does c l e a r l y have some i n f o r m a t i o n ,
and thus one can hope t o f i n d t h i s decomposition u s e f u l i n some circumstances.
An example o f t h i s appears i n r e c e n t work o f Furstenberg.
Szemeredi r e c e n t l y succeeded i n proving a c o n j e c t u r e o f Erdos which a s e r t s t h a t every s e t o f p o s i t i v e i n t e g e r s o f p o s i t i v e upper d e n s i t y c o n t a i n s a r i t h m e t i c progressions o f a r b i t r a r y ( f i n i t e ) l e n g t h .
I n c171, Furstenberg
gave another p r o o f o f Szemeredi's theroem, f i r s t by c o n v e r t i n g t h i s t o a statement about measure p r e s e r v i n g i n t e g e r a c t i o n s , and t h e n p r o v i n g t h e 1a t t e r by p r o v i n g i t f i r s t f o r a c t i o n s w i t h generalized d i s c r e t e spectrum, and then showing t h e p r o p e r t y i s preserved by passing t o r e l a t i v e l y weakly m i x i n g extensions.
5. Amenability The n o t i o n o f an amenable group can be described i n a v a r i e t y o f ways. Here, we s h a l l focus on t h e f i x e d p o i n t property. L e t E be a separable Ranach space, E* t h e dual, E; and Iso(E) t h e group o f i s o m e t r i c isomorphisms o f E. i s a r e p r e s e n t a t i o n o f G on E, and t h a t A C E; i n v a r i a n t set.
t h e u n i t b a l l i n E*, Suppose
n:
+
i s a compact convex G
(Yere 6 a c t s on E* v i a t h e a d j o i n t r e p r e s e n t a t i o n ,
n * ( g ) = (n(g-'))*-
6
-
Iso(E)
D e f i n i t i o n 5.1.
G i s amenable if f o r a l l
and 4 as ahove, t h e r e i s a
n
f i x e d p o i n t f o r G i n A. F o r example,
i f 6 i s amenable and 6 a c t s c o n t i n u o u s l y on a compact m e t r i c
We s i m p l y
space X, t h e n t h e r e i s a G - i n v a r i a n t p r o b a b i l i t y measure on X.
a p p l y t h e d e f i n i t i o n t o E = C ( X ) where A c C(X)* i s t h e s e t o f p r o b a b i l i t y I n f a c t a s t a n d a r d c o n v e x i t y argument shows t h a t G i s amenable if
measures.
and o n l y i f t h e r e i s a G - i n v a r i a n t measure on e v e r y compact m e t r i c G-space. A b e l i a n groups a r e amenable by t h e Markov-Kakutani f i x e d p o i n t theorem, and compact groups a r e e a s i l y seen t o be amenable.
If
O
+
A
+
R
+
C
+
0
is
an e x a c t sequence, t h e n B i s amenable i f and o n l y i f A and C a r e amenable. Thus, groups w i t h a cocompact s o l v a b l e normal subgroup a r e amenable.
Every
connected amenable group i s o f t h i s form, h u t t h i s i s no l o n g e r t r u e among a l l d i s c r e t e groups [187. We now w i s h t o d e f i n e t h e n o t i o n o f an amenable e r g o d i c a c t i o n o f a group, o r i g i n a l l y i n t r o d u c e d i n r51'.
This w i l l include a l l actions o f
amenable groups, as w e l l as some a c t i o n s o f non-amenable groups.
We b e g i n by
d e s c r i b i n g c e r t a i n c l a s s e s o f < - i n v a r i a n t compact convex s e t s t h a t a r i s e from an e r g o d i c G-space S. a: S x G + I s o ( E )
So suppose S i s an e r g o d i c G-space and cocycle.
We t h e n have t h e a d j o i n t c o c y c l e
a*-twisted
f
E
a c t i o n on
L-(s,E*).
L-(s,E*),
We observe t h a t
a dual space.
g i v e n by L-(s,E*)
(g
f ) ( s ) = a*(s,g)f(sg),
-
L-(?,A).
S
+
SO
that
for
L-(s,E*)
One n a t u r a l p o s s i b i l i t y i s t o t a k e AC a*(s,g)A
A ) w i l l be a compact convex G
= A.
-
Then F(S,A)
i n v a r i a n t set i n
Yowever, i t i s a l s o p o s s i b l e t o v a r y t h e s e t 4 as we move f r o m
p o i n t t o p o i n t i n 5.
Thus, suppcse
is
i n v a r i a n t compact convex s e t s
compact, convex, and s a t i s f y i n g t h e c o n d i t i o n
(= measurable f u n c t i o n s
=
(L' (s,E))*,
=
We want t o d e s c r i b e c e r t a i n G
i n t h e u n i t b a l l o f t h i s dual space.
E;
a*(s,g)
is a
1 * (a(s,g)- ) , and t h e
{As)
i s a c o l l e c t i o n o f compact convex
subsets
*
A S C El,
which v a r y measurably i n s, and s a t i s f y i n g t h e c o n d i t i o n Then
* ( s , g ) ~ ~ A~ . = convex G
-
F(S,{As))
L.(~,E*)I
f(s)
As)
t
i s a compact
C a l l a s e t of t h e f o r m
F(S,{As})
a compact convex
An e r g o d i c a c t i o n o f G on S i s c a l l e d amenable i f e v e r y compact
s e t o v e r 5.
-
'
i n v a r i a n t set.
n e f i n i t i o n 5.2 C511
convex G
= {f
i n v a r i a n t s e t o v e r S has a f i x e d p o i n t .
