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University of Groningen

Convolution on homogeneous spacesCapelle, Johan

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Chapter V

The Homogeneous Space G/N

V.1 Set-up

In this chapter we consider the following example of a non-symmetric homogeneous space G/H, with

non-compact H. The group G will be a real non-compact connected semi-simple Lie group with finite

centre, with Cartan involution ∆. Let G=KAN be an Iwasawa decomposition for G. We study the

homogeneous space ≈=G/N.

First we describe the N–orbit structure (Theorem V.2.5). Then we determine explicitly the

algebra å of G–invariant operators in ∂æ(≈): it turns out that å is isomorphic to the convolution

algebra of compactly supported distributions on the subgroup MA (Theorem V.3.1 and Corollary).

Using some theory from Chapter IV we then reduce the existence problem of fundamental solutions

for the operators in å to the similar problem on MA, which is a well-studied problem (Theorem

V.4.1). Finally, in Section V.5 we look at decomposition and multiplicity problems for invariant

Hilbert subspaces of ∂æ(≈). The result is that (G,N) is not a Gelfand pair. We then try to obtain a

Gelfand pair by a natural group extension as in Section IV.14, but whether this always leads to a

Gelfand pair remains undecided: we can only reduce this question to another, well-known, open

problem (Proposition V.5.9).

We formally choose a point p∑≈, p=eN. Furthermore, we fix the following notations and recall a

number of well-known facts [82] .

Denote the Lie-algebras of G, K, A and N by g, k, a and n respectively. Let aæ denote the

dual of a. Let Í≤aæ denote the root system associated to the pair (g,a), and for å∑Í let gå denote the

corresponding root space. Let Í+ denote the subset of positive roots associated with the choice of N,

82 We refer to the first chapter of Garth Warner, Harmonic Analysis on Semi-Simple Lie Groups I

(Berlin:Springer-Verlag, 1972).

139

— Chapter V The Homogeneous Space G/N —

so n=å¶∑Í

@+gå. Let ® denote half of the sum over the positive roots, counted with multiplicity, that is

to say that ®=2å¶∑Í

@+(dimgå).å.

Let aæÇ be the complexified dual of a. Its elements are called weights, and they correspond

to the character group of a, through the definition

(V.1.0.a) a~:=e~(H) ~∑aæÇ, and a∑A, H∑a such that a=exp(H)

Choose Haar measures dk, da, dn on K, A, and N, with dk having total measure 1. Then with respect to

the Iwasawa decomposition dg=dk¤a™®da¤dn is a suitable choice of Haar measure. It follows that

G/N possesses an invariant measure. With respect to the coordinates KAë(k,a) éêâ kap∑G/N we

choose µ:=dk¤a™®da. Consequently, the theory developed in Chapter IV applies.

As to other coordinates, let Nä denote the nilpotent group opposite to N, so Nä=∆(N). It

follows directly from the general theory [83] that the set NäMAp is an open and dense subset of G/N

whose complement is a finite union of lower-dimensional submanifolds (so it has measure 0), and that

for a suitable choice of Haar measure dn_ on Nä one has

(V.1.0.b) ªG/N

Ï(x)µ(dx)=ªK*A

Ï(kap)dka™®da

=ªäN

*M*AÏ( n_map)d_ndma™®da Ï∑∂(≈).

Let W denote the Weyl group of the root system. Let M be the centralizer of A in K. Let M* denote

the normalizer of A in K. Then M is obviously normal in M*, and through the adjoint representation

M*/M acts on a. The group M*/M is finite, and through the adjoint representation M*/M is a

manifestation of the Weyl group. For each w∑W we choose an element m*w∑M* such that the

restriction of Adm*w to a corresponds to the action of w on a.

We mention the fact that MA and M*A are closed subgroups of G, equalling respectively the

centralizer and the normalizer of A. The Weyl group W also equals the quotient M*A/MA. Let B

denote the minimal parabolic subgroup B=MAN. Pick m*w∑M*. Then M being contained in B, the

double coset Bm*wB depends only on w. The important Bruhat Lemma [84] states that

(*) G is the disjoint union of the double cosets Bm*wB, for w ranging over the Weyl group.

A consequence of immediate importance to us is the fact that B=MAN is the whole normalizer of N83See, for example, Nolan R. Wallach, Harmonic Analysis on Homogeneous Spaces (New York: Marcel

Dekker, 1973), Section 7.6.84 Garth Warner, Harmonic Analysis on Semi-Simple Lie Groups I, Theorem 1.2.3.1.

140

— V.1 Set-up —

(that B normalizes N is obvious). Indeed, suppose g normalizes N, and write g=b¡m*wb™, b¡,b™∑B,

w∑W. Since B normalizes N, so does m*w. Equivalently, w maps Í+ into itself, so that w is the

identity, i.e., m*w∑M, yielding g∑B.

Since B=MAN is the semi-direct product of MA and N, where N is normal, it follows that

B/N equals MA. The results in Section IV.8 then yield that:

Proposition V.1.1 The G–invariant diffeomorphisms of ≈=G/N are of the form

gp éêâ g(©p) g∑G,

for ©∑MA. This defines a right action of MA on ≈, for which ≈ is a principal MA–bundle.

Section IV.8 also shows

Proposition V.1.2 The set MAp is a closed regular submanifold of ≈, consisting of one-point

N–orbits.

We need more information on the N–orbit structure.

V.2 N˚ –orbit structure

The Bruhat Lemma implies that the B–orbits on the homogeneous space G/B (which is the same as

the N–orbits on G/B) are indexed by the Weyl group. There is a well-known refinement of the Bruhat

Lemma describing these orbits as regular submanifolds of G/B, so-called Schubert cells, parametrized

by particular subgroups of N. In this section we modify these results to obtain similar descriptions for

the N–orbits in G/N. Since we are interested in invariant operators in ∂æ(≈), or, equivalently, in

compactly supported N–invariant, i.e. zonal, distributions, the main purpose of this exercise is to

determine which orbits are bounded.

Let Nä be the nilpotent subgroup of G opposite to N, that is to say that Nä is the image of N under the

Cartan involution ∆. It turns out that for an understanding of the N–orbit structure on G/N it is useful

to study the subset äNäN of G. The following properties of NäN are essential:

141


Proposition V.2.1

(i) äNN§M*A=”e’

(ii) NäN is closed.

To see this we prove a few lemmas. The first is purely algebraic.

Lemma V.2.2 Let X and Y be two subgroups of a group G, and let Z be their common

normalizer (i.e. the intersection of the normalizers of X and Y). Then for x∑X, y∑Y, the product xy

normalizes Z if and only if both x and y do so.

Proof Take z∑Z and consider the element zx (that is, x¡zx). Since z normalizes X, so does zx.

Furthermore xy normalizes Z, so that zxy belongs to Z. Since therefore zxy normalizes Y, so does

(zxy)y¡=zx. Consequently, zx normalizes both X and Y, so that it belongs to Z. All this is valid for

arbitrary z∑Z, so x normalizes Z, and likewise does y !

Lemma V.2.3 Let X and Y be two closed subgroups of a Lie group G. Let X§Y be compact.

Then the set XY is closed in G if and only if the product map (x,y) éêâ xy is proper as map from

X*Y to G.

Corollary In addition, let Z be a compact normal subgroup in G, such that XY§Z=e. Then

XY is closed in G if and only if its image is closed in G/Z.

Proof Let the Lie group X*Y act on G through (x,y)g=xgy¡, x∑X, y∑Y, g∑G. Consider the set XY as the

orbit through e under this action, with stability group K:=”(z,z)»z∑X§Y’. When this orbit is closed, it

is a regularly imbedded submanifold of G (see [85] ). The map (X*Y)/K ììâ G, (x,y)K éêâ xy¡, is

therefore a homeomorphism with closed image, so that it it is proper. Since K is compact, the map X*Y

ììâ G, (x,y) éêâ xy¡ is proper as well, and this of course implies the same for the product map. The

converse statement in the lemma is trivial !

85 See V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations (Berlin: Springer Verlag,

1974), Theorem 2.9.7.

142

— V.2 N–orbit structure —

Proof of the corollary Let π denote the canonical map G ììâ G/Z. Since Z is compact, π is

proper. So when XY is closed, π(XY) is so too. In particular, π(X) and π(Y) are closed subgroups of

G/Z. Moreover, since (XY§Z)=e it follows that π is one-to-one on XY, which implies among other

things that π(X)§π(Y) equals π(X§Y), so that it is compact. According to the lemma, when

π(XY)=π(X)π(Y) is closed, the multiplication map π(X)*π(Y) ììâ G/Z is proper. Since π is proper, this

implies that the map X*Y ììâ G/Z, (x,y) éêâ π(x)π(y)=π(xy) is proper. This implies that the

multiplication (x,y) éêâ xy∑G is proper, so XY is closed.

The converse is immediate !

Lemma V.2.4 Let T(p) be the subgroup of GL(p;Â) formed by the upper triangular matrices

with all diagonal elements equal to 1. Let Tä(p) be the group of lower triangular matrices with

diagonal elements equal to 1. Then Tä(p)T(p) is a closed subset of GL(p;Â).

Proof By simply looking at what elements xy, x∑Tä(p), y∑T(p), look like one can easily convince

oneself that Lemma V.2.4 is true. Here is a formal proof.

Let å be the map GL(p;Â)êêêâT(p),

(»»»»»»»9

1 -g¡™ -g¡£ . . -g¡p )»»»»»»»0

0

å(g) =0 1p–1

0

where 1p–1 denotes the identity in GL(p–1;Â), and let å_: GL(p;Â)êêêâäT(p) be the counterpart to å,

(»»»»»»»9

1 0 0 . . 0 )»»»»»»»0

-g™¡

å_(g) = 1p–1-g£¡

..

–gp¡

143


It is quickly seen that for x∑Tä(p), y∑T(p), g∑GL(p;Â), one has å(xg)=å(g), å_(gy)= å_(g). Furthermore,

for y∑T(p), y.å(y) belongs to T(p–1), when the latter is seen as subgroup of GL(p;Â), in keeping with

the imbedding GL(p–1;Â) ≤ GL(p;Â),

(»»»»»»9

1 0 0 . . 0 )»»»»»»0

.

0g éêâ 0 g

0

This implies that under the map g éêâ å_(g).g.å(g), the set Tä(p)T(p) is mapped not only into itself

(obviously), but actually into Tä(p–1)T(p–1) (as subspace in GL(p-1;Â)). Proceeding by induction, one

obtains continuous maps †:G ììâT(p), and †_:G ììâTä(p) such that for g∑Tä(p)T(p) one has

†_(g).g.†(g)=1. This implies what we want. In fact this shows even more, that is, that the imbedding of

Tä(p)T(p) into GL(p;Â) has a continuous left inverse !

Proof of Proposition V.2.1 A proof of (i) can be found in Warner [86] . We see (i) essentially

a case of Lemma V.2.2: take X=Nä, Y=N. Since Nä§MAN = e, Z becomes MANä§MAN=MA, with

normalizer M*A. Lemma V.2.2 yields that ( NäN)§(M*A)=(Nä§M*A).(N§M*A). Iwasawa

decompositions show plainly that both Nä§M*A and N§M*A equal e. This proves statement (i) in

the proposition.

As to the second statement, let Ad denote the adjoint representation of G on g. Since G is

assumed to be connected, the kernel of Ad equals the centre Z of G, which is finite by assumption.

Furthermore, G being connected and semi-simple, Ad(G) is the connected component of the identity in

Aut(G), and therefore closed in GL(g). So we may identify Ad(G) with G/Z. Moreover, statement (i) in

the proposition implies that since Z is contained in M, one has NäN§Z≤ NäN§M=e. Finally,

Nä§N=e, so that the corollary to Lemma V.2.3 applies. Therefore, the closedness of NäN in G is

equivalent to the closedness of Ad(NäN) in GL(g).

Let p be the dimension of G. It is well-known (as a standard ingredient in proofs of the

Iwasawa decomposition) that for a suitable choice of basis in g, Ad(N) is a closed subgroup of T(p).

