University of Groningen

Convolution on homogeneous spacesCapelle, Johan

IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.

Document VersionPublisher's PDF, also known as Version of record

Publication date:1996

Link to publication in University of Groningen/UMCG research database

Citation for published version (APA):Capelle, J. (1996). Convolution on homogeneous spaces. s.n.

CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).

Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.

Download date: 02-02-2021

Chapter VII

SL(2;å) acting on å™*

VII.1 Zonal Distributions

Invariant Kernels

This chapter deals with the Harmonic Analysis on the space ™*, that is, the Euclidean plane ™

without the origin, seen as a homogeneous space when acted on by SL(2;Â), with Lebesgue measure

as invariant measure. On the one hand we specify what has been mentioned in Chapter VI, that is that

the case n=2 is exceptional among the homogeneous spaces Ân* acted on by SL(n;Â). On the other

hand this is an example of the space dealt with in Chapter V, that is the space G/N, where G is a

connected, real semi-simple Lie group with finite centre, and N maximal nilpotent subgroup. In this

low-dimensional case it is possible to fully determine some of the objects dealt with in a more general

fashion in Chapter V, such as the space of (not necessarily compactly supported) N–invariant, or

zonal, distributions concentrated on the submanifold MAp (Proposition VII.1.2).

Our approach to the Harmonic Analysis on this space is in line with Chapters V and VI, so

we look at zonal distributions and their associated G–invariant kernels, and we try to understand the

convolution structure that comes with these. The analysis in terms of G–invariant kernels makes this

approach, though self-contained, fit in with the treatment of the representation theory of SL(2;Â) by

Gelfand, Graev, and Vilenkin [118] . Their results go back to V. Bargmann [119] , who was the first

to classify all unitary representations of SL(2;Â). Nowadays the representations of SL(2;Â) are well

–known. More specifically, the series of representations as found in Theorems VII.7.3 and VII.7.8 is

well-known, as are some of the formulas pertaining to these (e.g. VII.7.8.a,b). The result is that all

118 I. M. Gel’fand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions, Vol 5: Integral Geometry and

Representation Theory (New York: Academic Press, 1966), Chapter VII.119 V. Bargmann, “Irreducible Unitary Representations of the Lorentz Group.,” Annals of Mathematics 48

(1947), nº 3, 568-640.

192

— VII.1 Zonal Distributions; Invariant Kernels —

unitary representations of SL(2;Â) are realized in Hilbert subspaces of ∂æ(Â*™), though not in a unique

way. This was more or less to be expected in view of the Subrepresentation Theorem of Casselman.

According to that theorem every irreducible (g,K)–module is a subrepresentation of an induced

representation H≈,µ (for the theorem and the relevant definitions, see [ 120] ). However, that theorem

does not directly concern Hilbert subspaces of distributions, so it does not seem to apply in a direct

way.

Our purpose, however, is not the generation of these representations as such but to deal

with the Harmonic Analysis of Â*™ in the spirit of the preceding chapters. We start from scratch, so to

speak, and using the convolution product we end up with a list of all minimal invariant Hilbert

subspaces (Theorem VII.8.2). We distinguish different types of subcones of the cone of all

G–invariant Hilbert subspaces, in at least one of which we show that the integral decompositions are

unique (Theorem VII.7.3). The main ingredient that enters into it which has not been used so far

(except briefly in Section V.5) is the theory of co–ç°–vectors (see Section VII.6).

We use coordinates ·st‚ for ™*. Choose the point p=·0

1‚ ∑™*. Its stability group under the natural

action of SL(2;Â) is

N=”· 01 t

1‚»» t∑Â’.

The normalizer B of N is the (parabolic) subgroup of determinant 1 upper triangular matrices, with

decomposition MAN, where

M=”±I’

A=”· 0r 0

r¡‚»» r>0’.

Furthermore, one has the Iwasawa decomposition G=KAN, with K the group of rotations around the

origin,

K=”· csoins∆∆ –

csoins∆∆‚»»

∆∑“0,™π)’

To make the treatment fit in with Chapter V, choose da=î™îrîπ dr as invariant measure on A, and

dk=䙡äπd∆ as invariant measure on K, then the Lebesgue measure dsdt equals

(VII.0.1.a) d∆rdr=䙡äπd∆¤r™î™îrî

π dr=dk¤a™®da,

120 Nolan R. Wallach, Real Reductive Groups I. Pure and Applied Mathematics Vol. 132 (New York:

Academic Press, 1988), p. 98, Theorem 3.8.3.