Thus w h i l e a m e n a b i l i t y o f G demands a f i x e d p o i n t i n e v e r y compact convex s e t , a m e n a b i l i t y o f t h e a c t i o n demands a f i x e d p o i n t o n l y i n compact convex sets over t h e action. p o i n t s i m p l y means
We a l s o remark t h a t t h e c o n d i t i o n t h a t one has a f i x e d
a*(s,g)
f(sg) = f(s)
for
f : S + E*,
f(s)
As.
As an
example o f how one can u s e t h i s c o n d i t i o n , suppose S i s an amenable G-space and t h a t X i s a compact m e t r i c G-space. measures on X. .cocycle
a:S
x
We have a r e p r e s e n t a t i o n G
+
Iso(C(X))
L e t Y(X) b e t h e space o f p r o b a b i l i t y r: G
+
by r e s t r i c t i o n , i.e.
Iso(C(X))
and hence a
a ( ~ , g ) = n(g).
M(X) w i l l
be a G - i n v a r i a n t compact convex set, and t h u s we can t a k e As = M(X) f o r a l l
s.
(So f o r t h i s example, we d i d n ' t have t o v a r y t h e compact convex s e t i n
going from p o i n t t o point.) i s a function
f:S
+
such t h a t
a*(s,g)f(sg)
= f(s),
i.e.,
S w i t c h i n g t o a r i g h t a c t i o n on Y(X), we o b t a i n t h a t
n*(g)f(sg) = f(s). f(sg) = f(s)
M(X)
Amenability o f t h e a c t i o n then implies t h a t t h e r e
Thus, we conclude t h a t i f S i s an amenable e r g o d i c G
g.
space, X a compact m e t r i c G-space, t h e n t h e r e i s a measurable G f:S
+
-
-
map
M(X).
We now l i s t some b a s i c p r o p e r t i e s .
P r o o f s can be found i n C511, C521.
P r o p o s i t i o n 5.3. a)
I f G i s amenable, e v e r y e r g o d i c G-space i s amenable.
b)
I f S i s an amenable 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, t h e n G i s amenable.
c)
I f S = G/Y,
then
i s an amenable 6-space i f and only if
H i s amenable. d)
I f S i s an amenable e r g o d i c G-space,
and
C G
i s a closed
r
subgroup, then t h e r e s t r i c t i o n o f t h e a c t i o n on S t o
is
amenable.
Example 5.4.
r
r
Let
C SL(2 ,R) be a l a t t i c e and consider t h e ergodic a c t i o n o f
on t h e boundary c i r c l e o f t h e Poincare disk.
on S1 (2,R)/P the
r
where P i s t h e upper t r i a n g u l a r subgroup.
r
Since P i s amenable,
a c t i o n on t h e boundary c i r c l e i s amenable by ( c ) and (d) o f t h e above
proposition.
Yore g e n e r a l l y , l e t G be a semisimple L i e group,
l a t t i c e , and P C G a minimal p a r a b o l i c subgroup.
r
This i s j u s t the action o f
r
CG
a
Then P i s amenable and so
a c t i n g on G/P i s amenable. T h i s example i n d i c a t e s how n a t u r a l and important examples o f actions o f
non-amenable groups are amenable.
A s s e r t i o n (c) of t h e above p r o p o s i t i o n
shows t h a t any group has amenable t r a n s i t i v e actions.
"lre
g e n e r a l l y , we have
the following. P r o p o s i t i o n 5.5 r 5 2 1
I f H C G i s a closed subgroup and S i s an ergodic
H-space, then S i s an amenable H-space i f and only i f
G indH(S)
i s an amenable
G-space. T h i s p r o p o s i t i o n r a i s e s t h e f o l l o w i n g question.
Although one can have
amenable a c t i o n s o f non-amenable groups, does every such a c t i o n come i n a simple way from an a c t i o n o f an amenable subgroup, namely j u s t by inducing? I n f a c t , t h e answer i n general i s no. l a t t i c e , then
r
One can show t h a t i f
r
i s amenable on CP' ( t h i s i s j u s t example 5.4)
a c t i o n i s not induced from an a c t i o n o f an amenable subgroup. f a c t i s not t r i v i a l , and a proof i s given i n '52'. have t h e f o l l o w i n g .
C SL(2,C)
is a
but that t h i s This l a t t e r
On t h e o t h e r hand, we do
Theorem 5.6 r 5 2 1 , C55l
Let G be a connected group.
Then every amenable
ergodic action of 6 i s induced from an a c t i o n of an amenable subgroup. Droof. and P
We give t h e proof f o r G semisimple.
G be a minimal parabolic subrougp.
Let S be an amenable G-space
Since G/P i s compact, by t h e re-
marks following i l e f i n i t i o n 5.2, t h e r e i s a measurable G
-
map
where t h e l a t t e r i s t h e space of probability measures on G/P.
f:S
+
The proof will The f i r s t
now follow from two b a s i c r e s u l t s about t h e action of G on M(G/P). i s due t o t h e author, t h e second t o C.C. Theorem 5.7 r 5 2 1 , C55l
-
Moore r331.
Let G be a connected semisimple Lie group and
P C I7 any parabolic subqroup. closed o r b i t under t h e G
Then every element i n M(G/P) has a l o c a l l y
action.
Theorem 5.5 (C.C. Woore [331.)
Let G be a connected semisimple Lie group
with t r i v i a l c e n t e r and P C G a minimal parabolic subgroup. 2nd
G
t h e s t a b i l i z e r of
u
M(G/P),
u
i n G.
Then
G
u
Let
u
E
W(G/P)
i s an amenable a l g e b r a i c
group.