Likewise, Ad(Nä) becomes a closed subgroup of Tä(p). The set Tä(p)T(p) is closed according to Lemma

V.2.4. Since Tä(p)§T(p)=e a straighforward use of Lemma V.2.3 shows that this implies that

indeed Ad(NäN)=Ad(Nä)Ad(N) is closed !86 Garth Warner, Harmonic Analysis on Semi-Simple Lie Groups I, p. 59, proof of Proposition 1.2.3.6.

144

— V.2 N–orbit structure —

Remark Let å:G ììâ A denote the projection onto the A–component in the Iwasawa decomposition

G=KAN. Lemma V.2.3 can easily be shown to imply that the closedness of äNN is equivalent to the

properness of the restriction of å to Nä. So we have now shown the second property by proving the

first.

To introduce the theorem on the N–orbit structure, a few remarks.

From Proposition V.1.2 one obtains that M*Ap is a closed and therefore regular

submanifold of ≈. It is the finite union of translations of MAp=pMA, that is, M*Ap=w

¶∑W

mw* pMA,

where for w∑W, the element mw* ∑M* is associated with w, as in Section V.1. Each ©*∑M*A is

associated to a particular w∑W, since M*A/MA=W.

Let w be an element in the Weyl group. Define two subalgebras of n, by setting

nw=n§Admw*(n), uw=n§Admw*( nä) These are complementary in n (because Admw* permutes the

root spaces). Moreover, they depend only on w. Define Nw=exp(nw), and Uw=exp(uw).

Theorem V.2.5 . N–orbit structure

(i) The N–orbits in ≈ are indexed by M*A, so ≈=©*∞∑M*A N©*p , with N© *

¡p=N© *™p if

and only if ©*¡=©*™.

(ii) The N–orbits are closed regular submanifolds of ≈.

(iii) Let ©*∑M*A be associated to w∑W. Then N©*p is homeomorphic to Uw, under the

map Uwëu éêâ u©*p.

Proof It is well-known [87] that Nw and Uw are closed simply connected subgroups of N, and that

N decomposes as

(V.2.5.a) N=Nw.Uw=Uw.Nw, with Nw§Uw=e.

Let Nw¡ denote the group mw*Nmw* ¡, then Nw=N§Nw¡. Since Nw¡ equals Gmw*p (the stability

group of the point mw*p∑≈), Nw equals the stability group within N of mw* p. Decomposition (V.2.5.a)

therefore implies that the N–orbit through m*wp is parametrized by the group Uw, that is to say, the

map Uwëu éêâ um*wp is one-to-one map onto the N–orbit through m*wp.

Being G–invariant, the right action of MA permutes the N–orbits. More precisely, take

87 Garth Warner, Harmonic Analysis on Semi-Simple Lie Groups I, p. 37, Proposition 1.1.4.5 and preceding

paragraphs.

145


©*∑M*A, and assume that ©*=m*w.©, for w∑W, ©∑MA. Then N©*p=Nm*w©.p=Nm*w.p©=(Nm*w.p)©,

implying that Uwëu éêâ u©*p is a parametrization of the N–orbit through ©*p, for all ©*∑M*A, with

w the Weyl element associated to ©*. This yields a parametrization for every orbit. Indeed, the Bruhat

decomposition for G immediately implies that G is the union of double cosets N©*N, ©*∑M*A, which is

just another way of saying that G/N is covered by the orbits N©*p, ©*∑M*A.

The orbit N©*p can now be viewed as Uwm*w.p©=m*w(Uw)wp©=©*(Uw)wp, where

(Uw)w=(m*w)¡(Uw)m*w equals ∆(Uw¡), which is a closed subgroup of Nä. And so, N©*p is contained

in ©*Näp. Proposition V.2.1.(ii) means precisely that Näp is a closed, and therefore regular, submanifold

of ≈. Since Nä§N=e, the map Näën_éêân_p is a regular imbedding with closed image. The set (Uw)w

being closed in Nä, this implies that N©*p=©*(Uw)wp is closed in ≈. This proves (ii). This also proves

(iii), that is, the parametrization is regular because the image is closed.

Finally, when ©¡*, ©™*∑M*A, the orbits N©¡*p and N©™*p are equal if and only if ©¡*p belongs

to N©™*p. Since N©™*p is contained in ©™*Näp, this can happen only if (©™*)¡©¡* belongs to NäN. Proposition

V.2.1.(i) then implies that ©¡*=©™*. This takes care of (i) !

Using the action of the Weyl group, it is now easy to show that M*Ap indexes the Nw–orbit structure

on ≈ for any Weyl-conjugate of N. Since there is always a (unique) element wº∑W with Nwº=Nä, in

particular M*Ap indexes the Nä–orbit structure. It also follows from the proof that the N-orbits of

highest dimension (equal to the dimension of N) are those of the form N©*p, for ©* associated with

wº. Of more immediate interest to us is:

Corollary V.2.6 Every N–orbit outside MAp is unbounded.

Proof Let ©* be associated with w. Since Uw is simply connected, the orbit N©*p is homeomorphic

to the linear space uw. So for N©*p to be bounded it must be a single point. As argued in Section V.1,

this means w=1, i.e. ©*∑MA!

We defined a space as being weakly symmetric when the map F éêâFfi maps bounded zonal sets to

bounded zonal sets (Definition IV.5.1).

Corollary V.2.7 G/N is weakly symmetric.

146

— V.2 N-orbit structure —

Proof In view of Theorem V.2.5 the zonal subsets of G/N correspond to subsets of M*A

(algebraically only, since N\G/N is non-Hausdorff). Under this correspondence the fundamental

reflection F éêâ F% agrees with the group inversion in M*A, that is, (N©*p)fi=(N(©*)¡p) for ©*∑M*A.

Let F be a compact zonal subset of G/N. Then, in view of Corollary V.2.6, the set F is a

compact subset of MAp, and so the inverse set Ffi is also compact. So, the map F éêâ F% maps compact

zonal sets to compact sets, and so bounded zonal sets are mapped to bounded zonal sets !

V.3 Convolution Operators in ¶¾(G/N)

Again, let G be a connected real semi-simple Lie group with finite centre, with Iwasawa

decomposition G=KAN. The purpose of this section is to determine explicitly the algebra of

G–invariant operators in the distributions on the homogeneous space G/N.

First recall some facts. Let ≈ denote the homogeneous space G/N, with origin p=eN. Let

M be the centralizer of A in K. Then the group of G–invariant diffeomorphisms equals MA, identified

with the submanifold MAp of single-point N–orbits (see Section V.1). Let ^:MA ììâ ≈, ^(©)=©p, be

the (formal) imbedding of MA. Then according to Section IV.9 every compactly supported

distribution S on MA is the propagator of a unique convolution operator in the distributions,

uT=T*^*S T∑∂æ(≈).

Such operators are called induced (from MA), see Definition IV.9.5.

Theorem V.3.1

All G–invariant operators in the distributions on ≈ are induced from MA.

Corollary V.3.2 The map that associates to every S∑ æ(MA) the operator of convolution by ^*S%

is an is an isomorphism from the convolution algebra æ(MA) onto the algebra å of G–invariant

operators in ∂æ(G/N).

Before giving the proof we first reformulate the problem, and prove some lemmas.

Set ≈:=G/N. Let u be a G–invariant operator in ∂æ(≈). Since ≈ is weakly symmetric

147


(Corollary V.2.7), u is propagated by a zonal distribution U on ≈ with compact support. In view of

Corollary V.2.6, this means that supp U is contained in MAp. So, U∑´æMAp(≈)N. What we have to

show is that the N–invariance of U implies that U is of the form ^*S, S∑´æ(MA), in other words, that U

is of transversal order 0. Note furthermore that N is connected, so that a distribution is zonal, i.e. fixed

by N, if and only if it is killed by n.

In view of Theorem III.4.1, as a topological Ë(g)–module,

(V.3.2.a) ´æMAp(≈) = Ë(g) Ë ¤(b)

´æ(MA).

Here b equals the Lie algebra m@a@n of the minimal parabolic subgroup B=MAN. The

representation of Ë(b) in ´æ(MA) is the infinitesimal quasi-regular representation, where MA is seen

as MAN/N, a homogeneous space under the left action of MAN. Since N is normal in MAN, the action

of N is trivial in ´æ(MA). Moreover, ´æMAp(≈) is a B–module simply because MAp is a B–orbit. Let †õ

denote the quasi-regular representation of B in ´æ(MA). Then according to Proposition III.4.4 the

natural (Ë(g),B)–module structure of ´æMAp(≈) is realized in Ë(g)Ë

¤(b )´æ(MA) by imposing the

representation Ad¤†õ of B defined by

(Ad¤†õ)b(¨Ë

¤(b)

S)=·Adb¨‚Ë

¤(b)

†õbS, b∑B, ¨∑Ë(g), S∑´æ(¥).

So, reformulating the problem, what we have to show is that in the (Ë(g),B)–module Ë(g)Ë

¤(b )

´æ(MA) the zonal, or N–invariant elements (equivalently, elements killed by n) are those of

transversal order 0. We first deal with the analogous problem for a familiar class of (Ë(g),B) modules.

Let ≈ be an irreducible (and therefore finite dimensional) unitary representation of the compact group

M, on the Hilbert space Ó≈. Let aæÇ denote the complexified dual of a, i.e., the space of complex-

valued Â–linear (restricted) weights. Remember that for ~∑aæÇ, the character a éêâ a~ is defined on

A by

(expX)~=e~(X) X∑a, ~∑aæÇ.

The space Ó≈ becomes a B–representation space, by defining the action of N to be trivial, and by

letting A act by multiplication by the character a~. Let (≈¤a~¤1) denote this representation, so

(V.3.2.b) (≈¤a~¤1)manv=a~. ≈mv, ≈∑ Mfl, ~∑aæÇ, m∑M, a∑A, n∑N, v∑Ó≈.

Let Ó≈~ be Ó≈

equipped with this representation. Then Ó≈~ is a Ë(b)–module as well as a B–module.

Define the module

148

— V.3 Convolution Operators in ∂æ(G/N) —

(V.3.2.c) √≈~:=Ë(g)

Ë¤(b )Ó≈

~ ≈∑MÀ, ~∑aæÇ.

This type of module is called a generalized Verma module (when M is discrete, so m=(0), it is an

ordinary Verma module)[88] . It is a (Ë(g),B)–module in the usual way, that is, B acts by

b(¨Ë

¤(b)

v)=·Ad(b)(¨)‚Ë

¤(b)

(≈¤a~¤1)bv, b∑B, ¨∑Ë(g), v∑Ó ≈~, cf. Definition III.4.3 and Proposition

III.4.4.

Since the action of N on Ó≈~ is trivial, it is obvious that transversal order 0 elements, i.e.,

those of the form 1Ë

¤(b)

v, v∑Ó≈~, are fixed by N. (In the theory of Verma modules such elements are

usually called highest weight (see the proof of the following lemma), but in the present context of

distributions with transversal derivatives we prefer the expression 'transversal order 0').

Lemma V.3.3 The generalized Verma module √≈~ is irreducible if and only if all

n–annihilated elements are of transversal order 0.

In fact we need only the "only if" direction of this lemma. For the sake of completeness we prove the

lemma in both directions.

Proof Fix ≈ and ~, and set √=√≈~.

Let n_ denote the nilpotent subalgebra of g opposite to n, opposite with respect to the

Cartan involution, so g=n_@m@a@n=n_@b. Let √(+) denote the subspace n_√ of √, and let √(º)

denote the space of transversal order 0 elements in √. Then it is easy to see that √(º) and √( +) are

complementary:

(V.3.3.a) √= √(º) @ √(+) .