193

— Chapter VII SL(2;Â) acting on ™* —

where a™® equals the character ·0r 0

r¡‚ éêâ r™ on A. On M choose dm equal to half the counting

measure, so that ªM

dm=1. In keeping with this, the invariant measure used on MA=”·· s00s¡‚»s∑Â*’ will

be dmda= îπ|s|ds.

According to Section IV.8 and Proposition V.1.1 the group Ì of G–invariant

diffeomorphisms of ™* can be identified with MA, operating from the right on ™

* through (gp)©=g©p,

g∑G, ©∑MA. More concretely, one finds that

·st‚· ©00©¡‚=· st©

©‚ s,t∑Â, ·st‚≠0, ©∑Â*.

In other words, the only SL(2;Â)–invariant diffeomorphisms of ™* are the obvious ones, i.e.

multiplications by scalars.

We now go on to determine which distributions are zonal, i.e. N–invariant.

The N–orbit structure is simple enough, but essentially different from the orbit structure

encountered in Chapter VI (see the end of Section VI.1). We describe the orbit structure in the terms

used in Theorem V.2.5.

The group M*, the normalizer of A in K, equals the group of four rotations around the

origin over multiples of πì™. Since M=”±È’ the Weyl group W=M*/M has two elements, so according

to Theorem V.2.5.(iii) there are two types of orbits, those of dimension 0 and those of dimension 1.

These orbits are indexed by the group M*A, which can be identified with its orbit in ™*, that is, with

the union of the standard coordinate axes (without the origin). So, there are the single point orbits,

together forming the submanifold MAp (that is, the s–axis ), and there are orbits of dimension 1, one

through each point of the t–axis.

It is a trivial matter to verify these facts without any regard to the group structure, but it is

the way the Weyl group figures here that sheds light on some of the later results, such as Theorem

VII.3.4. That is, the nature of the orbit structure leads one to expect that zonal distributions are

characterized by a pair of distributions, one on the s–axis, and one on the t–axis, so one distribution

for each Weyl group element. This is roughly speaking so, and Theorem VII.3.4 will present a precise

statement.

To obtain more definite results one has to determine which zonal distribution defined

outside the s–axis can be extended to all of ™*, and one has to determine whether there exist zonal

distributions concentrated on the s–axis of strictly positive transversal order, and whether this

194

— VII.1 Zonal Distributions; Invariant Kernels —

transversal order is always finite. We treat these problems in a very formal way, partly using results

from Chapter V.

The union of the orbits of highest dimension we denote by ™**=™

*-–MAp, so ™** is the

N–invariant set of points outside the s–axis. Define ˙:™* ìêâ Â, ˙(s,t)=t. Then ˙ is a submersion.

Moreover, ˙ is N–invariant, that is, ˙(nx)=˙(x), n∑N, x∑™*. Let U∑∂æ(≈) be zonal, and let U

û be its

restriction to ™**. Let ˙û be the restriction of ˙ to ™

**. Since ˙û indexes the N–orbits in ™**, it is

essentially the quotient map ™** ììâ N\™

** = Â*. Since ˙û is a submersion this means that all zonal

distributions on ™** can be described as pull-backs by ˙û (see [ 121] )). So,

(VII.0.1.b) Uû=( ˙û)*(T

û )

for a uniquely determined Tû ∑∂æ(Â*). This is more or less obvious. The following is not.

Lemma VII.1.1

In (VII.0.1.b), whenever Uû

∑∂æ(™**) is the restriction of a zonal distribution U∑∂æ(™

*), then

Tû∑∂æ(Â*) can be extended (if not uniquely) to a distribution on Â.

Proof The submersion ˙ gives rise to a pull-back ˙*: ∂æ(Â)ìâ∂æ(™*). Define M:∂(™

*) ììâ ∂(Â) by

(MÏ)(t)=ªÂ

Ï(st)ds, Ï∑∂(™*), t∑Â.

One checks that <¥û˙,Ï>=<¥,MÏ> for ¥∑∂(Â). It follows that ˙* is the transpose of M. Note that if Ï

belongs to ∂(™**), then if MÏ=0 it follows from (VII.0.1.b) that <U,Ï>=0 for any zonal U∑∂æ(Â*

™).

Take a test-function ∫∑∂(Â) with support disjoint from 0, and with ª∫(s)ds=1. Define

R:∂(Â)êêâ∂(™*),

(VII.1.1.a) (Rƒ)· ts‚=∫(s)ƒ(t), for ·t

s‚≠0.

Then R is a right inverse for M. One shows that RM:∂(™*) ììâ ∂(™

*) maps ∂(™**) into itself.