(By a l g e b r a i c , we mean t h e i n t e r s e c t i o n of G with an a l g e b r a i c subroup
of
where t h e l a t t e r i s an algebraic subgroup containing 6 a s a subgroup
E,
of f i n i t e index.) To conclude t h e proof of Theorem 5.6 given t h e s e r e s u l t s , we simply observe t h a t Theorem 5.7 a s s e r t s t h e smoothness of t h e G-action on W(G/P), so i f m denotes t h e measure on S, then f*(m) is quasi-invariant and ergodic on M(G/P) and hence supported on an o r b i t . f:S
+
G/Y,
Thus we can consider f as
where H i s t h e s t a b i l i z e r of an o r b i t .
R u t by theorem 5.8, Y i s
amenable so t h e r e s u l t follows from Proposition 1.11. Without giving a proof of Theorems 5.7 and 5.8,
l e t us a t l e a s t t r y t o
give some indication of why they are true. To t h i s end, we s t a t e t h e following lemma of Furstenberg [15?.
Lemma 5.9 and suppose
p
.
(Furstenberg)
. gn
+
v
Suppose gn
f o r some
,
gn
Then
v
SL(n ,R)
6
v c b!(pn-l).
.
+
Let
p c
M(P~-~)
i s supported on a
union o f two proper p r o j e c t i v e subspaces.
Remarks
Since t h e set o f measures supported on a union o f two proper
i)
p r o j e c t i v e subspaces i s closed, t h i s shows t h a t t h e o r b i t o f any supported i s l o c a l l y closed.
u
not so
This i s t h e beginning o f an i n d u c t i v e procedure
f o r p r o v i n g theorem 5 -7.
If H
ii )
c SL(n,R) leaves a measure f i x e d , then H i s e i t h e r
compact o r leaves t h e union o f two subspaces i n v a r i a n t . Roth o f these c o n d i t i o n s are "a1 gebraic"
.
Proceeding i n d u c t i v e l y , one should o b t a i n e i t h e r
compactness o r f u r t h e r spl i t t i ng. This should r e s u l t i n an a1 gebraic o b j e c t which i s a compact extension o f a s o l v a b l e group. suggests Theorem 5.8 as we1 1 Proof o f Lemma 5.9. can assume
hn
+
Let
h, h
+
.
hn = gn/llgnn.
0,
I n t h i s way, L e n a 5.9
nhns = 1,
Then
as
c [Nl,
support (pl)
n
+ m,
hn(x)
+
v = limp
l i m u2
gn
0,
so we
[N]
Write
.
If x
p = f
pn-l
+ u2
-
[N]
, then
Passing t o a subsequence, we can w r i t e
g = limu n
and t h e previous sentence i m p l i e s have
-
support (p2) c p n - l
h(x) E [V].
+
L e t V = range ( h ) , N = k e r ( h ) , [V],
det(h) = 0.
[N] t h e corresponding p r o j e c t i v e subspaces i n p n - l . where
det(hn)
9,
+
limp2
l i m p2
gn
supported on [Wl where
1 im
gn,
i s supported on [V!.
[Nl
We a l s o
gn = [My.
We now t u r n t o t h e theory o f o r b i t equivalence f o r amenable actions.
We
begin w i t h an observation. P r o p o s i t i o n 5.10. o f o r b i t equivalence.
For f r e e ergodic actions a m e n a b i l i t y i s an i n v a r i a n t
The c i r c l e o f i d e a s c o n c e r n i n g o r b i t e q u i v a l e n c e began i n t h e l a t e 1 9 5 n ' s w i t h t h e work o f H. n y e
[lo!
[ill and f o r amenable a c t i o n s has been b r o u g h t t o
complete form v e r y r e c e n t l y . Theorem 5.11 Z-actions
We h e g i n w i t h t h e fundamental theorem of Dye. A1 1 f i n i t e measure p r e s e r v i n g ( p r o p e r l y ) e r g o d i c
(Dye [ l o ] )
are o r b i t equivalent.
T h i s was l a t e r extended t o t h e Theorem 5.12
A1 1
(Krieger)
o - f i n i t e case by K r i e g e r [ 2 5 1 .
o - f i n i t e ( b u t n o t f i n i t e ) measure p r e s e r v i n g
Z-actions are o r b i t equivalent. K r i e g e r a l s o extended t h e theorem t o t.he case o f q u a s i - i n v a r i a n t measure w i t h o u t e q u i v a l e n t i n v a r i a n t measure.
He showed t h a t a measurement of t h e
e x t e n t t o which t h e a c t i o n f a i l s t o he measure p r e s e r v i n g i s a c o m p l e t e i n v a r i a n t o f o r b i t equivalence. r:X
G
x
+
Namely, l e t X b e an e r g o d i c G-space,
R+ b e t h e Radon-Nikodym cocycle.
f u n c t i o n o f G , and l e t t h e modular cocycle.
m:X
x
G
+
Let
R+ be m(x,g)
A:G
+
and
R+ be t h e modular
= ~ ( x , ~ ) A ( ~ ) - ' . We c a l l m
The Mackey range o f t h i s c o c y c l e w i l l be an e r g o d i c R+-
a c t i o n , which we c a l l t h e modular f l o w o r t h e modular range.
For unimodular
groups, t h e modular f l o w w i l l be t r a n s l a t i o n o f R+ on R+ i t s e l f i f and o n l y i f t i l e r e i s an i n v a r i a n t measure f o r t h e a c t i o n ( f o r t h e n t h e Radon-Nikodym c o c y c l e i s t r i v i a1 ( P r o p o s i t i o n 3.6)). Theorem 5.13.