For example, it follows from the direct sum decomposition g=n_@b that the canonical map Ë(g)¤Ó≈~

ììâ Ë(g)Ë

¤(b )Ó≈

~ = √ is an isomorphism when restricted to Ë(_n)¤Ó≈

~ (to see this, set ¬=Ë(n_) in

Proposition I.7.5). Accordingly, the decomposition Ë(n_)=Ç.1@ n_Ë( n_) induces the decomposition

(V.3.3.a). The subspace √( +) can also be described as the sum over the weight spaces for the action

of a for weights other than ~.

n_ is an invariant subspace for the adjoint representation of M. And so, for m∑M, X∑_n, v∑√,88 In the theory of Verma modules usually a ®–shift is involved, that is, ~ is replaced by ~+®, where ® is half

the sum of the positive weights (counted with multiplicity). At this point the ®–shift is irrelevant.

149


mXv=·Ad(m)X‚(mv)∑ n_√=√( + ), so that √( + ) is M–invariant. Moreover, set b_=n_@m@a, then

[_b,nä]≤nä, so that b_√( +)=_bnä√≤nä√=√( +). In other words,

(V.3.3.b) √(+) is a (Ë(b_),M)– submodule of √.

Now assume that √ is irreducible. Assume that v∑√ is an n–annihilated element, or, equivalently,

N–fixed. Decompose v=vû+v+ in accordance with (V.3.3.a). Since vû is of transversal order 0, it is

n–annihilated. Therefore, so is v+. Since g=b_@n is an AdM–invariant decomposition, the

(Ë(g),M)-submodule of √ generated by an n–annihilated element equals the (Ë(b_),M)–submodule

generated by that element. In view of (V.3.3.b) this implies that the (Ë(g),M)–submodule of √generated by v+ is contained in √( +). And so this submodule cannot be all of √. Since √ is

irreducible it must be (0). This forces v+=0, in other words, v=vû is of transversal order 0. This

proves the "only if" direction of the lemma.

To show the reverse implication in the lemma we demonstrate that for every generalized

Verma module √,

(V.3.3.c) every non-zero (Ë(g),M)–submodule of √ contains a non-zero n–annihilated vector.

To see that this will do, assume, as in the lemma, that all n–annihilated vectors have transversal order

0. Then (V.3.3.c) implies that every non-zero submodule of √ contains a transversal–order–0 vector

1Ë

¤(b)

h, h∑Ó≈~, h≠0. But such a vector generates all of √, as (Ë(g),M)–module. This is because Ó≈

~ is

by definition an irreducible M–module, so the M–module generated by h≠0 is all of Ó≈~, and Ó≈

~

generates all of √=Ë(g)Ë

¤(b )Ó≈

~. In other words, every non-zero (Ë(g),M)–submodule of √contains a vector that generates all of √, so √ is indeed irreducible.

We prove (V.3.3.c) as follows. One way of expressing the triviality of the action of n on

Ó≈~ is through the equation X

Ë¤(b)

h=0, for all X∑n, h∑Ó≈

~. This leads to

(V.3.3.d) X(¨Ë

¤(b)

h)=(adX(¨))Ë

¤(b)

h , X∑n, ¨∑Ë(g), h∑Ó≈

~.

The adjoint representation of n on g is nilpotent, that is to say, the derivations adX, X∑n, operating on

g, are nilpotent endomorphisms. This is easy to see, for example by considering the root space

decomposition of g. It follows that the adjoint representation of n on the universal enveloping algebra

of (the complexification of) g is nilpotent on every subspace Ë(m)(g) of elements of order not

exceeding m. Therefore (V.3.3.d) implies that the action of n on √ is nilpotent on every subspace

√(m) of elements of transversal order not exceeding m (cf. Definition I.7.8). Every √(m) being

150


finite dimensional, and every v∑√ being of finite transversal order, it follows that for every v∑√,

v≠0, the submodule Ë(n)v≤√ is a finite dimensional vector space in which n operates by nilpotent

endomorphisms. Then apply the following theorem: for a Lie-algebra of nilpotent endomorphisms

operating on a finite-dimensional vector space there exists a non-zero vector annihilated by all these

endomorphisms. This theorem is one of the main ingredients in the proof of Engel's Theorem, see

[89] . This then proves (V.3.3.c) !

As to the irreducibility of generalized Verma modules we need only the following global property,

which is known in the literature on the subject [90] :

Lemma V.3.4 For every ≈∑MÀ the set of ~∑aÇæ such that √≈~ is irreducible is an open and

dense set in aæÇ.

Let V be a finite dimensional B–module. Set √=Ë(g)Ë

¤(b)

V, and let W be a finite dimensional

vector space. Then V¤W becomes a B–module by trivial extension of the action of B, that is, by

setting b(v¤w):=(bv)¤w, b∑B, v∑V, w∑W. Moreover, (Ë(g),B)–induction commutes with this

operation, that is,

(V.3.4.a) Ë(g)Ë

¤(b)

“V¤W‘= “Ë(g)Ë

¤(b)

V‘¤W.

As to the set of n–annihilated elements, it is easy to see that

(V.3.4.b) ·Ë(g)Ë

¤(b)

“V¤W‘‚n= “Ë(g)Ë

¤(b)

V‘n¤W

(where the subscript n has the meaning ‘subspace of elements annihilated by n‘). We use the

following consequence:

Lemma V.3.5 When the module √ =Ë(g)Ë

¤(b)

V has the property that the n–annihilated

elements are those of transversal order 0, then so has every module √¤W, for W any finite

dimensional complex vector space W.

89 James E. Humphreys, Introduction to Lie Algebras and Representation Theory (New York: Springer

Verlag, 1972), Sections 3.2 and 3.3.90 See Nolan R. Wallach, Real Reductive Groups II (New York: Academic Press, 1992), Pure and Applied

Mathematics Series Vol. 132-II, appendix to Chapter 11, p. 127, Lemma 11.A.3.2. This result (rather than the

better-known result for the complex case, or for m=(0)) was kindly pointed out to me by Dr. E.P. van den Ban.

151


Finally we recall a few well-known facts about the representation theory of compact and abelian

groups.

Consider again the representation ≈∑MÀ, on the space Ó≈. Let (.|.) denote an invariant inner

product on Ó≈, with respect to which ≈ is a unitary representation. Let ˜≈ be the complex span of the

matrix coefficients Mëméêâ·≈mx»y‚, x,y∑Ó≈. Then ˜≈ is a finite dimensional subspace of the

ç°–functions on M, and a closed Hilbert subspace of L™(M), depending only on the equivalence

class of ≈.

Let Ó… …≈ denote the Hilbert space conjugate to Ó≈, so Ó… …≈ is the linear space conjugate to

Ó≈, with inner product (x_|y_)Ó…… ≈

=(y|x)Ó≈, and equipped with the representation ≈_ defined by

≈_mx_=≈î …mx_, m∑M, x∑Ó≈. The span of matrix coefficients ˜≈ can be identified with the tensor product

Ó… –≈¤Ó≈, by setting (y_¤x)(m)=·≈mx»y‚Ó≈, x, y∑Ó≈, m∑M. This isomorphism intertwines the

representation ≈_¤≈ of M*M on Ó……≈¤Ó≈ with L¤R, where L and R are the left and right regular

representations of M in the functions on M. The space Ó… …≈¤Ó≈ has its own tensor product Hilbert

structure, and the identification ˜≈=Ó… …≈¤Ó≈ is isometric up to a factor only. More precisely,

˜≈=d≈“ Ó… …≈¤Ó≈‘, where d≈ equals the dimension of ≈. In these terms the Peter-Weyl Theorem and

the Schur orthogonality relations for the compact group M can be formulated as

(V.3.5.a) L™(M)=≈¶∑MÀ

@d≈“ Ó… …≈¤Ó≈‘

Let Ô≈ be the character of ≈, so

(V.3.5.b) Ô≈(m)= i¶d

=

≈

1 ·≈mei»ei‚ m∑M,

for ·ei‚i=1,Ú,d≈ an orthonormal basis of Ó≈. Then the orthogonal projection L™(M) ììâ ˜≈ is

convolution by d≈Ô≈. So, when seen as a Hilbert subspace of ∂æ(M)=´æ(M), ˜≈ has reproducing

distribution d≈Ô≈. So (V.3.5.a) can be reformulated as a decomposition of the Dirac delta distribution

at the origin of M:

(V.3.5.c) ∂eM=≈¶∑MÀ

d≈Ô≈ in ´æ(M).

All this is well-known [91] .

Since ˜≈ is contained in ´(M), its reproducing operator (i.e., convolution by d≈Ô≈) extends

91 See e.g. Charles F. Dunkl & Donald E. Ramirez, in their Topics in Harmonic Analysis, Appleton-Century

Mathematics Series (New York: Meredith Corporation, 1971), Chapters 7 and 8.

152


continuously to ´æ(M), with unchanged (because finite dimensional) image, so

(V.3.5.d) ´æ(M) * d≈Ô≈=˜≈ ≈∑MÀ.

Take a weight ~∑aÇæ . Let ˜≈~ denote the space of matrix coefficients associated to the

representation ≈¤a~¤1, defined at (V.3.2.b). Let Óä≈–~ denote Óä≈ equipped with the representation

≈_¤a–~¤1. Then ˜≈~ equals Ó…≈

–~¤Ó≈~, when y_¤x is identified with the map MANëman éêâ

(≈mx»y)Ó≈a~, and under this identification the representation L¤R of B*B on ˜≈

~ corresponds to the

representation “≈_¤a–~¤1‘¤“≈¤a~¤1‘ on Ó…≈–~¤Ó≈

~. Furthermore, since the matrix coefficients that

belong to ˜≈~ are derived from a representation of B=MAN that is trivial on N, they are themselves

right N–invariant, so ˜≈~ can be identified with a left B–invariant submodule of ∂æ(MA). More

precisely, this identification is ˜≈~=˜≈¤“a~‘≤∂æ(MA).

Let Ô≈~ denote the character

(V.3.5.e) Ô≈~(ma)=Ô≈(m).a~

corresponding to i¶d

=

≈

1e_i¤ei, where ·ei‚i=1,Ú,d≈ is an orthonormal basis of Ó≈

~. When ~ is purely

imaginary, ˜≈~ is a bi-invariant Hilbert subspace of ∂æ(MA), with reproducing distribution d≈Ô≈

~.

Moreover, since its image is contained in ´(MA), and is finite dimensional, convolution by d≈Ô≈~

extends continuously from ∂(MA) to ´æ(MA), with identical image, so

(V.3.5.f) ´æ(MA) * d≈Ô≈~=˜≈

~ = Ó…≈–~¤Ó≈

~.This is in fact true for all ≈∑Mfl, ~∑aÇæ

. The difference when ~ is not purely imaginary is merely that in

that case the bi-invariant space ˜≈~ has no invariant Hilbert structure for the action of A.

For S∑´æ(MA) define its Fourier transform SÀ as the map

(V.3.5.g) SÀ: MÀ*aÇæ ììâ ∂æ(MA)

SÀ(≈,~)=S*Ô ≈~ ∑ ˜≈

~ [92]

What we need is the fact that the map S éêâ SÀ is one-to-one. This follows from the Plancherel

formula for the group MA, seen as a decomposition of the Dirac delta distribution at the origin of MA

by means of characters Ô≈~:

92 These are ad hoc definitions. The Fourier transform on M alone, for example, would normally be defined

as a field over Mfl with values in L(Ó≈) [the space of linear operators on Ó≈,equipped with Hilbert-Schmidt

norm]. There is no essential difference, however. The map L(Ó≈)ëA éêâ ·g éêâ tr(A≈g)‚∑˜≈ leads from

one situation to the other. For the Fourier transform on MA, see Section V.4.

153


(V.3.5.h) ∂eMA= ≈¶∑MÀ

d≈ ªiaæ

d~ Ô ≈~

for a suitable choice of Lebesgue measure d~ on iaæ.

With these facts available, we can give the

Proof of Theorem V.3.1 Take ≈∑MÀ,~∑aæÇ. Consider the distribution ^*Ô≈~, where Ô≈

~ is defined as

above, at (V.3.5.e). Since this is a zonal distribution, it propagates a convolution operator ´æ(≈) ììâ

∂æ(≈), according to Theorem IV.2.2. Moreover, when U belongs to ´æMAp(≈), then according to

Proposition IV.4.5 supp (U*^*Ô≈~) is contained in supp(U).supp (^*Ô≈

~), which, being the product of

two subsets of MAp, is itself contained in MAp. So,

´æMAp(≈) * ^*Ô≈~ ≤ ∂æMAp(≈).