Let L:∂æ(™*) ììâ ∂æ(Â) be the transpose of R. Then L is a left inverse for ˙*. For zonal U,

and any Ï∑∂(™*), one has <(˙*LU)–U,Ï>=<U,(RMÏ–Ï)>. Assume that Ï∑∂(™

**), then the same will

be true for RMÏ–Ï. But since M(RMÏ–Ï)=0 this means <U,RMÏ–Ï>=0 for any zonal U∑∂æ(Â*™). So,

for any Ï∑∂(™**) one has <˙*LU–U,Ï>=0, which is saying that ˙*LU and U coincide on ™

**. This

implies that T:=LU is an extension of Tû !

121 Mannes Poel and E. G. F. Thomas, “Pullbacks en Invariante Distributies,” Report (University of Groningen

(the Netherlands): Department of Mathematics, 1987).

195

— Chapter VII SL(2;Â) acting on ™* —

Assume U∑∂æ(™*)N, so, U is zonal. According to the lemma, there exists T∑∂æ(Â) such that U

coincides with ˙*T on ™**. Choose such a T. Then, ˙ being N–invariant, ˙*T is zonal. So, U–˙*T is

a zonal distribution supported by the s–axis. Considering the s–axis as the submanifold MAp, this

can be denoted

U–˙*T∑∂æMAp(™*)N,

as in Section V.3.

To determine ∂æMAp(™*)N, we use the shape

∂æMAp(™*)=Ë(g)

ˤ(Àb)

∂æ(MA) ,

as in (V.3.7.b). Here b is the Lie-algebra of B=MAN. One has b=m@a@n, where m is (formally) the

Lie algebra of M (which is (0)), a the Lie algebra of A (one-dimensional and spanned by · ¡º¡º‚), n the

Lie algebra of N (one-dimensional, and spanned by ·ººº¡‚). Furthermore, Nä will denote the nilpotent

group of strictly lower diagonal matrices. Its Lie algebra is denoted by än, and is spanned by ·º¡ºº‚.

In this low-dimensional case it is easy to determine the zonal elements in ∂æMAp(™*)

explicitly. We do this formally. Let å be the single positive root, so

[H,X] := å(H)X for H∑a, X∑n

[H,Y] := –å(H)Y for H∑a, Y∑än.

Fix H:=· ¡º¡º‚

X:=· ººº¡)

Y:=· º¡ºº)

so that “X,Y‘=H, å(H)=2, ®(H)=1

The weight ® is defined as in Section V.1, for a=· 0r 0

r¡‚∑A one has a®=r. This is in keeping with

(VII.0.1.a).

Use the fact that

(VII.1.1.b) (ad(X))(Yn)=nYn–1(H–(n–1)) n∑ˆ, n>0,

as is easily verified. As in the proof of Proposition V.3.7 for U∑∂æMAp(™*) one can use the unique

decomposition

196

— VII.1 Zonal Distributions; Invariant Kernels —

(VII.1.1.c) U= n ¶°

=ºY

n

ˤ(b)

a®Sn

with the (Sn)n∑ˆ a locally finite sequence in ∂æ(Â*) [122]. When U is zonal it follows that for X and

Y as defined above

(VII.1.1.d) 0 = XU= n¶°

=ºXYn

ˤ(b)

a®Sn= n¶°

=º(adX(Yn))

ˤ(b)

a®Sn

= n¶°

=¡nYn-1

ˤ(b)

(H–(n–1))(a®Sn).

The uniqueness of the decomposition yields that

(VII.1.1.e) H·a®Sn‚ = (n–1)·a®Sn‚ for n˘1,

whereas Sº can be any distribution on MA (this is only natural, Sº being the transversal order 0 part).

The general solution of (VII.1.1.e) is given by characters of MA:

(VII.1.1.f) Sû∑∂æ(MA)

Sn=An·1dm¤a–n®da‚+Bn·Ídm¤a–n®da‚ n>0, An,Bn∑Ç.

Sn=0 except for a finite number of n.

Here 1 and Í denote respectively the trivial and the non-trivial character of the two-point group

M=”±È’, and dm equals 2 the counting measure on M, and da denotes the Haar measure d·0r 0

r¡‚=

™πr¡dr on A, as at (VII.0.1.a). The support of each Sn∑∂æ(Â*) is either Â* or Â>0 or Â<0 (depending

on An and Bn). Since a priori the series (VII.1.1.c) is locally finite it now follows that it must be

globally finite as well.