(Yrieger)
F o r Z - a c t i ons w i t h q u a s i - i n v a r i a n t measure,
and
n o t possessing (an e q u i v a l e n t ) f i n i t e i n v a r i a n t measure, t h e modular f l o w i s a c o m p l e t e i n v a r i a n t o r o r b i t equivalence. A good account o f K r i e g e r ' s work i s '441.
Example 5.14. Let
r
C SL(2,R)
L e t us see how t o compute t h e modular f l o w i n some examples. a l a t t i c e , P t h e upper t r i a n g u l a r subgroup.
The Radon-
Nikodym cocycle f o r t h e a c t i o n o f G = SL(2,R) a:G/P
x
G + R'
corresponding t o t h e homomorphism P
modular f u n c t i o n o f P.
r a c t i o n on G/P
r
x
Clearly +
+
G/P
x
as a ker
R+ given by
+
r:G/Px r + R + i s j u s t
UJG/P
r o f t h e 6 a c t i o n G/P
x a
-
R+ i s s u r j e c t i v e , as i s w e l l known, t h i s i s t h e 6 .P/ker
bp
r-space,
R'
x
i s j u s t the action of
R'.
Yince
kerbp.
r on Glker
i s n o t compact, by Moore's e r g o d i c i t y theorem ( s e c t i o n 2)
ergodic on t h i s space.
r.
the P' Now the
action
on which I; acts t r a n s i t i v e l y w i t h s t a b i l i z e r 5/P
x
A
R+ t h a t appears i n t h e c o n s t r u c t i o n o f t h e Yackey
range i s j u s t t h e r e s r i c t i o n t o A:P
on G/P i s t h e cocycle
A ~ .
Thus Since
r is
Thus t h e r e i s o n l y one ergodic component, and so t h e
modular f l o w i s t h e a c t i o n o f R+ on a p o i n t .
This computation can c l e a r l y he
c a r r i e d o u t on any semisimple non-compact L i e group.
If
r i s a lattice i n
such a group 6 , and P C G i s a minimal p a r a b o l i c , then t h e modular f l o w of t h e action of
r
on G/P w i l l be t h e a c t i o n o f R+ on a p o i n t .
The Dye-Crieger theorems were extended over t h e years by a number o f persons t o i n c l u d e w i t h i n i t s framework a c t i o n s o f l a r g e r classes o f groups. ( I n f a c t Dye d i d n o t r e s t r i c t h i m s e l f t o t h e integers.)
T h i s work has
r e c e n t l y culminated w i t h t h e f o l l owing theorems. Theorem 5.15 i)
(Connes-Feldman-9rnstei n-Wei ss) C81, [371 A f r e e p r o p e r l y ergodic a c t i o n o f a d i s c r e t e group i s
amenable i f and o n l y i f i t i s o r b i t e q u i v a l e n t t o a Z - a c t i on. ii)
The Dye-Krieger theorems (5.11-5.13)
hold f o r the class o f
amenable p r o p e r l y ergodic a c t i o n s o f d i s c r e t e groups. Theorem 5.16
i)
(Connnes-Feldman-Ornstein-$lei ss) [81, c371 A f r e e p r o p e r l y ergodic a c t i o n o f a continuous group i s
amenable i f and o n l y i f i t i s o r b i t e q u i v a l e n t t o an Raction.
For such actions, t h e modular f l o w i s a complete j n v a r i a n t
ii)
o f o r b i t equivalence.
I n p a r t i c u l a r , any two f r e e p r o p e r l y
ergodic a c t i o n s o f continuous amenahle unimodular groups w i t h i n v a r i a n t measure are o r b i t equivalent. Fxarnple 5.17.
If
C,
i s a semisimple non compact L i e group,
r
CG
a lattice,
P C G a minimal p a r a b o l i c , we saw i n Example 5.14 t h a t t h e modular f l o w o f on G/P i s independent o f G and
r.
r
Since these a c t i o n s a r e amenable, theorem
5.15 says t h a t they a r e a l l o r b i t equivalent.
6.
9igidity:
The Yostow-Margulis
Theorem and a G e n e r a l i z a t i o n
to
i r g o d i c Actions.
I n t h i s l e c t u r e we s h a l l d e s c r i b e t h e p r o o f o f t h e Yostow-Margulis r i g i d i t y theorem f o r l a t t i c e s i n semisimple L i e groups and i n d i c a t e how t h i s r e s u l t can be extended t o y i e l d r e s u l t s about o r b i t e q u i v a l e n c e f o r e r g o d i c a c t i o n s o f s e m i s i m p l e groups and t h e i r l a t t i c e s . Theorem 6.1.
(Mostow-Margulis R i g i d i t y )
L e t G, G' b e connected semi-
r, r '
s i m p l e L i e groups w i t h f i n i t e c e n t e r , no compact f a c t o r s , and d u c i b l e l a t t i c e s i n G, G' r e s p e c t i v e l y . i)
If
r
and
Suppose R-rank(G)
_>
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
>_
<
an
+
The a c t i o n of G on S has p r o p e r t y ( T ) i f
I means
a
-8
I
dere I i s t h e one dimensional
for n s u f f i c i e n t l y large.
t r i v i a l c o c y c l e and
p o i n t w i s e on
where
~(s,g)v = v
f o r some non-
zero v e c t o r v. We t h e n have t h e f o l l o w i n g r e s u l t s . Theorem 7.7
157'.