More explicitly, convolution being G–equivariant, it is Ë(g)–equivariant. So, for ¨∑Ë(g), U∑´æ(Ì),

V∑∂æ(Ì), one obviously has (¨^*U)*(^*Ô≈~ )=¨^*(U*Ô≈

~ ) (compare Proposition IV.9.4.(b)).

Therefore, in terms of the algebraic-topological form for ´æMAp(≈) and ∂æMAp(≈) as given in

Theorem III.4.1, the operator Ù of convolution by ^*Ô≈~ allows the following simple description:

(V.3.5.i) Ù : Ë(g)Ë

¤(b)´æ(MA) ììâ Ë(g)

Ë¤(Àb)∂æ(MA)

(¨Ë

¤(b)

S) *(^*Ô≈~ ) = ¨

Ë¤(b)

(S*Ô≈~).

For U∑´æMAp(≈) define its transform UÀ, as follows:

(V.3.5.j) UÀ: MÀ*aæÇ ììâ ∂æMAp(≈)

UÀ(≈,~)=U* ^*Ô≈~ .

It is easy to see that UÀ is continuous. The main thing needed for this is the separate continuity of the

convolution product as map ´æ(≈)*∂æ(≈)N ììâ ∂æ(≈) (Proposition IV.4.5.(i)). Moreover, (V.3.5.g)

and (V.3.5.i) show that for each ≈ and ~

(V.3.5.k) UÀ(≈,~) ∑Ë(g)Ë

¤(b)

˜≈~ ≤ Ë(g)

Ë¤(Àb)

∂æ(MA).

Here the (Ë(g),B)–module structure on Ë(g)Ë ¤(b)

˜≈~ is brought about by the imbedding Ë(g)

Ë ¤(b)

˜≈~

≤Ë(g)Ë ¤À(b)

∂æ(MA), which means that in its own right Ë(g)Ë ¤(b)

˜≈~ is the (Ë(g),B)-module induced

from ≈~ as (left) B–module. In view of the identification ˜≈

~=Ó…≈–~¤Ó≈

~ one obtains, in keeping with

(V.3.4.a):

154


(V.3.5.ñ) Ë(g)Ë

¤(b)

˜ ≈~=Ë(g)

Ë¤(b)

“ Ó…≈–~¤Ó≈

~‘

=“Ë(g)Ë

¤(b)

Ó…≈–~‘¤Ó≈

~

=√ ≈–_

~¤Ó≈

~ ≤ ∂æMAp(≈) .

where √≈–_

~ is a generalized Verma module as defined at (V.3.2.c).

Let U∑´æMAp(≈) be the propagator of a convolution operator, so U is zonal. Since

convolution is G–equivariant, it follows that UÀ(≈,~) is a zonal element in √ ≈–_

~¤Ó≈

~, for every ≈∑Mfl

and every ~∑aÇæ . Assume that ≈ and ~ are such that √≈–_

~ is irreducible. Then according to Lemmas

V.3.3 and V.3.5, the zonal elements in √≈–_

~¤Ó≈

~ are of transversal order 0. In view of Lemma V.3.4

this means that on a dense subset of Mfl*aÇæ the transform UÀ takes its values in the subspace of

transversal order 0 elements in ∂æMAp(≈). But UÀ is continuous, and the transversal order 0 elements

form the closed subspace ^*∂æ(MA) of ∂æMAp(≈) (cf. Proposition II.6.2.(ii)). And so UÀ takes all its

values in ^*∂æ(MA). This is the key point.

To round off the proof, choose a linear complement l of b in g, choose a basis X¡,Ú,Xñ of l,

and decompose uniquely U=å˚˚¶∑ˆ

ñX¡

å¡X™å™ÚXñ

åñË

¤(b)

Uå. Then (V.3.5.i) shows that

Ufl(≈,~)=å¶∑ˆ

ñX¡

å¡X™å™ÚXñ

åñ

Ë¤(b)

Uflå(≈,~).

The fact that Ufl maps into ^*∂æ(MA) means precisely that Uflå(≈,~)=0, for all ≈ and ~, and for all å≠0.

The injectivity of the Fourier transform (argued at (V.3.5.h)) implies that Uå=0, for å≠0. So U is

indeed of transversal order 0 !

Proof of Corollary V.3.2 Compare Section IV.9 to see that the map ∫: ´æ(MA) ììâ å,

(∫(S))(T)=T*(^*S%), is an injective algebraic isomorphism. It is onto in view of Theorem V.3.1 !

By looking only at invariant operators whose propagators are concentrated at p one obtains as a

special case of Corollary V.3.2:

Corollary V.3.6 The algebra of G–invariant differential operators on ≈ is isomorphic to the

universal enveloping algebra of MA.

155


Theorem V.3.1 can be seen as a generalization of this result. Corollary V.3.6 is not new, and was

obtained by for example by Koornwinder in [93] , in developing work by Helgason [94] . The

method used by Koornwinder exists essentially in reducing a problem involving non-commuting

vectorfields to one with commuting vectorfields by an efficient handling of the infinitesimal

G–action, and by involving the symmetrizer map. His method cannot, however, be adapted to yield a

proof of Theorem V.3.1. We do have proofs that use only systematic calculations in the non-

commuting vector fields derived from G, but only under the condition that the root system allow a

total ordering of a very particular nature. We have not been able to demonstrate the existence of such

an ordering for the E and F root systems. So, Verma modules appear to be a necessary ingredient in

the proof.

The fact that propagators of G–invariant operators in ∂æ(≈) must have compact support is

essential in the proof of Theorem V.3.1. Indeed, there do exist zonal distributions concentrated on

MAp (necessarily of unbounded support) of strictly positive transversal order. The proof shows that

there are in fact lots of these. Indeed, ∂æMAp(≈) contains the modules Ë(g)Ë

¤(b)

˜≈~, ≈∑MÀ, ~∑aÇæ (see

(V.3.5.k)), and it is a part of the proof that when such a module is reducible it contains contains

N–fixed elements of strictly positive transversal order, which happens when the corresponding Verma

module √≈–_

~ is reducible. We will not be concerned with the question when exactly this happens: this

can be found in various texts, such as [95] . A concrete example will emerge in Section VII.7, where

zonal distributions concentrated on MAp of strictly positive transversal order (and therefore

necessarily of unbounded support) emerge as essential constituents of reproducing distributions of the

discrete series of SL(2;Â). See Theorem VII.7.8.A.(ii).

In turn, however, when the invariance requirements are stepped up, one can obtain the

following result for not necessarily compactly supported distributions concentrated on MAp.

93 T.H. Koornwinder, “Invariant Differential Operators on Non-Reductive Homogeneous Spaces,” Report,

ZW 153/81 (Amsterdam: Mathematical Centre, 1981), Theorem 3.2.94 S. Helgason, “Duality and Radon Transform for Symmetric Spaces,” Amer. J. Math.85 (1963), pp. 667-

692. See also his “Invariant Differential Operators and Eigenspace Representations,” Representation Theory

of Lie Groups; London Mathematical Society Lecture Note Series 34 (Cambridge: Cambridge University

Press, 1979), pp. 236-286, see Lemma 3.3 on page 267.95 Jens Carsten Jantzen, Moduln Mit Einem Höchsten Gewicht Lecture Notes in Mathematics Vol. 750

(Berlin:Springer Verlag,, 1979).

156


Proposition V.3.7 A distribution concentrated on MAp is strongly zonal if and only if it is of

transversal order 0, and derives from a central distribution on MA.

The proof is straightforward, as follows.

Proof The only real problem is to show that when a distribution is concentrated on MAp is strongly

zonal, then it must be of transversal order 0, so of the form ^*S. It is easily seen that a distribution of

the form ^*S, which is already zonal, is strongly zonal if and only if S is central on MA. For this it is

sufficient to consider the commutation relations with respect to other distributions of the form ^*V,

and to use the fact that ^* is an injective homomorphism for the convolution product (Proposition

IV.9.4).

Let N<> be the stability group of p for the extended group G<>:=G*MA, as defined in Section

IV.14. From (IV.14.1.a) one sees that N<>=”(man,ma)»man∑MAN’. So we can and will view the action

of N<> as a diagonal action of B=MAN, which we denote by ÿ. From (IV.14.0.b) it follows that the

explicit form for this diagonal representation of B is

ÿmanU=†manU*∂pm¡a¡ m∑M, a∑A, n∑N, U∑∂æ(≈).

MAp being left MAN– and right MA–invariant, N<> operates on ∂æMAp(≈). In view of (III.4.4.b) its

action can be expressed as

(V.3.7.a) ÿman“¨^*S‘=“(Adman¨)^*LmaS‘*^*∂m¡a¡=(Adman¨)^*LmaRmaS

=(Adman¨)^*LmRmS m∑M, a∑A, n∑N, ¨∑Ë(g), S∑∂æ(MA),

where L and R denote respectively the left and right regular representations of MA in the distributions

on MA.

In view of Theorem III.4.1 we have the isomorphism

(V.3.7.b) ∂æMAp(≈)=Ë(g)Ë

¤(Àb)∂æ(MA).

Let ñ be the dimension of nä. For every negative root –å choose a basis of the corresponding root

space g–å. Unite these bases to form a basis Y¡,…,Yñ for n_ . Use Corollary III.1.6 to see that in terms

of the isomorphism (V.3.7.b) every U∑∂æMAp(≈) allows a decomposition (unique with respect to the

basis chosen)

157


(V.3.7.c) U=µ¶∑ˆ

ñY¡

µ¡Y™µ™ÚYñ

µñË

¤(b)

Sµ

with the (Sµ)µ∑ˆñ a locally finite collection in ∂æ(MA). Assume that Yi belongs to g–åi (some of the

–åi may coincide, depending on the multiplicities of the –åi, but that does not matter for the present

argument). It follows that Ada(Y¡µ¡Y™

µ™ÚYñµñ)=a–µ¡å¡–µ™å™–...–µñåñ.Y ¡

µ¡Y™µ™ÚYñ

µñ, for a∑A. Restrict

the diagonal representation ÿ to A. In view of (V.3.7.a) and Proposition III.4.4 one obtains the

following expression:

ÿaT=µ¶∑ˆ

ñY ¡

µ¡Y™µ™ÚYñ

µñË

¤(b)

a–µ¡å¡–µ™å™–...–µñåñSµ a∑A.

The –åi being negative roots, the weight –µ¡å¡–µ™å™–...–µñåñ will be zero if and only if µ=0.

From the uniqueness of decomposition (V.3.7.c) it follows that U is ÿA–invariant if and

only if Sµ=0 for all µ≠0. So it must be of transversal order 0 !

We end up by discussing some easy consequences of Theorem V.3.1 for the space G/N acted on

bilaterally by G<>=G*MA.

Proposition V.3.8 Consider ≈=G/N as acted upon bilaterally by the extended group

G<>=G*MA. Then the algebra å <> of bilaterally invariant convolution operators in ∂æ(≈) is

isomorphic to the convolution algebra of central compactly supported distributions on MA. The

isomorphism is the restriction of the isomorphism in Theorem V.3.1.

Proof This is straightforward by now. One has å<>=´æ(≈)N<>, on the basis of Corollary V.2.7 and

Proposition IV.5.2. Moreover, Theorem V.3.1 yields that ´æ(≈)N<>=´æMAp(≈)

N<>=·^*´æ(MA)‚N<>.

This space equals ^*“·´æ(MA)‚central‘, according to the remarks in the first paragraph of the proof of

Proposition V.3.7!

Recall that for a weakly symmetric homogeneous space ≈ the centre Ω is the algebra of those

convolution operators ∂(≈)æîâ∂æ(≈) that commute with all convolution operators ∂(≈)îâ∂æ(≈)

(Proposition IV.14.5). Let Ω<> denote the centre of ≈ as a homogeneous space under the bilateral

action of G< >, so the algebra of those bilaterally invariant convolution operators ∂(≈)æîâ∂æ(≈) that

commute with all bilaterally invariant convolution operators ∂(≈)îâ∂æ(≈). In general one has

158


(V.3.8.a) Ω≤Ω<> ≤å<>≤å.