Let ^ denote the embedding of Â*=MA into G/N=™**, ^(s)=(s,o), s≠0. One calculates that

(VII.1.1.g) Yn ^*·1dm¤a–n®a®da‚=·-s îÿ

ÿ êt

‚n·|s|–nπds¤∂‚

=π(-1)n(sign(s))nds¤∂(n)

Yn ^*·Ídm¤a–n®a®da‚=π(-1)n(sign(s))n+¡ds¤∂(n).

Here ds¤∂(n) should be interpreted as the restriction to ™* of the distribution ds¤∂û

(n) on ™ (see

Section VII.2 for a discussion of this notation). To describe in a formally correct manner the

122 We have inserted the character a® merely to make the present argument fit in with unitary induction, it is

not essential at this point.

197

— Chapter VII SL(2;Â) acting on ™* —

distributions (sign(s))ds¤∂(n) which emerge here, introduce the twisted pull-back ˙*Í:∂æ(Â) ììâ

∂æ(Â*™) as the transpose of the map

(MÍ(Ï)‚(t)=ªÂ

sign(s)Ï(st)ds, Ï∑∂æ(Â*™), t∑Â.

A distribution ˙*ÍZ is zonal only if suppZ=”0’, and ˙*

Í is one-to-one because MÍ maps onto ∂(Â).

The formal description of sign(s)ds¤∂(n) is now as ˙*Í∂(n).

The result is that

(VII.1.1.h) Yn ^*·1dm¤a–n®a®da‚=π(-1)n˙*

Ín∂(n)

Yn ^*·Ídm¤a–n®a®da‚=π(-1)n˙*

Ín+¡∂(n)

In these notations the consequence of (VII.1.1.f) is that every U∑∂æMAp(™*) can be represented in the

form

(VII.1.1.i) ^*a®Sû+˙*·n¶>º

ån ∂(n)‚ + ˙Í*·n

¶>º

∫n ∂(n)‚.

The whole argument so far shows that every zonal distribution is of the form ˙*T+^*a®Sû+

˙*·n¶>º

ån ∂(n)‚ + ˙Í*·n

¶>º

∫n ∂(n)‚, with the sums finite, but this is clearly not a unique

decomposition. With the notational conventions introduced so far it is possible to describe the non-

uniqueness exactly.

Proposition VII.1.2 Zonal Distributions

Let N be the nilpotent group N=”· 01u

1‚, u∑Â’.

Let ∂æº(Â) denote the space of distributions on  concentrated at the origin of Â.

Then every zonal distribution U allows a decomposition

U=^*S+˙*T+ *ÍTû, S∑∂æ(Â*), T∑∂æ(Â), Tû∑∂æº(Â).

with the non-uniqueness of the decomposition described by

^*ds=˙*∂

^*(signs)ds=˙ *Í∂ .

More precisely, the map

198

— VII.1 Zonal Distributions; Invariant Kernels —

Á: ∂æ(Â*)@∂æ(Â)@∂æº(Â) ììâ ∂æ(™*) ,

Á (S, T, Tû)=^*S+˙*T+ *ÍTû ,

is a continuous linear map onto the space ∂æ(™*)N of zonal distributions on ™

*, with a two-

dimensional kernel ~ spanned by (ds,–∂,0) and (sign(s)ds,0,–∂).

Á is a topological homomorphism, that is, ∂æ(™*)N is isomorphic as a topological vector

space to the quotient “∂æ(Â*)@∂æ(Â)@∂æº(Â)‘ /~.

Comment The decomposition as such is given, in different notation, and without proof, by

Harzallah, in [123] . The occurrence in the proposition of the twisted pull-back ˙*ÍTû is due to the

missing origin. Furthermore, it is justified, and in some ways preferable, to view the pair ·T,Tû‚ as a

single distribution on the non–Hausdorff manifold  with a doubled origin, more easily understood

by considering the pair ·T+Tû,T–Tû‚ as a pair of distributions on  coinciding outside the origin.

The twisted pull-backs ˙*ÍTû of strictly positive transversal order are exceptional in the

sense that they cannot be approached by zonal measures concentrated on single N–orbits. The closure

of the span of the latter is ^*∂æ(Â*)+˙*∂æ(Â).

Proof The considerations preceding the proposition prove that Á is onto. Take n=0 in (VII.1.1.h) to

obtain the identities in the proposition that describe the non-uniqueness. To see that the

decomposition is unique up to these identities assume that ^*S+˙*T+ *ÍTû=0. Then

˙*T+ *ÍTû=–^*S is a transversal order 0 distribution concentrated on the s–axis, forcing T and Tû

to be a multiple of ∂û. The fact that the images of ˙* and ˙*Í have intersection (0) then settles the

matter.