I f G has p r o p e r t y (T),
a)
and S has a f i n i t e i n v a r i a n t
measure t h e n S has p r o p e r t y ( T ) . b)
I f S has p r o p e r t y (T), f i n i t e i n v a r i a n t measure and i s weak m i x i n g (i.e.
t h e r e a r e no f i n i t e dimensional i n v a r i a n t subspaces i n
L ~ ( s ) @C ) ,
t h e n G has
p r o p e r t y .(T). c)
F o r f r e e a c t i o n s o f d i s c r e t e groups, p r o p e r t y ( T ) i s an i n v a r i a n t o f o r b i t
equivalence. Combining 7.5 and 7.7,
we have t h e foll,owing,
showing t h a t , i n f a c t ,
o r b i t e q u i v a l e n c e i s q u i t e s e n s i t i v e t o t h e way H i s c o n s t r u c t e d f r o m N and
HIN. C o r o l l a r y 7.8[57!.
r2 =
SL(n,Z)
aZ. n
Let Then
n > _ 3 , and
rl
and
r2
rl=SL(n,Z)
X Z
n
,
do n o t have f r e e o r b i t e q u i v a l e n t
weakly m i x i n g a c t i o n s w i t h f i n i t e i n v a r i a n t measure. We s h a l l now d e s c r i b e a g e o m e t r i c i n t e r p r e t a t i o n o f t h e r i g i d i t y theorem f o r ergodic actions.
We b e g i n by r e c a l l i n g t h e g e o m e t r i c f o r m u l a t i o n o f t h e
Yostow-Yargulis theorem.
L e t 6 be a connected semisimple 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 , Y C 6 a naximal colnpact subgroup, and
r C G a t o r s i o n free l a t t i c e .
Then G/K i s a Riemannian symmetric space
( d i f f e o m o r p h i c t o E u c l i d e a n space), and
r operates properly discontinuously
on X = S/K. nl(r\x)
r\X
Thus
i s l o c a l l y symmetric soace o f f i n i t e volume, and
r.
z
Theorem 7.9 (Mostow-Margulis r i g i d i t y , geometric form).
Let
"1,
Y2 be
1ocall.y symmetric Riemannian manifolds o f f i n i t e volume whose u n i v e r s a l
X 2 are symmetric spaces o f p u r e l y non-compact type, and whose
covers, XI,
fundamental groups
ni (Mi)
act as i r r e d u c i b l e groups o f i s o m e t r i e s o f Xi.
5uppose f u r t h e r t h a t t h e rank o f M1 i s a t l e a s t 2.
Then any isomorphism
nl(M1)
+
+
n1(Y2)
i s induced by a diffeomorphisrn
M1
Y2
t h a t i s an isometry
modulo n o r m a l i z i n g s c a l a r m u l t i p l e s . Roughly speaking, t h i s asserts t h a t f o r a p a r t i c u l a r c l a s s o f Riemannian manifolds, i.e.
s u i t a b l e l o c a l l y symmetric spaces, t h a t a p u r e l y t o p o l o g i c a l
i n v a r i a n t , namely t h e fundamental group, determines t h e Riemannian structure.
We now d e s c r i b e an analogous geometric i n t e r p r e t a t i o n o f t h e
r i g i d i t y theorem f o r a c t i o n s which w i l l make an a s s e r t i o n about f o l i a t i o n s by symmetric spaces. L e t G be a connected semisimple non-compact L i e group w i t h f i n i t e c e n t e r
(S,u)
and no compact f a c t o r s , K C G a maximal compact subgroup, and e r g o d i c S-space w i t h f i n i t e i n v a r i a n t measure.
L e t Y = S/Y.
a free
Then because Y
i s compact, Y i s a standard Rorel space, and t h e o r b i t s i n S y i e l d an equivalence r e l a t i o n 3 on Y i n which each equivalence c l a s s can be i d e n t i f i e d w i t h G/K.
Thus, (Y ,a ) i s a "9ielnannian measurable f o l i a t i o n " ,
i.e.,
a
measure space w i t h an equivalence r e l a t i o n T i n which each equivalence c l a s s ( o r " l e a f " ) has t h e s t r u c t u r e o f a
cW-
Riemannian manifold, so t h a t these
s t r u c t u r e s vary measurably i n a s u i t a b l e sense r53!
over Y.
Given Y, Y'
, two
spaces supporting Riemannian measurable f o l i a t i o n s , we c a l l them i s o m e t r i c i f t h e r e i s a measure space isomorphism between Y and Y ' t h a t c a r r i e s leaves o n t o leaves i s o m e t r i c a l l y ( p o s s i b l y a f t e r d i s c a r d i n g n u l l s e t s of leaves).
By a
t r a n s v e r s a l f o r (Y, 1 ) ,we mean a sore1 s e t i n t e r s e c t i n g almost every l e a f i n
a countable set.
Such a s e t T w i l l have a n a t u r a l e q u i v a l e n c e r e l a t i o n on i t
w i t h c o u n t a b l e e q u i v a l e n c e c l a s s e s (namely X l T ) , and a n a t u r a l measure c l a s s v
s a t i s f y i n g t h e c o n d i t i o n t h a t f o r B C T,
u n i o n o f t h e l e a v e s i n t e r s e c t i n g R has
v(B) = 0
11-measure
i f and o n l y i f t h e
0 r l 3 1 , [41].
We c a l l two
Riemannian measurable f o l i a t i o n s t r a n s v e r s a l l y e q u i v a l e n t i f t h e y have i s o morphic t r a n s v e r s a l s .