Proposition V.3.9 For ≈=G/N one has Ω<> =å<>.

This is equivalent to saying that the that convolution product of strongly zonal distributions is

commutative, that is U*V=V*U for U and V strongly zonal, and with U or V having compact support.

It is therefore not so unreasonable to suppose that the bilateral action of G*MA in the distributions is

multiplicity free. But things are not that simple, see the discussion following (V.5.5.a), where

Proposition V.3.9 is used.

Proof Take a convolution operator a∑å. According to Theorem V.3.1 its propagator E=a∂p is of the

form ^*S, S∑´æ(MA). By Definition IV.13.4 this implies that E*U=U*E for every strongly zonal

distribution U. This means that every convolution operator a:∂æ(G) îâ ∂æ(G) commutes with every

bilaterally invariant convolution operator u:∂(≈) îâ ∂æ(≈). Assume that the convolution operator a

is also bi-laterally invariant to see that å<>≤Ω<> . This will do, in view of (V.3.8.a) !

V.4 Fundamental Solutions

Since all G–invariant operators on ≈=G/N are induced from MA, Section IV.10 now applies to the

whole algebra å of G–invariant operators in ∂æ(G/N). This means that problems concerning the

existence of fundamental solutions on G/N can be reduced entirely to MA. This section makes

explicit how this is done, and explains some relevant results from the literature.

Let åº be the algebra of left MA–invariant operators in ∂æ(MA). In view of Theorem V.3.1 the

algebras å and åº are isomorphic. More precisely (according to Proposition IV.10.1), this

isomorphism is determined by the intertwining relation

(V.4.0.a) u^*=^*uº u∑å, uº∑åº,

where ^* is the push-forward ∂æ(MA) êêâ ∂æ(G/N) under the imbedding ^(©)=©p, ©∑MA.

159


Theorem V.4.1 Fundamental Solutions on G/N; Reduction to MA.

Let u be a convolution operator ∂æ(G/N) êêâ ∂æ(G/N).

Then u is induced from MA, and

i) u has a fundamental solution in ∂æ(G/N) if and only if its inducing operator uº has a

fundamental solution in ∂æ(MA).

ii) If u has a fundamental solution in ∂æ(G/N), it also has one that is zonal.

iii) If u is a bilaterally invariant convolution operator, then if it has a fundamental solution, it

also has one that is strongly zonal.

Reminders A bilaterally invariant convolution operator on G/N is one that is invariant for the group

G <>=G*MA acting bilaterally. Strongly zonal is invariant for the stability group N<>:=G<>p, so the

minimal parabolic subgroup B=MAN acting diagonally through (man,x) éêâ manxm¡a¡.

Corollary V.4.2 When M is discrete (i.e.: finite), every non-0 invariant differential operator on

G/N has a strongly zonal fundamental solution.

Proof of the Theorem i) and ii) are a rendering of Theorem IV.10.2 in this specific situation, with

Theorem V.3.1 in mind. To prove iii), note that when u is bilaterally invariant invariant, uº will be

bi–invariant (compare Proposition V.3.7). When u has a fundamental solution, uº has one too

(according to (i)). But uº being bi-invariant, and MA being what it is (the direct product of a compact

group and, essentially, a Euclidean space), a fundamental solution of a bi-invariant operator can be

integrated over M to yield a central fundamental solution E (central in ∂æ(MA), that is). But then ^*E is

a strongly zonal fundamental solution for u in ∂æ(G/N) !

Proof of Corollary V.4.2 When u is a differential operator, the propagator of the inducing operator uºis a distribution Uº concentrated in eMA (the unit of MA). M being discrete, this can be seen as a

distribution Uºº concentrated in eA. Let uºº be the operator in ∂æ(A) of convolution by Uºº. Then uººcan be seen as a differential operator with constant coefficients on Ân, and so, according to the

Theorem of Malgrange-Ehrenpreis, it has a fundamental solution in ∂æ(A) [96] . Since M is discrete,

96 Lars Hörmander, The Amalysis of Partial Differential Operators; Part I: Distribution Theory and Fourier

Analysis (Berlin: Springer Verlag, 1983).

160

— V.4 Fundamental Solutions —

A is the connected component of MA, so Eºº can be extended trivially (i.e., by 0) to a fundamental

solution for uº in ∂æ(MA) !

As to other interesting convolution operators, consider difference-differential operators, so those

whose propagators have finite supports. Every (non-0) invariant difference-differential operator uû on

A is known to have a fundamental solution (since A=Ân, see the first paragraph of Section IV.10).

When M is discrete one would expect the same conclusion, but this is not in general true when the

support of the propagator of uû is not contained in A. The following simple argument gives a precise

criterion.

When M is discrete consider a distribution U on MA as a function m éêâ Um on M with

values in the distributions on A, so U=m

¶∑M

∂m¤Um. With these conventions the convolution product

on MA can be expressed as a two-step convolution product:

(E * U)k=m

¶∑M

Em*Um¡k E, U ∑ ´æ(MA), k∑M.

This implies

(V.4.2.a) (E * U)n¡k=m

¶∑M

En¡m*Um¡k k, n ∑M.

For E a distribution on MA let Eõ be the convolution kernel Eõ(n,k)=En¡k on M*M. Then (V.4.2.a) reads

as

(V.4.2.b) (E*U)õ= Eõ*Uõ,

where the right-hand side should be interpreted as a matrix product. By convolution U propagates a

left MA–invariant operator uû on MA, and E will be a fundamental solution for uû if and only if Eõ*Uõ

equals the identity, that is, the element I¤∂eA. This will happen if and only if and only if the operator

propagated by Det(Uõ) has a fundamental solution. Indeed, this condition is necessary, because

Eõ* Uõ=I¤∂eA implies that Det(Eõ)*Det( Uõ)=Det(I¤∂eA)=∂eA. Moreover, let co(Uõ) denotes the

transposed cofactor matrix of Uõ, so that

Uõ*co(Uõ)=co(Uõ) * Uõ =Det(Uõ)*·I¤∂eA‚.

Then if F∑∂æ(A) satisfies F*Det(Uõ)=∂eA, a solution for Eõ * Uõ =I¤∂eA is given by Eõ=F*co(Uõ).

For a map Ï defined on M*M with values in an abelian algebra å define DetÏ in the

obvious way, that is, as the element in å with formula ß∑P¶erm(M) (–1)|ß|

m•∑M

Ï(m,ß(m)). When uûis a convolution operator with propagator U=m

¶∑M∂m¤Um, Um∑´æ(A), let Uõ:M*M ììâ´æ(A) be the

161


convolution kernel Uõ(n,m)=Un¡m, n,m∑M. Let Det(uû) denote the operator of convolution by

Det( Uõ).

With these notations one we have the following.

Proposition V.4.3 M discrete; reduction to A

Let M be discrete. Let uû be a left invariant operator in the distributions on MA.Then uû has a

fundamental solution in ∂æ(MA) if and only if Det(uû) has a fundamental solution in ∂æ(A).

In particular, a difference-differential operator on uû on MA has a fundamental solution

if and only if Det(uû)≠0.

Later on we prove a similar result in a concrete context, using another approach (Corollary VI.3.3).

As to the existence of fundamental solutions for differential operators on MA for general M (no longer

discrete), precise criteria (in the form of growth estimates on Fourier coefficients) have been obtained

by Cerezo and Rouvière [97] . Their results being so directly relevant, we give a brief summary in a

way that fits in with the notation as used in the introduction to the proof of Theorem V.3.1.

For the estimates, one first constructs a series of bi-invariant differential operators on M of

increasing order. Being reductive, the Lie algebra m is the sum of its centre and its semi-simple

derived algebra [m,m]. Choose an orthonormal basis of [m,m] (orthonormal with respect its Killing

form), and extend this to a basis X¡,ÚXm of m. Set D:=i¶=

m

1 Xi™, a bi-invariant differential operator on M.

For k∑ˆ set

Dk:=0¯

¶q¯k

(-D)q

Being bi-invariant, Dk operates by scalars on the traces Ô≈, ≈∑Mfl, say DkÔ≈=dk(≈)Ô≈. The dk(≈) are

strictly positive numbers, and increasing in k (for ≈ fixed). They function as a kind of basic

polynomials in ≈, of degree k. For S a continuous function on M its Fourier transform is defined as a

field of Fourier coefficients Sfl(≈) with Sfl(≈)∑¬(Ó≈), the space of linear operators on the representation

space Ó≈, equipped with the Hilbert-Schmidt inner product (A»B)HS=trace(B*A) [98] . The

97 André Cerezo and François Rouvière, “Solution Élémentaire d’un Opérateur Différentiel Linéaire Invariant

a Gauche Sur un Groupe de Lie Réel Compact et Sur un Espace Homogène Réductif Compact,” Ann. Scient.

Éc. Norm. Sup. 4th series, part 2 (1969), pp. 561-581. In the paper the compact group is assumed to be

connected. The results that we need are just as much valid when M is non-connected. See note to page 562 in

the paper.

162

— V.4 Fundamental Solutions —

coefficient Sfl(≈) is obtained by integrating ªMdm. S(m)≈m¡. As remarked at (V.3.5.d) Fourier

transformation extends to the distributions. One shows that a field (a≈)≈∑MÀ is the Fourier transform of

a (uniquely determined) distribution if and only if it is “of no more than polynomial growth”, in the

sense that

‰k∑ˆ ‰C>0 Å≈∑ Mfl »»a≈»»¯Cdk(≈).

It follows from this that a left invariant differential operator u¡ on M has a fundamental solution in

∂æ(M) if and only if the Fourier coefficients of its propagator u¡∂ are invertible, and satisfy

(V.4.3.a) ‰k∑ˆ ‰C>0 Å≈∑ Mfl »»“ u fl¡∂(≈)‘¡»» ¯ Cdk(≈).

That all the Fourier coefficients should be invertible (equivalently, that the differential operator

should be injective) is an obvious condition. In the simplest case of a one-torus (i.e., the circle)

invertibility of the Fourier coefficients implies (V.4.3.a), but already on the two-torus there are

examples of first order differential operators where all the coefficients are invertible (non-zero, that is,

the group being abelian), but where this growth condition is not fulfilled. This is not obvious, and the

clue is an arithmetic subtlety. A first order polynomial like (p–åq+2), p,q∑Û™, with å irrational,

though nowhere vanishing, will still approach 0 on certain sequences in Û™. Moreover, depending on

å, the rate of this approach may on certain sequences be faster than any negative power of 1+p™+q™.

For such å the differential operator that has p–åq+2 for its Fourier polynomial has therefore no

fundamental solution, at least not in the distributions [99].

Next consider compactly supported distributions on the product M*A as compactly

supported distributions on A with values in the distributions on M. It follows that the Fourier

transform of a distribution U∑ æ(MA) can be seen as a function on the dual Afl=aæÇ with values in the

fields in ·¬(Ó≈)‚≈∑Mfl. More precisely, let Uû denote the element in ∂æ(A;∂æ(M)) corresponding to U,

defined by <<Uû,¥>,ƒ>=<U,ƒ¤¥>, ƒ∑∂(M), ¥∑´(A), then flU(≈,~)=< U

û,a–~>À(≈), ≈∑Mfl, ~∑aæÇ . So for

fixed ≈ one has a function Ufl≈ on aæÇ with values in ¬(Ó≈). This is implicit in (V.3.5.f).

When U is the propagator uº.∂eMA of a left invariant differential operator uº on MA, the

maps Ufl≈ are polynomial maps with values in ¬(Ó≈). They are described succinctly by

Ufl≈(~)=(uº(≈¤a~))(eMA), ≈∑ flM, ~∑aæÇ .

For a polynomial map P on aæÇ define the norm ||P||=”(9å¶∑ˆr» îîîÿÿ

åî~îå

P(0)»™0)»

™. Similarly, for a polynomial

98 ¬(Ó≈)= Ó… … ≈¤Ó≈, the form we used at (V.3.5.a).99 André Cerezo and François Rouvière, page 570.