From the proof of Lemma VII.1.1 we know that ˙* has a left inverse L. If in (VII.1.1.a) one

takes ∫ with support in Â+ the same map L will serve as a left inverse for ˙*Í. Moreover, according

the proof of Theorem IV.10.2 there exists a left inverse for ^*. The upshot is that there exists a

continuous linear left inverse for Á, so that Á is a topological homomorphism !

By Proposition IV.2.1 the zonal distributions correspond to G–invariant kernels on ™**™

*, leading to

a more transparent decomposition theorem. For a zonal distribution of the type ^*a®S the

123 Khélifa Harzallah,”Distributions Invariantes: Une Introduction,” Deux Courses d’Analyse Harmonique,

École dÉté d’Analyse Harmonique de Tunis, 1984 (Basel: Birkhäuser, 1987), p. 249.

199

— Chapter VII SL(2;Â) acting on ™* —

corresponding kernel is given by (IV.11.2.a). For the theorem we use another form.

We need a few notations. Let j denote the imbedding j:Â*™*Â*

îâ Â*™*Â*

™,

j(x,©)=(x,x©).

The range of j equals the set Z of zeros of the determinant function on Â*™*Â*

™, and this set has two

connected components Z+:=”(x,x©)» x∑™*, ©>0’ and Z–”(x,x©)» x∑™

*, ©<0’, distinguishable for

example by the sign of the inner product. Let ƒ denote a smooth function that is constant in a

neighbourhood of Z, while on Z+ it is 1, on Z– it is -1. Let Det*:∂æ(Â) îâ∂æ(Â*™*Â*

™) denote the

pull-back by the determinant function Det, well-defined because Det is a submersion on Â*™. For a

distribution Tû concentrated at 0 the notation DetÍ* Tû will be used for the product ƒ.Det*Tû. As

long as Tû is concentrated at 0, the distribution DetÍ* Tû does not depend on the particular choice of

ƒ, because for such Tû one has Supp(Det*Tû)≤Det–1”0’=Z.

Theorem VII.1.3 SL(2;å)–Invariant Kernels on å*“

*å*

“˚.

Every SL(2;Â)–invariant kernel on Â*™*Â*

™ allows a decomposition

˚=j*(dx¤S)+Det*T+ DetÍ* Tû ,

with S∑∂æ(Â*), T∑∂æ(Â), Tû∑∂æº(Â).

The decomposition is unique modulo the identities

j*dx¤ds=Det*∂

j*dx¤sign(s)ds=DetÍ* Tû .

Proof From Proposition IV.9.4 one sees that the convolution kernel associated to ^*S equals

j*(dx¤S). Furthermore, recall that ˙(s,t)=t, s,t∑Â, (s,t)≠0. For g¡,g™∑G one has Det(g¡p,g™p)=

Det(p,g¡¡g™p)=˙(g ¡

¡g™p), so that for any smooth function å on  one has (åûDet)(g¡p,y))=

(åû˙)(g ¡¡y)=(†g¡

Ï)(y), y∑™*, and therefore (åûDet)(x,y)=(∂x*(åû˙))(y), y∑™

*. So <(åûDet),ϤÁ>=

<Ï*(åû˙),Á>, showing that the kernel associated to åû˙=˙*å equals Det*å. By definition of pull-

back this relationship extends to T∑∂æ(Â). The fact that the distribution kernel associated to ˙*Í Tû

equals DetÍ* Tû can be shown by using the fact that ˙*

Í Tû coincides with respectively ˙* Tû and –˙*

Tû on the two connected components of Â**”0’≤Â*™ .!

200

— VII.1 Zonal Distributions; Invariant Kernels —

Finally, we describe the fundamental involution U éêâ U% in the zonal distributions, so the involution

determined by (see Proposition IV.3.1.B.(ii)):

<Ï*U,Á>=<Ï,Á*U> Ï,Á∑∂(™*)

Proposition VII.1.4

Let the involutions Séâ fiS, TéâfiT, TûéâfiTû in ∂æ(Â*), ∂æ(Â), and ∂æº(Â) be those brought

about by the group reflections © éêâ ©¡, t éêâ –t, and t éêâ –t respectively.

Then the fundamental involution in the zonal distributions is given by

·^*|s|S+˙*T+ *ÍTû‚%=^*|s|fiS+˙*fiT+ *

ÍfiTû , S∑∂æ(Â*), T∑∂æ(Â), Tû∑∂æº(Â).