Ry an isomorphism o f t r a n s v e r s a l s , we mean isomorphism
as measure spaces w i t h e q u i v a l e n c e r e l a t i o n s , i.e.,
a measure space isomorph-
i s m c a r r y i n g one e q u i v a l e n c e r e l a t i o n o n t o t h e o t h e r . measure-theoretic i n v a r i a n t o f t h e f o l i a t i o n .
This i s a purely
The f o l l o w i n g i s t h e geometric
v e r s i o n o f Theorem 6.3. Theorem 7.10
[55]
( R i g i d i t y f o r f o l i a t i o n s by symmetric spaces).
6 , G 8 , S , S 8 be as i n Theorem 6.3.
L e t Y = S/K,
a r e t h e maximal compact subgroups.
Let
Y ' = S 1 / K ' where K , K 1 CG,G'
L e t ( Y , I; ) , ( Y '
,a' )
Riemannian measurable f o l i a t i o n s by symmetric spaces.
t h e associated
If the f o l i a t i o n s are
t r a n s v e r s a l l y e q u i v a l e n t , t h e n t h e y a r e i s o m e t r i c , modulo n o r m a l i z i n g s c a l a r mu1t i p l e s (independent o f t h e 1eaves). Thus, r o u g h l y speaking, f o r s u i t a b l e f o l i a t i o n s i n which t h e l e a v e s a r e s y m n e t r i c spaces, a p u r e l y measure t h e o r e t i c i n v a r i a n t , namely t h e measure t h e o r y o f t h e t r a n s v e r s a l , determines t h e Riemannian s t r u c t u r e on almost e v e r y leaf. As we have a l r e a d y remarked, t h e r i g i d i t y theorem f o r l a t t i c e s h o l d s i n t h e R-rank 1 c a s e as we1 1 as l o n g as
G + PSL(2 ,R) , a l t h o u g h t h e p r o o f we have
g i v e n i n s e c t i o n 6 does n o t apply, and one must use o t h e r t e c h n i q u e s , f o r example t h o s e o f MDstow [36]
and Prasad [39].
It i s n a t u r a l t o e n q u i r e as t o
what e x t e n t 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 h o l d s i n t h e R-rank 1 case as w e l l .
I n [58!
we proved t h e f o l l o w i n g r e s u l t i n t h i s d i r e c t i o n , apply-
i n g b a s i c r e s u l t s o f b s t o w [35]
on quasi-conformal mappings.
Theorem 7.11. 8;:!
L e t S, S ' be f r e e ergodic
w i t h f i n i t e i n v a r i a n t measure, and assume
n
_>
SO(1 ,n)/{+,I}-spaces L e t (Y, '2 ) , (Y:
3.
a ' ) be
t h e associated measurable f o l i a t i o n s by hyperbol i c space (as i n t h e d i s c u s s i o n preceding Theorem 7 . l o ) .
I f (Y, 2 ) and (Y'
,T')
are quasi-conformally
e q u i v a l e n t , then they a r e i s o m e t r i c (modulo a n o r m a l i z i n g s c a l a r independent o f t h e l e a f ) , and t h e a c t i o n s o f
on S and S ' a r e
SO(l,n)/(fI}
automorphical l y conjugate. Here, o f course, quasi-conformal equivalence a s s e r t s t h e e x i s t e n c e o f a measure space isomorphism t a k i n g (almost a1 1) leaves t o leaves quasiconformally.
We remark t h a t t h e analogous statement f o r R~ a c t i o n s can be
shown t o be f a l s e by many counterexamples.
8.
Margul is ' F i n i t e n e s s Theorem. I n s e c t i o n 6, we saw how t h e a n a l y s i s o f t h e ergodic a c t i o n o f
l e d t o Margulis' p r o o f o f t h e r i g i d i t y theorem.
r
on G/P
Margulis has a l s o
demonstrated some o t h e r deep p r o p e r t i e s o f t h i s ergodic a c t i o n and used t h i s t o o b t a i n very s t r o n g r e s u l t s about t h e s t r u c t u r e o f
r.
b r e p r e c i s e l y , he
has shown the f o l l o w i n g . Theorem 8.1 (Margulis [30] ,[31]).
L e t G be a connected semisimple 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 , and assume R-rank(G) Let
r C G be an i r r e d u c i b l e l a t t i c e , and H
group.
= r/N
> 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 <
r,
-
and
[I'' :
r nrlJ <
m.
r l C G , I' i s a l a t t i c e and
r, r'
are
r1 i s a lattice.
F o r example, g i v e n t h a t SL(n,z)
i s a l a t t i c e , { a c SL(n,Z) ( a = I mod p
f o r a g i v e n p r i m e p } i s a commensurable l a t t i c e . Here i s a n o t h e r way t o g e t new l a t t i c e s . P r o p o s i t i o n 9.4.
If
r
C H
i s a lattice,
homomorphism w i t h compact k e r n e l t h e n
(p
(I')
(p
:H
+
G
a surjective
i s a l a t t i c e i n G.
Margulis' theorem says t h a t aside from these two types o f r a t h e r t r i v i a l m o d i f i c a t i o n s , every l a t t i c e i n a semi simple L i e group o f h i g h e r R-rank a r i s e s as i n Theorem 9.1.
More p r e c i s e l y , l e t us make t h e f o l l o w i n g d e f i n i t i o n .
(If
H i s a group, HO denotes t h e t o p o l o g i c a l l y connected component o f t h e i d e n t i t y .) L e t G be a connected semisimple L i e group w i t h t r i v i a l
D e f i n i t i o n 9.5.
c e n t e r and no compact f a c t o r s .
r CG be a l a t t i c e .