163


map P≈ on aæÇ with values in Ó≈ define ||P≈||=”(9å¶∑ˆr»» îîîÿÿ

åî~îå

P≈(0)»»™Ó≈0) »

™. Let DetP≈ denote the

Determinant of P≈, a polynomial on aæÇ . Finally, let coP≈ denote the transposed cofactor matrix of

P≈, so that P≈.coP≈=coP≈.P≈=Det(P≈).Id≈.

Theorem V.4.4 Cerezo-Rouvière

Let uº be a left invariant differential operator on MA, with propagator U. Then uº has a

fundamental solution in ∂æ(MA) if and only if for every ≈∑Mfl the polynomial Det( flU≈) is not

identically 0, and

‰k∑ˆ ‰C>0 Å≈∑ Mfl

»î »î»»î

D

c î

e

oî

t

Uîfl

flUîî≈î

≈îî»»

»

» ¯Cdk(≈).

Finally a somewhat similar result concerning differential operators on the group G. By means of a

(generalized) Radon transform François Rouvière has succeeded in reducing bi–invariant differential

equations on the group G to bi-invariant differential equations on MA, under the assumption that G

have only one conjugacy class of Cartan subalgebras [100] . To compare that result, let z be a bi-

invariant differential operator on G. Being in the centre Z(g) of the universal enveloping algebra, it

acts as a bilaterally invariant differential operator õz on G/N. Proposition V.3.8 implies that õz is

induced from a bi-invariant operator zº in MA. This yields an algebra homomorphism from Z(g) to

Z(m@a). It is actually into: this is because Z(g)§Ë(g)n=(0), so that the map z éìâzº is one-to-one

[101] . The homomorphism thus arising coincides with the map † introduced by Rouvière, defined as

the restriction to Z(g) of the projection onto the first summand in the direct sum decomposition

Ë(g)=Ë(m@a) @ Ë(m@a@_n)_n @ Ë(g)n [102]

Rouvière’s result is that (again, supposing G has only one conjugacy class of Cartan subalgebras) the

existence of a fundamental solution for zº ensures the existence of a central fundamental solution for

z. This result suggest a relationship between the existence of fundamental solutions on G/N and the

existence of fundamental solutions on G, but it is not clear how the actual solutions themselves should

be related.

100 François Rouvière, “Invariant Differential Equations on Certain Semi-Simple Lie Groups” Transactions

of the American Mathematical Society Vol. 243, September 1978, pp. 97-114, Theorem IV.2101 Garth Warner, Harmonic Analysis on Semi-Simple Lie Groups, Lemma 2.3.3.5. The kernel of the map

éêâ ¨∂p is Ë(g)n, see our Example III.4.2.ii.102 François Rouvière, “Invariant Differential Equations on Certain Semi-Simple Lie Groups,” Lemma 2.3.

164

V.5 Plancherel Formulas;

(G,N) a Generalized Gelfand Pair ?

In the preceding sections some questions in Harmonic Analysis on G/N have been reduced to similar

questions on MA. In the present section we induce Hilbert subspaces of ∂æ(G/N) from Hilbert

subspaces of ∂æ(MA). This is based on Sections IV.11-13. We then deal with the question whether the

representation of G in G/N is multiplicity free.

The modular functions introduced in Section IV.11 take the following form. One calculates that the

modular function of the right action of MA on G/N is given by ®µMA(ma)=a™®, m∑M, a∑A (use

(IV.11.0.b), and µ=dk.a™®da). Since MA is unimodular it follows that ®MA=1, and çµ(ma)=a™®,

m∑M, a∑A. According to Section IV.12 a left MA–invariant Hilbert subspace Ó of ∂æ(MA) give rise to

a G–invariant Hilbert subspace Óind of ∂æ(G/N). To obtain an explicit expression for the latter space

one can choose a smooth strictly positive measure on ≈/MA and then calculate the associated

modular function ç~µ, determined by (IV.11.1.a). Since ≈/MA=G/B=K/M an obvious choice is the

uniquely determined K–invariant measure dk$ with total mass 1. It follows from the expression

(V.5.0.a) ª≈Ï(x)dx=ªK/M

dk$ªMA

Ï(kmap)a™®dmda Ï∑∂(≈),

that ç~µ(kap)=a™®, k∑K, a∑A. Denote the function “ç~

µ(x)‘™ by x®, so (kap)®=a®. For a smooth

function on a smooth manifold let µå denote the operator in the distributions of multiplication by å.

The expressions in Theorem IV.12.1 then turn out as follows:

(V.5.0.b) Óind= µx®ª@

K/Mdk$†k^*Ó

=ª@

K/M dk$†k^*µa®Ó

for Ó a left MA–invariant Hilbert subspace of ∂æ(MA). Equally, however, (V.1.0.b) can be used to

derive the expressions

(V.5.0.c) Óind= µ ∫ ª@

Nädn_† n_^*Ó

=ª@

Nädn_† n_^*µa®Ó

where ∫ denotes the function ∫(n_map)=a®.

165


Use the Plancherel formula for the group MA, as described in the proof of Theorem V.3.1. That is,

choose Lebesgue measure d~ on iaæ so that

(V.5.0.d) ∂eMA= ≈¶∑MÀ

ªiaæ

d~ d≈ Ô≈~ (compare (V.3.5.h)).

The corresponding direct integral decomposition of L™(MA) in Hilb˚MA*MA(∂æ(MA)) is then

(V.5.0.e) L™(MA)=≈¶∑MÀ

@ ª

@

iaæd~ ˜≈~ ,

where ˜≈~ equals the space ˜≈¤“a~‘ of matrix coefficients of the representation ≈¤a~.

Decomposition (V.5.0.e) is a direct integral decomposition by means of minimal bi-invariant Hilbert

subspaces of ∂æ(MA). According to Corollary IV.13.7 this implies that

(V.5.0.f) L™(≈;µ)=≈¶∑MÀ

@ d≈ ª

@

iaæd~ “˜≈

~‘ind

is a direct integral decomposition of L™(≈;µ) by means of bilaterally invariant Hilbert subspaces of

∂æ(G/N).

There are a number of ways of obtaining a more explicit description of “˜≈~‘ind. One way

is to work out the integral (IV.11.2.a). We use the following argument, which requires no

calculations, and has some use in Chapter VII.

The reproducing distribution of “˜≈~‘ind is ^*d≈Ô≈

~+®, and this is a strongly zonal

distribution (Definition IV.13.4). According to Proposition IV.14.2 a strongly zonal distribution is

one that is invariant for N<>, the stability group of p in the extended group G<>=G*MA. According to

Proposition IV.14.1 this means that the convolution operator that ^*d≈Ô≈~+® propagates can be

determined by considering G/N as the quotient G<>/N<>. But by the same token, the convolution

product can be determined by considering G/N as a homogeneous space for any closed subgroup

G¡<G<>, as long as this group operates transitively on ≈=G/N. A particularly suitable subgroup is

G¡:=K*A≤G*MA (so, with A operating from the right). Its action on G/N allows a very simple

expression in the KAp coordinates. That is,

(V.5.0.g) (k,a)(kºaºp)=k(kºaº)pa¡ = (kkº)(aºa¡)p.

The result is that the bilaterally invariant convolution operator propagated by a strongly zonal

distribution can be calculated as if G/N were the group K*A. Explicitly, on the space of strongly

zonal distributions the left translation †ka equals Lk¤La, k∑K, a∑A, with Lk and La denoting the left

166

— V.5 Plancherel Formulas; (G,N) a Generalized Gelfand Pair ? —

regular translations through k and a in ∂æ(K) and in ∂æ(A) respectively. This almost settles it, except

that the G–invariant measure dk.a™®da on G/N is not the Haar measure for the K*A–group structure.

The result is that the G–invariant operator u:∂(≈)îîâ∂æ(≈) propagated by U is

uÏ=(Ï.(dk.a™®.da))*U.

Taking this into account, one obtains that the kernel associated to U is determined by

(V.5.0.h) <uÏ»Á>=<((a®Ï).(dkda))*(a–®U)»a®Á> .

This yields

Proposition V.5.1 Positivity of Strongly Zonal Distributions

A strongly zonal distribution U on G/N is of positive type if and only if a–®.U is a distribution of

positive type as a distribution on the group K*A. The same is true for a zonal distribution U merely

satisfying

(V.5.1.a) (^*S)*U=U*(^*S) ÅS∑´æ(A).

Since (V.5.1.a) will be true in particular for every distribution U induced from MA, Proposition V.5.1

implies and improves on Proposition IV.11.1 (for this particular space).

The second statement in the proposition, stronger than the first, can be proved in a similar

fashion, using only that AN normalizes N. A consequence of (V.5.0.h) is that when U is a strongly

zonal distribution of positive type satisfying (V.5.1.a), then the Hilbert subspace of ∂æ(G/N)

reproduced by U is the image under multiplication by a® of the Hilbert subspace of ∂æ(K*A)

reproduced by a–®U. As a result, the Hilbert subspace “ ≈~‘ind of ∂æ(G/N), reproduced by

^*d≈Ô≈~+®= a®^*(d≈Ô≈

¤a~), is

(V.5.1.b) “˜≈~‘ind= “˜≈‘ind¤“a~+®da‘ in ≈=KAp coordinates.

Here “˜≈‘ind is the Hilbert subspace of ∂æ(K) induced by the Hilbert subspace ˜≈ of ∂æ(M) spanned

by the matrix coefficients of ≈. That is, if j is the (formal) imbedding M≤K, then “˜≈‘ind equals

(V.5.1.c) “˜≈‘ind=ª@

K/M dk$ †kj*˜≈ in Hilb K*M(∂æ(K)).

Using Proposition V.5.1 one shows that more in general for ˚ and Ó left invariant Hilbert subspaces

of ∂æ(M) and ∂æ(A) respectively one has:

167


(V.5.1.d) “˚¤À™Ó‘ind= ˚ind¤À™µa®Ó.

It is well-known that the Plancherel decomposition for L™(M;dm):

L™(M;dm)=≈¶∑MÀ ˜≈

can be seen as the isotypical decomposition for the left regular representation of M in L™(M;dm). Each

˜≈ allows further direct sum decompositions

(V.5.1.e) ˜≈ =k

d¶=

≈

1@

˜(≈k)

into minimally left invariant Hilbert subspaces of ∂æ(M), with each ˜(≈k) equivalent to _≈. A way of

expressing this is by writing ˜≈=d≈(Ó≈_¤Ó≈), with decomposition (V.5.1.e) corresponding to a

choice of basis in Ó≈. Decomposition (V.5.1.e) is highly non-unique, except, of course, in the trivial

case d≈=1.

By means of (V.5.1.d) and (V.5.1.e) one shows that when the bilaterally invariant Hilbert

subspace “ ≈~‘ind is considered as a (left) G–invariant Hilbert subspace of ∂æ(≈) it allows a further

decomposition

(V.5.1.f) “˜≈~‘ind=k

d¶=

≈

1@

·˜(≈k)¤“a~da‘‚ind

=k

d¶=

≈

1@

“˜(≈k)‘ind¤“a~+®da‘.

Since for fixed ≈,~ the spaces ˜(≈k)¤“a~da‘, k=1,...,d≈, are equivalent as left representation spaces for

MA, it follows that the induced spaces are also equivalent as G–representation spaces.

We still assume ~ is purely imaginary, that is, that a~ is unitary. It is not difficult to show

(for instance by using (V.5.1.f)) that as a G–representation space each of the · (≈k)¤“a~da‘‚ind is

equivalent to a unitary principal series representation of G. More precisely, · (≈k)¤“a~da‘‚ind is

equivalent to the parabolically induced representation

π_≈,–~=MAInNdàG _≈¤a–~¤1.

We mention this fact, without defining parabolic induction, a subject on which there is a vast

literature. See, e.g., Warner [ 103] . We cannot, however, avoid using some known results in this field

to draw conclusions on (V.5.0.f).