Proof The equality (^*a®S)%=^*a®Sfi is valid in the general setting of Section IV.11, and is

immediate from (IV.11.2.a) and Proposition IV.3.1.B.(ii),(iii).

The function Det is anti-symmetric, i.e. Det(y,x)=–Det(x,y). So the pull-back by Det is

intertwining between the involution in ∂æ(™**™

*) defined by S¤T éêâ T¤S and the involution in

∂æ(Â) brought about by t éêâ –t, in other words (TûDet)%=(T%ûDet). Since according to the proof of

Theorem VII.1.3 the distribution TûDet is the kernel associated to ˙*T, it follows from Proposition

IV.3.1.B.(ii),(iii) that (˙*T)%=˙*T%.

Finally, the sets Z+

and Z—

as defined in the introduction to Theorem VII.1.3 are invariant

under (x,y) éêâ (y,x), so that (DetÍ*Tû)%=·ƒ.Det*Tû‚%=ƒ.·Det*Tû‚%=ƒ.·Det*T%û‚=DetÍ

*T%û, so that

(˙*Í Tû)%=(˙*

Í T%û) !

VII.2 Some Notations

For the formal proofs in Section VII.1 we have used very formal notations. In this section we

introduce notations which are slightly corrupt, but convenient. We introduce some other notations as

well.

For an imbedded distribution ^*V, V∑∂æ(Â*) we simply write V¤∂. Strictly speaking this

tensor product denotes a distribution on Â**Â, but it has a unique N–invariant, or zonal, extension Â*™.

To avoid possible confusions about measures, we sometimes specify these, so for ƒ∑´(Â*) the

201

— Chapter VII SL(2;Â) acting on ™* —

distribution îπ1^*a®ƒdmda is denoted by ƒds¤∂ rather than by ƒ¤∂. A distribution ˙*T is denoted

1¤T or ds¤T. Strictly speaking this denotes a distribution on ™, but we mean its restriction to ™*. .

Similarly, ˙*ÍTº is denoted ͤTº.

Repeatedly we identify ≈=™* with KA through (k,a)+kap. So we shall denote the unit

circle by K. The positive s–axis is identified with A. Apart from the tensor products in

(s,t)–coordinates we also freely use the tensor product with respect to KA, that is, polar coordinates

(∆,r), with the usual choice of measures such that dx=d∆¤rdr=䙡äπd∆¤r™î™îrî

π dr. An equality like

sin∆d∆¤r™dr=1¤t should cause no confusion, even though the tensor products involved are very

different.

The group M has two characters, denoted by 1 and Í, so 1(1)=Í(1)=1=1(-1), Í(-1)=-1.

We also use these symbols to denote respectively the trivial and the signum character of Â*, so

1(©)=1, Í(©)=sign(©), ©∑Â*. Furthermore, general characters of Â* will be denoted (somewhat

loosely) as ©~≈, ≈∑”1,Í’, ~∑Ç, meaning

©~≈=≈(©).|©|~ ©∑Â*.

A similar notation is used on the unit circle. That is, by sin≈~∆ will mean the function ≈(sin∆).|sin∆|~.

For Âe~¯-1 this needs to be defined more carefully, since then ≈(sin∆).|sin∆|~d∆ cannot be viewed as

a regular distribution. Such problems will be dealt with in Section VII.4. One has the equality

sin≈~–¡∆d∆¤r~dr=1¤t≈

~–¡ ≈∑”1,Í’, ~∑Ç, Âe~>-1

VII.3 Weyl Types

Roughly speaking, the zonal distributions come in two types, differing in their behaviour with respect

to the left and right actions of MA=Â*.

Recall that a zonal distribution is said to be strongly zonal when

(VII.3.0.a) E*U=U*E ÅE=S¤∂, S∑´æ(Â*)

(Definition IV.13.4). It turns out that zonal distributions of the form U=1¤T+ͤTû have the

property that

(VII.3.0.b) E*U=U* E% ÅE=S¤∂, S∑´æ(Â*).

An easy way to show this is to use Proposition VII.3.3.(ii) below.

202

— VII.3 Weyl Types —

As shown in Section VII.1 the Weyl group W for SL(2;Â) consists of two elements, which we denote

by 1 and -1. It operates on MA=Â*. This operation is given by ©1=©, ©–1=_©1 , ©∑Â*. Since W operates

on MA, it operates on ´æ(MA). Since according to Theorem V.3.1 the algebra å of convolution

operators in the distributions is isomorphic to ´æ(MA), the Weyl group also operates on å. In

defining this we involve the ®–shift, so by definition (^*a®S)w:=^*a®Sw. In (s,t) coordinates one

has

(VII.3.0.c) (S¤∂)–1=(S¤∂)%, S∑´æ(Â*).