Let
Then
a r i t h m e t i c i f t h e r e e x i s t s an a l g e b r a i c group H d e f i n e d over s u r j e c t i v e homomorphism
i) ii)
(G)
G
i s called
9, and
a
such t h a t
kernel ( cp ) i s compact;
n HO)
( P ( H ~
Theorem 9.6 R-rank
Ip: HE +
r
R
i s a l a t t i c e i n G commensurable w i t h
( M a r g u l i s [281).
> 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 <
completing t h e proof.
This i n t u r n implies t h a t
m.
r r)
p(YZ)
p(HZ) i s a l a t t i c e i n
i s an i n c l u s i o n o f l a t t i c e s , we a l s o have
T h i s shows commensurability o f
r and p(HZ),
References L. Auslander, L. Green, F. Hahn, Flows on Homogeneous Spaces, Annals o f Math. S t u d i e s , no.53, P r i n c e t o n , 1963
L. Auslander, An E x p o s i t i o n o f t h e S t r u c t u r e o f S o l v m a n i f o l d s , B u l l Amer. Math. Yoc., 79(1973), 227-285.
.
A. B o r e l , L i n e a r A l g e b r a i c Groups, Benjamin, New York, 1969. A. B o r e l , D e n s i t y P r o p e r t i e s f o r C e r t a i n Subgroups o f Semi simple L i e Groups Without Compact F a c t o r s , Annal s o f Math., 72(1960), 179-188. A. B o r e l , Harish-Chandra, A r i t h m e t i c Subgroups o f A1 g e b r a i c Groups, Annal s o f Math., 75(1962), 485-535.
.
A. B o r e l , J .P Serre, ~ h e o r s m e s de F i n i t u d e en Cohomol o g i e G a l o i s i e n n e , Comm. Math. H e l v 39(1964), 111-164.
.,
J. B r e z i n , CC. b o r e , Flows on Homogeneous Spaces: A New Look, preprin t
.
A. Connes. J. Feldman, 8. k i s s , Amenable e q u i v a l e n c e r e l a t i o n s are generated by a s i n g l e t r a n s f o r m a t i o n , p r e p r i n t . C. Delaroche, A K i r i l l o v , Sur Les R e l a t i o n s E n t r e L'Espace Dual d ' u n Groupe e t 1 a S t r u c t u r e de ses Sous-Groupes ~ e r m g s , Semi n a i r e Bourbaki , no.343, 1967/68. H.A.
Dye, On Groups o f R a s u r e P r e s e r v i n g T r a n s f o r m a t i o n s , I , Amer. 81(1959), 119-159.
J . Math.,
H .A. Dye, On Groups o f R a s u r e P r e s e r v i n g T r a n s f o r m a t i o n s , 11, Amer. J Math., 85(1963), 551-576. E .G. E f f r o s , T r a n s f o r m a t i o n Groups and c *gebras, - ~ l Annals o f h t h . , 81(1965), 38-55.
J . Feldman, P. Hahn, C.C.
b o r e , O r b i t S t r u c t u r e and Countable S e c t i o n s f o r A c t i o n s o f Continuous Groups, Advances i n Math., 28(1978), 186-230. H. Freudenthal , Topologische Gruppen mi t Genugend V i e l e n F a s t p e r i o d i s h e n Funktionen, Annals o f Math., 37(1936), 57-77.
H. Furstenherg, A Poisson Formula f o r Semisimple L i e Groups, Annals o f Math., 77(1963), 335-383. H. Furstenberg, The S t r u c t u r e o f D i s t a l Flows, Amer. J. Math., 85(1963), 477-515. H. Furstenberg, E r g o d i c B e h a v i o r o f Diagonal R a s u r e s and a Theorem o f Szemeredi on A r i t h m e t i c Progressions, J. Analyse Math., 3 1 (1977), 204-256.
F .P. Greenleaf, I n v a r i a n t R a n s on T o p l o g i c a l Groups, Van Nostrand, New York, 1969.
Y. G u i v a r c ' h , Croissance P o l y n o m i a l e e t P e r i o d e s des F o n c t i o n s Harmonique, B u l l Path. Soc. France, 101 (:973), 333-379
.
G . Hedlund, The Dynamics o f Geodesic Flows, B u l l . Amer. FBth. Soc., 45(1939), 241-260
E . Hopf, S t a t i s t i k d e r ~ o s u n ~ egne o d a t i s c h e r probleme typus, Math. Ann. 117(1940), 590-608.
vom u n s t a b i l e n
R. Howe and C.C. b o r e , Asymptotic P r o p e r t i e s o f U n i t a r y Representations, J Func, Anal ,32 (1979), 72-96.
.
.
J .W. J e n k i n s , Growth o f Connected L o c a l l y Compact Groups, J. Funct. Anal 12(1973), 113-127.
.,
D. Kazhdan, Connection o f t h e Dual Space o f a Group w i t h t h e S t r u c t u r e o f i t s Closed Subgroups, Funct. Anal. Appl 1(1967), 63-65.
.,
W. K r i e g e r , On E r g o d i c Flows and t h e Isomorphism o f F a c t o r s , Math. Ann. 223(1976), 19-70. G.W. k c k e y , E r g o d i c T r a n s f o r m a t i o n Groups w i t h a Pure P o i n t Spectrum, I l l i n o i s J . Math 8(1964), 593-600.
.,
G .A. Margul i s , Non-uniform l a t t i c e s i n Semi s i m p l e A l g e b r a i c Groups, i n L i e Groups and t h e i r Representations, ed. I.V. Gelfand, Wiley, New Y ork
.