103 Garth Warner, Harmonic Analysis on Semi-Simple Lie Groups,Vol I, Chapter V

168


In the first place, it is well known that π_≈,–~ is generically irreducible. This can be made explicit in

terms of the action of the Weyl group on the representations of MA. That is, the Weyl group acts by

automorphisms on MA, and so by composition it acts on the representations of MA, say by

(wR)(©)=R(©w), w∑W, ©∑MA, R a representation of MA. It is a result of Bruhat’s theory of sesquilinear

pairings that the G–representation induced by ≈¤a~ can only be reducible (though it may well happen

that it is not) when there exists a w∑W, w≠1, such that the representation ≈¤a~ is equivalent to

w(≈¤a~) [104] . This can happen only for (≈,~) on a set of measure 0 in Mfl¤iaæ (measure 0 in terms

of the product of counting measure and Lebesgue measure). And so (V.5.0.f) and (V.5.1.f) provide a

way of giving a direct integral decomposition of L™(≈;µ) in HilbG (∂æ(≈)):

(V.5.1.g) L™(≈;µ)=≈¶∑MÀ

@ d≈ ª

@

iaæd~k

d¶=

≈

1@

“˜(≈k)‘ind¤“a~+®da‘

with almost all “(≈k)‘ind¤“a~+®da‘ irreducible. This result (in various shapes) has been derived by

N. Wallach, by G. van Dijk, and by Mannes Poel [ 105] . It can be seen as a refinement of the

bilateral decomposition (V.5.0.f), the refinement taking place after restriction of a representation to a

smaller group. Only decomposition (V.5.0.f) is canonical (see Theorem V.5.3).

One sees immediately that when M is not abelian, some of its representations are of

dimension more than one. But this implies that the representation of G is not multiplicity free, the

“˜(≈k)‘ind¤“a~+®da‘ being equivalent when k varies, and ≈ and ~ are fixed. This is in keeping with

Theorem V.3.1. That theorem (or rather its corollary) implies that when M is non-abelian, the algebra

å of G–invariant operators in the distributions is non-abelian as well. For (G,N) to be a Generalized

Gelfand Pair å would have had to be abelian, essentially because in a multiplicity free situation åoperates by scalars on every irreducible Hilbert subspace (Corollary IV.14.10)

However, even when there is no such obvious obstruction, so when M is abelian, there is

multiplicity still. This is because of a second result in Bruhat theory: when ≈¡¤a~¡ is equivalent to

104 F. Bruhat, “Sur les Représentations Induites des Groupes de Lie,” Bull. Soc. Math. France 84 (1956), pp.

97-205.105 G. van Dijk, “A Plancherel Formula for the Isotropic Cone” Proceedings of the Koninklijke Nederlandse

Akademie van Wetenschappen Series A, Vol . 91 (March 28, 1988) no. 1. This paper in fact concerns a

generalization of a Plancherel formula for the cone, which is G/MN rather than G/N, but the paper includes a

formula for the latter space.

Nolan R. Wallach, Real Reductive Groups II (New York: Academic Press, 1992), Pure and Applied

Mathematics Series Vol. 132-II, Theorem 15.1.3.

Mannes Poel, unpublished work..

169


w(≈™¤a~™) for some w∑W, the representations induced by ≈¡¤a~¡ and w(≈™¤a~™) respectively are

unitarily equivalent. As Wallach points out, this means that at the regular points the multiplicity in

(V.5.1.g) is described by (≈,~) éêâ d≈ »W», with »W» denoting the order of the Weyl group.

The result is that

Theorem V.5.2

(G,N) is not a Generalized Gelfand Pair.

Formula (V.5.1.g) is nevertheless a Plancherel formula, in the sense that the integrand is reducible

only on a set of measure 0.

As such there is nothing remarkable about this. As argued in Section IV.14 similar things happen

when one considers the left regular representation in the distributions of a non-abelian unimodular

group. The noticeable thing is rather that when M is abelian one has here an example of a pair (G,H),

H=N, that fails to be a Generalized Gelfand Pair in spite of the fact that the algebra of invariant

operators in the distributions is abelian. (Note that N is not compact. Otherwise this phenomenon

could not occur, because when H is compact the commutativeness of å guarantees that (G,H) is a

Generalized Gelfand Pair. This is shown in [106]).

One way of dealing with the non-uniqueness of decomposition (V.5.1.g) is to group

equivalent representations together, and to integrate over a suitable set of representatives of the

W–orbits in MÀ*iaæ. This is done by Wallach [107]. We prefer the approach of M. Poel, that is, to

consider the bilateral action of G*Ì, with Ì the group of G–invariant diffeomorphisms, Ì=MA. That

this is a ‘natural’ approach has been argued in Section IV.14. In this particular case this leads to the

following:

Theorem V.5.3 The direct integral decomposition in Hilb˚G*MA(∂æ(G/N)):

(V.5.3.a) L™(G/N;dx)=≈¶∑MÀ

@ d≈ ª

@

iaæd~ “˜≈

~‘ind

is the unique, multiplicity free, bilateral Plancherel decomposition.

106 E.G.F. Thomas, “ An Infinitesimal Characterization of Gelfand Pairs,” Contemporary Mathematics Vol

26: Conference in Modern Analysis and Probability, (Providence, Rhode Island: American Mathematical

Society, 1984), pp. 379-385.107 Nolan R. Wallach, Real Reductive Groups II, Section 15.1.4.

170


Proof That the decomposition is multiplicity free has already been argued in the proof of Corollary

IV.13.7, the point being that this decomposition is isotypical for the right action of Ì=MA. Every

“˜≈~‘ind=“˜≈‘ind¤“a~+®da‘ is the finite sum of spaces “

(≈k)‘ind¤“a~+®da‘. Under the right

action of M any of the left invariant spaces ˜(≈k) generates ˜≈

, and this implies that under the right

action of M any of the spaces “(≈k)‘ind¤“a~+®da‘ generates all of “ ≈

~‘ind. This implies that the

integrand in (V.5.3.a) is G*MA–irreducible whenever “˜(≈k)‘ind¤“a~+®da‘ is irreducible. This means

we can refer back to Theorem V.5.2 !

Theorem V.5.3 is closely related to Theorem V.3.1. The proof of the latter theorem can be described

as being based essentially on the determination of the Fourier coefficients with respect to

decomposition (V.5.3.a) of distributions with compact support. More precisely, the reproducing

operator of a Hilbert subspace Ó of ∂æ(≈), an operator from ∂(≈) into Ó, always allows an extension

as operator from ´æ(≈) to Ó–°. Therefore, the Plancherel decomposition (V.5.3.a) gives the

possibility of associating with every compactly supported distribution S a field of distributions

(S ≈~)≈∑MÀ,~∑ia, with S≈

~∑·“˜ ≈~‘ind‚–°, such that S=

≈¶∑MÀ

d≈ ªiaæ

d~ S≈~ . This is essentially what is

done in the proof of Theorem V.3.1.

In general, when a homogeneous space is weakly symmetric the existence of a multiplicity free

Plancherel decomposition L™(≈;dx)=ª@Ó¬ d¬ is enough to ensure that the algebra of convolution

operators in the distributions is abelian. Indeed, let u be a convolution operator in the distributions.

Then since the space is weakly symmetric, u maps ∂(≈) into itself (Corollary IV.5.3). Let uû be the

restriction of u to the maximal domain in L™(≈;dx), that is to say, the operator whose graph is the

intersection of the graph of u with L™(≈;dx)*L™(≈;;dx). Then uû is a densely defined closed operator

in L™(≈;dx) commuting with the action of G. Such an operator can be shown to disintegrate as

uû=ª@uû¬d¬ (see [108] ). Since two such operators both map ∂(≈) into itself, they commute on the

subdomain ∂(≈). But that of course means they commute on all of ∂æ(≈).

Therefore, G/N being weakly symmetric, and given the existence of the multiplicity free

bilateral Plancherel decomposition (V.5.3.a), it follows that the algebra of bilaterally invariant

convolution operators in the distributions is abelian. This means that Theorem V.5.3 is in agreement

with Proposition V.3.8.

108 Erik G. F. Thomas, “Symmetric Closed Operators Commuting with a Unitary Type I Representation of

Finite Multiplicity are Self-Adjoint,” Illinois Journal of Mathematics 36, (1992), nº 4, pp. 551-557.

171


However, the existence of a multiplicity free (bilaterally invariant) Plancherel decomposition does not

guarantee that the (bilateral) group representation in the distributions is multiplicity free. One might

well suppose, for example, that there exist invariant Hilbert subspaces that do not at all occur in the

Plancherel formula, and that some of these are equivalent. Or one might suppose that one of the

individual reducible members in the decomposition is not multiplicity free.

We first present cases where it takes only a simple argument to prove absence of

multiplicity. For the general case we then see how this question (whether the bilateral representation

of G*MA in ∂æ(G/N) is multiplicity free) can be reduced to a standard question on the unitary

principal series.

In general, a pair (G,H) is a Generalized Gelfand Pair if and only if the representation of G is

multiplicity free in every G–invariant Hilbert subspace Ó≤â∂æ(G/H), that is, if the bounded linear

operators in Ó that commute with the action of G form an abelian algebra. This is one of a number of

equivalent definitions of a Generalized Gelfand Pair referred to in Section IV.1. From this criterion it

is easy to see that for a pair (G,H) to be a Generalized Gelfand Pair it is enough for there to exist a

subgroup G¡<G such that (G¡,G¡§H) is a Generalized Gelfand Pair. Consequently, for the

representation of G*MA in ∂æ(G/N) to be multiplicity free it is sufficient for the representation of the

subgroup K*MA in ∂æ(G/N) to be multiplicity free. And this is sometimes easy to demonstrate,

because of the following.

The action of K*MA on G/N is very simple when expressed in the KAp coordinates. That is

(generalizing (V.5.0.g)):

(V.5.3.b) (k,ma)(kºaºp)=(kkºm¡)(aºa¡)p k, kº∑K, m∑M, a, aº∑A

So, further developing the argument at (V.5.0.g) we get that the representation of K*MA in ∂æ(G/N),

unitarized with respect to the measure dk.a™®.da, is equivalent to the regular representation of

(K*M)*A in ∂æ(K*A) with respect to the Haar measure dk.da (here K*M is understood to operate

bilaterally on K). In the case of G=SL(2;Â), the action of K*A is already multiplicity free, since K is

abelian. This simple argument yields:

Proposition V.5.4

For G=SL(2;Â) the bilateral action of the extended group G*MA in ∂æ(G/N) is multiplicity free.

172


More in general, the preceding argument shows that for the representation of G*MA in ∂æ(G/N) to be

multiplicity free it is enough for the bilateral representation of (K*M)*A on ∂æ(K*A) to be multiplicity

free. The latter statement is equivalent to the bilateral representation of K*M on ∂æ(K) being

multiplicity free [109] . One possibility of this occurring is the following.

For an irreducible representation ∂∑Kfl let ∂M denote the restriction to M.

Proposition V.5.5 Assume that for each ∂∑Kfl the restriction ∂M is a multiplicity free

representation of M. Then the bilateral action of G*MA in ∂æ(G/N) is multiplicity free.

Proof The condition in the proposition says that every irreducible representation ≈ of M occurs at

most once in the minimal decomposition of each ∂M. In view of the Frobenius reciprocity theorem for

the compact groups M and K, this implies that each ∂ occurs at most once in each IndKM(≈) (the

representation of K induced by ≈), so Ind KM(≈) is multiplicity free. The space [˜≈]ind induced by the

minimal bi-invariant Hilbert subspace ˜≈ of ∂æ(M) can be identified with the tensor product

[˜≈]ind =·Ó ≈_¤Ó≈‚ind =·Ind KM(≈_)‚¤·Ó≈‚

as tensor product representation for the bilateral action of K*M. When the K–representation in IndKM(≈_)

is multiplicity free, the representation of K*M in [˜≈]ind=·Ind KM(≈_)‚¤·Ó≈‚ is so too. Therefore each

K*M–irreducible representation ∂¤≈ will occur at most once in the decomposition

L™(K)=≈¶∑Mfl

[˜≈]ind =≈¶∑Mfl

·Ind KM(≈_)‚¤·Ó≈‚ !