Definition VII.3.1

Let w belong to the Weyl group W. A zonal distribution U will be said to be of Weyl type w,

w=±1, when

(VII.3.1.a) E*U=U*(Ew) ÅE=S¤∂, S∑´æ(Â*).

An SL(2;Â)–invariant Hilbert subspace of ∂æ(™*) will be said to be of Weyl type w if its

reproducing distribution is so. For w∑W the cone of SL(2;Â)–invariant Hilbert subspaces of type

w is denoted by Hilb

wSL(2;Â)∂æ(™

*).

Recall that å denotes the algebra of convolution operator operators ∂æ(™*)ììâ ∂æ(™

*). Let v denote

the unitarized right action of Â* in ∂æ(Â*™)

Proposition VII.3.2

Let U be a zonal distribution, and let u be the convolution operator u:∂(™*)ììâ ∂æ(™

*)

propagated by U. Then the following statements are equivalent:

i) U is of Weyl type 1, that is E*U=U*E for E=S¤∂, S∑´æ(Â*)

ii) The G–invariant kernel associated with U is right Â*–invariant, that is, it is

invariant under the representation Â*ë©éîâv©¤v©iii) u is bilaterally invariant (SL(2;Â) operating from the left, Â* from the right)

iv) ua=au, for all a∑å.

Moreover, if U is type 1, so is U%.

203

— Chapter VII SL(2;Â) acting on ™* —

Proof The proof is straightforward. The equivalence of (i) and (iii) is an application of Theorem

IV.13.5. The equivalence of (i) and (ii) follows easily if one uses the connection between U and ˚

given by

(VII.3.2.a) <Ï*U,Á>= <˚,ϤÁ> ·= <uÏ,Á>‚ Ï,Á∑∂(™*).

(i) and (iv) are trivially equivalent, because the whole of å is propagated by ´æ(Â*)¤∂. The final

statement is obtained from i) by using the identities (E*U)%= U%*E%, and (E%)%=E !

Proposition VII.3.3

Let U be a zonal distribution, and let u be the convolution operator u:∂(™*)ììâ ∂æ(™

*)

propagated by U. Then the following statements are equivalent:

i) U is of Weyl type -1, that is E*U=U* E% for E=S¤∂, S∑´æ(Â*)

ii) The G–invariant kernel associated with U is invariant under the representation

Â*ë©éîâv©¤v©¡

iii) The convolution operator u:∂(Â*™)ììâ ∂æ(Â*

™) propagated by U satisfies

uv©Ï=v©¡uÏ Ï∑∂(™*), ©∑Â*.

iv) u(ta)=au for all a∑å.

Finally, if U is type -1, so is U%.

Proof Again, the proof is easy. The equivalence of (i), (ii) and (iii) is shown using (VII.3.2.a). The

equivalence of (i) and (iv) again follows from the fact that å is propagated by ´æ(Â*)¤∂, and the fact

that if E propagates a, then E% propagates ta (Proposition IV.3.1). The final statement is again obtained

from i) by using the identities (E*U)%= U%*E%, and (E%)%=E !

Properties (VII.3.0.a,b) can be explained by considering the N–orbit structure as described in Section

VII.1, and its relationship with the Weyl group. This justifies the terminology. Moreover, the orbit

structure explains that one can very roughly expect that every zonal distribution is the sum of Weyl

types. We have no explanation for the fact that this should be precisely true:

204

— VII.3 Weyl Types —

Theorem VII.3.4 Weyl Type Decomposition

Every zonal distribution allows a decomposition

U=U1+U–1,

with U1 of Weyl type 1, and U–1 of Weyl type –1.

Proof Distributions of the type S¤∂ are strongly zonal, or Weyl type 1 zonal, and those of the type

1¤T+ͤTû are of Weyl type -1 zonal. The existence of the decomposition is then immediate from

Proposition VII.1.2 !

The following will be shown formally later on. See Corollary VII.4.4.

The zonal distributions that are of type 1 as well as -1 are those of the form

å1¤∂+∫ͤ∂+©1¤Pvt_1 , å,∫,©∑Ç.

Here Pvt_1 denotes a principal value distribution, see the next section.