G.A. Margulis, D i s c r e t e Groups o f b t i o n s o f Manifolds o f NonP o s i t i v e Curvature, Amer. Math. Soc. Trans1 a t i o n s , 109(1977), 33-45. G.A. Marguli s, A r i t h m e t i c i t y o f I r r e d u c i b l e L a t t i c e s i n Semisimple Groups o f Rank G r e a t e r t h a n 1, Appendix t o Russian T r a n s l a t i o n o f M. Ragunathan, D i s c r e t e Subgroups o f L i e Groups, Mir, b s c o w , 1 9 7 7 ( i n Russian). G .A. Margul is, F a c t o r Groups o f D i s c r e t e Subgroups, S o v i e t Math. Dokl. 19(1978), 1145-1149. G.A. Margulis, Q u o t i e n t Groups o f D i s c r e t e Subgroups and k a s u r e Theory, F u n c t Anal Appl , 12 (1978), 295-305.
.
.
.
C .C. b o r e , E r g o d i c i t y o f Flows on Homogeneous Spaces, Amer. J. Math , 88(1966), 154-178.
.
CC . b o r e , Amenable Subgroups o f Semi s i m p l e Groups and P r o x i m a l Flows, I s r a e l J. Math., 34(1979), 121-138. C.C. b o r e and R .J. Zimmer, Groups A d m i t t i n g E r g o d i c A c t i o n s w i t h General ized D i s c r e t e Spectrum, I n v e n t . Math., 51 (1979) , 171-188.
-
G.D. b s t o w , Quasi-Conformal k p p i n g s i n n Space and t h e R i g i d i t y -of H y p e r b o l i c Space Forms, Pub:. Math. I.H .E .S., (1967), 53-104.
G.D.
Fbstow, S t r o n g R i g i d i t y o f L o c a l l y Symmetric Spaces, Annals o f Math. S t u d i e s , no .78, P r i n c e t o n U n i v . Press, P r i n c e t o n , N.J 1973.
.
D. O r n s t e i n and B. Weiss, t o appear.
W. P a r r y , Zero E n t r o p y o f D i s t a l and R e l a t e d Transformations, i n T o p o l o g i c a l Dynamics, eds. J . Auslander, W. G o t t s c h a l k, Benjamin, New York, 1968. G. Prasad, S t r o n g R i g i d i t y o f Q-rank 1 L a t t i c e s , I n v e n t . FBth., 21(1973), 255-286. M. Ragunathan, D i s c r e t e Subgroups o f L i e Groups, S p r i nger-Verlag, New York, 1972.
A. Ramsay, V i r t u a l Groups and Group Actions, Advances i n Math., 6(1971), 253-322. A. r e 1 b e r g , On D i s c o n t i n u o u s Groups i n H i g h e r Dimensional Symmetric Spaces, I n t . Colloquium on F u n c t i o n Theory, Tata I n s t i t u t e , Rombay, 19613. Sherman, A Weight Theory f o r U n i t a r y R e p r e s e n t a t i o n s , Canadian J Math., 18(1966), 159-168.
T.
.
C. S u t h e r l and, O r b i t Equivalence: L e c t u r e s on K r i e g e r s Theorem, U n i v e r s i t y o f Oslo L e c t u r e Notes.
.
V .S Varadarajan, Geometry o f Quantum Theory, v o l P r i n c e t o n , N .J 1970.
.
. I 1 ., Van Nostrand,
S.P. Wang, On I s o l a t e d P o i n t s i n t h e Dual Spaces o f L o c a l l y Compact Groups, Math. Ann., 218(1975), 19-34. R.J. Zimmer, Extensions o f E r g o d i c Group A c t i o n s , I l l i n o i s J. h t h . , 20(1976), 373-409. R .J. Zimmer, E r g o d i c A c t i o n s w i t h G e n e r a l i z e d D i s c r e t e Spectrum, I 1 1 in o i s J Math., 20(1976), 555-588.
.
.
R .J Zimmer, O r b i t Spaces o f U n i t a r y Representations, E r g o d i c Theory, and Simple L i e Groups, Annals o f Math 106(1977), 573-588.
.,
R.J. Zimmer, U n i f o r m Subgroups and E r g o d i c A c t i o n s o f Exponential L i e Groups, Pac. J. Math., 78(1978), 267-272.
.
R .J Zimmer, Amenable E r g o d i c Group A c t i o n s and an Appl i c a t i o n t o Poisson Boundaries o f Random Walks, J Funct Anal , 27(1978), 350-372.
.
.
.
R .J. Zimmer, Induced and Amenable E r g o d i c A c t i o n s o f L i e Groups, Ann. S c i . Ec. Norm. Sup., 11(1978), 407-428.
.
R .J Zimmer, A l g e b r a i c Topology o f E r g o d i c L i e Group A c t i o n and Measurable F o l ia t ions, p r e p r i n t
.
54.
R .J. Z i m e r , An Algebraic Group Associated t o an Ergodic Diffeomorphism, Comp. FBth., t o appear.
55.
R.J. Zimmer, Strong Rigidity f o r Ergodic Actions of Semisimple Lie Groups, Annals of Math., t o appear.
56.
R.J. Zimner, Orbit Equivalence and Rigidity of Ergodic Actions of Lie Groups, p r e p r i n t .
57.
R.J. Zimmer, On t h e Cohomology of Ergodic Actions of Semisimple l i e Groups and Discrete subgroups, Amer. J . Math., t o appear.
58.
R.J. Zimmer, On t h e Mostow R i g i d i t y Theorem and Measurable F o l i a t i o n s by Hyperbolic Space, p r e p r i n t .