To treat the general case, we first show that there are no bilaterally invariant Hilbert subspaces outside

the Plancherel formula, in the sense that all minimal bilaterally invariant Hilbert subspaces are

subrepresentations of induced representations.

To specify and corroborate the last statement, we use Proposition V.3.9 (so one of the

consequences of Theorem V.3.1). Consider ≈=G/N as homogeneous space under the action of

G<>:=G*MA. According to Proposition V.3.9 one has

109 One argument that shows this is as follows. In general, (G,K) for K compact is classically called a Gelfand

Pair when the convolution algebra of K–bi–invariant functions is abelian, and for K compact that definition is

equivalent to the one we use for Generalized Gelfand Pairs. From that definition it is easy to show that the

direct product of two classical Gelfand Pairs is a classical Gelfand Pair. That applies directly to the case of

(K*M)*A acting on K*A.

173


(V.5.5.a) au=ua Åa∑ å <>=LG<>(∂æ(≈)), u∑LG<>(∂(≈),∂æ(≈)),

an identity between convolution operators ∂(≈) ììâ ∂æ(≈). This equality makes sense, because G/N

is weakly symmetric, so that a(∂(≈))≤∂(≈) (see Corollary IV.5.3). Now apply Corollary IV.14.8,

with the extended group G*MA instead of the original group. One obtains that for every minimal

bilaterally invariant Hilbert subspace Ó there exists a Hermitian character ∆:å <>îêâÇ such that

(V.5.5.b) ah=∆(a).h a∑ å<>, h∑Ó.

From this it follows that there exist ≈∑MÀ and ~∑ia, such that Ó is contained in ´æ(K)≈¤“a~+®da‘,

where ´æ(K)≈ can be described as the closure of “˜≈‘ind in ∂æ(K).

To argue further, we use some of the language of ç°–vectors. In general, let † be a

(continuous) unitary representation of the Lie group G on a Hilbert space Ó. Let Ó° denote the space

of ç°–vectors associated to †, that is, the space of vectors v in Ó for which the map võ, võ(g)=†gv, is a

smooth map from G to Ó. Since võ is uniquely determined by v, the map v éêâ võ embeds Ó° into the

Fréchet space of all smooth maps from G to Ó, and the range of this map being closed, Ó° becomes a

Fréchet space by transport of topological structure. Moreover, Ó° is a dense G–invariant subspace of

Ó, and by restricting † to Ó° one obtains a ç°–representation †°, so that Ó° also becomes a

representation space for the universal enveloping algebra Ë(g). Let Ó–° denote the anti-dual of Ó°,

equipped with the strong topology. Its elements are called co–ç°–vectors. The conjugate

contragredient representation of G on Ó–° is denoted by †–°. This leads to the dense and

continuous inclusions

(V.5.5.c) Ó° ≤â Ó ≤â Ó–°,

the second inclusion arising by transposing the first, and with †° and †–° restricting and extending

†. For more on these matters, see for example [110] .

What we need at this point is that for Ó=“˜≈~‘ind one has the following realizations:

(V.5.5.d) Ó°=´(K)≈¤“a~+®da‘

Ó=L™(K)≈¤“a~+®da‘

Ó–°=´æ(K)≈¤“a~+®da‘,

110 Pierre Cartier, “Vecteurs Différentiables dans les Représentations Unitaires des Groupes de Lie,”

Séminaire Bourbaki, 27e année, 1974/75, nû 454.

174


with anti-duality given by the G–invariant sesquilinear pairing

<S¤a~+®da»ƒ¤a~+®da>=<S»ƒ>, S∑´æ(K)≈, ƒ∑´(K)≈.

The G–invariant pairing between ´(K)≈¤“a~+®da‘ and ´æ(K)≈¤“a~+®da‘ is a particular case of a

general sesquilinear pairing occurring in the context of induction of (in this case finite dimensional)

unitary representations (for more details, see [111] ). Furthermore, when Ó is a unitarily induced

representation, then Ó° is contained in the space of ç°–functions in Ó, so in ´(K)≈¤“a~+®da‘, and

it is actually equal to ´(K)≈¤“a~+®da‘ because K/M is compact (this is based on a general theorem

by Poulsen [112] ). One may note that ^*Ô≈~+®, by our definition the reproducing distribution of

“˜≈~‘ind, is zonal and belongs to ´æ(K)≈¤“a~+®da‘, so it is an N–fixed co-ç°–vector associated to

“˜≈~‘ind. We will return to this in Sections VII.6-8.

It follows from (V.5.5.b) that

Proposition V.5.6 For every minimal bilaterally invariant Hilbert subspace of ∂æ(G/N)

there exist ≈∑MÀ and ~∑ia such that is a Hilbert subspace of ·“˜ ≈~‘ind‚–°.

We then use the following lemma, in addition to some general theory.

Lemma V.5.7 Let Ó° ≤â Ó ≤â Ó–° be as at (V.5.5.c).

Let Ó have a finite decomposition:

Ó=i¶=

n

1@

Ói, n<°, Ói irreducible and inequivalent.

Then any G–invariant Hilbert subspace of Ó–° is of the form

˚=i¶=

n

1@

¬iÓi,

for certain positive numbers ¬i, i=1,Ú,n.

Corollary V.5.8 Assume that Ó is the sum

(V.5.8.a) Ó=i¶=

n

1@

Ói

111 Garth Warner, Harmonic Analysis on Semi-Simple Lie Groups I, Chapter 5.112 Niels Skovhus Poulsen, “On ç°–Vectors and Intertwining Bilinear Forms for Representations of Lie

Groups,” Journal of Functional Analysis 9 (1972), pp. 87-120, Theorem 5.1.

175


with the Ói irreducible.

Then the representation †–° in Ó–° is multiplicity free if and only if (V.5.8.a) is

multiplicity free.

The lemma is quite what one would expect, but does require a proof. To see that it is not enough that

Ó is dense in Ó–° consider a homogeneous space ≈ with invariant measure dx such that L™(≈;dx) is

irreducible. (Such spaces exist, an example is Ân acted on by the semi-direct product of SL(n;Â) and

Ân). Then although L™(≈;dx) is dense in ∂æ(≈), there still exists the one-dimensional Hilbert subspace

[dx] of ∂æ(≈), which is not a multiple of L™(≈;dx). This example does not contradict Lemma V.5.7,

since dx does not belong to Ó–°. Roughly speaking, Lemma V.5.7 says that Ó–° fits Ó like a glove,

leaving no room for any G–invariant Hilbert subspaces other than the obvious ones.

Lemma V.5.7 is easily proved, using some well-known facts. One fact we use twice is that

if Ó is an irreducible Hilbert representation space, then the positive multiples of Ó are the only

G–invariant Hilbert subspaces of Ó. The reason is that due to Schur’s Lemma reproducing operators

of Hilbert subspaces of Ó must be trivial.

Proof First take n=1, so Ó is irreducible. Assume that ˚≠(0) is a Hilbert subspace of Ó–°. For a

testfunction ƒ consider the image of ˚ under the operator πƒ–°:=ª

Gƒ(g)π g–°dg. Choose ƒ so that

πƒ–°˚ is not (0). (This is always possible, otherwise varying ƒ to approximate ∂eG would show that

˚=(0)). It is well-known that the Gårding vectors πƒ°(v):=ª

Gƒ(g)πgvdg, ƒ∑∂(G), v∑V, belong to

Ó°. Since πƒ–° is the transpose of πƒõ

° it follows that πƒ–° maps Ó–° into Ó. So in particular

πƒ–°˚ is contained in Ó. On the other hand, ˚ is assumed π–°–invariant, and the restriction of

π–° to ˚ is a continuous representation. Therefore, πƒ–°˚ is also contained in ˚. This shows that

Ó§˚ is not 0. Now Ó§˚ is in its own right a Hilbert subspace of Ó–°, when equipped with the

norm ||T||™Ó§˚=||T||™Ó+||T|| ™˚ [113] . So one has the continuous inclusion (Ó§˚)≤âÓ, and Ó being

irreducible it follows that Ó§˚=¬Ó for some positive constant ¬. But this implies that Ó is contained

in ˚, that is, Ó≤â˚≤âÓ–°, both inclusions being continuous in view of the closed graph theorem.

This in turn implies that ˚ must be irreducible: indeed, if ˚ allowed a G–invariant decomposition

˚=˚¡@˚™, with neither ˚¡ nor ˚™ equal to (0), the preceding argument involving ˚ would apply

113 A standard construction: see Laurent Schwartz, "Sous-espaces Hilbertiens d'Espaces Vectoriels

Topologiques et Noyaux Associés (Noyaux Reproduisants)," Jour. Anal. Math. 13 (1964), p. 138, Proof of

Proposition 3.

176


to both ˚¡ and to ˚™, leading to the absurd conclusion that Ó is contained in both ˚¡ and ˚™. But

now that ˚ is irreducible, and Ó is a G–invariant subspace of , it follows that ˚ is a positive

multiple of Ó.

Now assume n>1. Then Ó–°=i•n

=1Ói

–°. Let pi–° denote the G–equivariant projection

Ó–°îâÓi–°. When ˚ is contained in Ó–°=

i•n

=1Ói

–°, it follows that for each index i the

projection pi–°(˚) is a G–invariant Hilbert subspace of Ói

–°. So, by the preceding argument, for

each index i, the projection pi–°(˚) is contained in Ói. This implies that ˚ is contained in Ó. This is

the real point, because the reproducing operator of ˚ as subspace of Ó now belongs to the

commutant of G in Ó, and the commutant is well-known to consist of diagonal maps ¶xi éêâ ¶¬ixi(again, this follows from Schur’s Lemma) !

Comment There are other proofs possible. One can show that ˚ is associated with a separately

continuous G–invariant sesquilinear form on Ó°*Ó°, positive on the diagonal. According to results

by Poulsen [114] , in the case of irreducible Ó any separately continuous G–invariant sesquilinear

form on Ó°*Ó° must be a multiple of (the restriction of) the inner product of Ó. The first part of our

proof shows this fact by a simple argument, but only for positive forms.

Proof of the Corollary Assume that Ó=i¶=

n

1@

Ói is multiplicity free. Then if ˚ is an irreducible

Hilbert subspace of Ó–° the lemma implies that it must be a (positive) multiple of one of the Ói.

This implies that two irreducible Hilbert subspaces of Ó–° are either proportional or inequivalent,

which is saying that †–° is multiplicity free.

The reverse inclusion is trivial !

We continue the argument leading up to the lemma.

Under the the bilateral representation of G*MA each “˜≈~‘ind decomposes into a finite

number of G*MA–spaces. This follows from the known fact that each member of the unitary principal

series of G decomposes into a finite number of irreducible G–spaces. More precisely, the number of

these constituents cannot exceed the order of the Weyl group [115] . However, as far as we are aware

it is not known whether the constituents (for fixed (≈,~)) are always pairwise inequivalent. This seems

114 Niels Skovhus Poulsen, “On ç°–Vectors and Intertwining Bilinear Forms for Representations of Lie

Groups,” Theorem 3.4 and Corollaries.115 Garth Warner, Harmonic Analysis on Semi-Simple Lie Groups I, Corollary 5.5.2.2 and comment.

177


to be a very difficult question. The question whether this decomposition is multiplicity free is

equivalent to the analogous question for the “˜≈~‘ind.

Now let ˚¡ and ˚™ be two equivalent minimal bilaterally invariant Hilbert subspaces of

∂æ(≈). Then they must have the same character in (V.5.5.b), so they are subspaces of one and the

same ·“˜ ≈~‘ind‚–°. So they can be equivalent only if there is multiplicity in ·“ ≈

~‘ind‚–° (for the

bilateral action of G*MA). Now use Corollary V.5.8 to obtain the equivalence:

Proposition V.5.9 The bilateral action of G*MA in ∂æ(G/N) is multiplicity free if and only if

the decomposition of each member of the unitary principal series of G is multiplicity free.

178

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