VII.4 Homogeneous Zonal Distributions

The purpose of section is to analyze the right action of MA=Â* on the zonal distributions. This leads

to a description of the zonal distributions that are homogeneous with respect to å. Also, to

understand more of their behaviour, we want to see the homogeneous zonal distributions organized

into holomorphic families. The result is a theorem that prepares for Sections VII.7 and VII.8.

We first sum up results on homogeneous distributions on the real line.

Let ¨ denote the action of MA=Â* on ∂(Â) defined by

(VII.4.0.a) (¨©Ï)(t)=Ï(t©), Ï∑∂(Â), ©∑Â*,t∑Â.

Let ¨ also denote the extension of this action to the distributions on Â. Let Ô be a character of Â*. A

distribution T∑∂æ(Â) is called homogeneous of degree Ô when

(VII.4.0.b) ¨©T=Ô(©)T Å ©∑Â*.

205

— Chapter VII SL(2;Â) acting on ™* —

This implies that T is a weight vector for the infinitesimal representation, so

tädd-tT=~T

with ~ equal to the derivative of Ô at ©=1.

To parametrize the character set of Â*, define for ≈∑”1,Í’, å∑Ç the character Ô≈~ by

(VII.4.0.c) Ô≈~(©)=© ≈

~ = ≈(©)|©|~ ©∑Â*.

This parametrization makes the character set of Â* into a holomorphic manifold, i.e. ”1,Í’*Ç.

Theorem VII.4.1 Homogeneous Distributions on å.

Let  À* denote the group of all (not necessarily unitary) characters of Â*. For Ô∑ À* let ∂æ(Â)Ô

denote the space of distributions homogeneous of degree Ô.

i) ∂æ(Â)Ô is one–dimensional for every character Ô∑ À*

ii) There exists a holomorphic map Ò: À*ììâ∂æ(Â) such that, for every Ô∑ À*, the

distribution Ò(Ô) spans ∂æ(Â)Ô.

This result can be gathered from [124] . An easy proof using the sophisticated language of

hyperfunctions can be found in [125] . In [126] we give a detailed proof, and deal with more general

families of homogeneous distributions, and some of their properties. Roughly speaking, the family is

obtained by meromorphic extension of a family of functions defined on an open set of the complex

plain, after which the (first order) poles are removed by dividing by meromorphic functions with

corresponding poles. This causes the residues at the singular points to emerge as regular members of

the new family. These residues are distributions supported by 0, as is easily explained. We briefly

discuss what we use.

For ~∑Ç with Âe~>0 define the distribution family t+~–1 as the locally integrable function

t~–1for t>0

0for t<0.

124 I.M.Gelfand and G.E. Shilow, Generalized Functions Vol.1: Properties and Operations, translated from

the Russian (New York: Academic Press, 1964), in particular Chapter I, Section 3.125 Henrik Schlichtkrull, Hyperfunctions and Harmonic Analysis on Symmetric Spaces, Series Progress in

Mathematics Vol. 49 (Boston-Basel-Stuttgart: Birkhaüser Verlag, 1984), pp 35-36.126 Johan Capelle, “Families of Homogeneous Distributions,” Report (University of Groningen (the

Netherlands): Department of Mathematics, 1996).

206

— VII.4 Homogeneous Zonal Distributions —

Taylor expansion yields the expression

<t+~–1,ƒ>=ª

¡

°ƒ(t)t~–1dt+ în1!

ªº

¡·ª

º

¡(1–u)nƒ(n+1)(tu)du‚ t

~+ndt+k¶=

n

0 îƒ

î(k

kî)

!

î(î0)

î~î+1ìk,

an expansion valid for n any nonnegative integer, and Âe~>0. This expression defines a meromorphic

extension with poles at ~¯0, ~∑Û.

Define t-~–1 as the image of t+

~–1 under the reflection t éêâ –t. For ≈∑”1,Í’ define

t ≈~–1:= t+

~–1 + ≈(–1).t–~–1

meromorphically extending the L1loc families |t|

~–1 and Í(t)|t|~–1 from Âe~>0. Then one obtains

a family with first order poles, and residues

(VII.4.1.a) Res~=–n t≈~–1=î≈î(î–î1î)

nî+

!î(-î î1)

n∂û(n) n∑ˆû, ~∑Ç, ≈∑”1,Í’.

This expression vanishes where ≈(-1)=(-1)n+1, in other words, where ≈=Ín+1. At those points the

singularity is merely apparent, and one comes upon the well-known distributions Pf˚˚t–n–1, with

explicit description for example

<Pft–n–1,ƒ>=´liè mº

“ ª|

°

t|˘ ´ƒ(t)t–n–1dt+

n

k¶=

-

1

0 ·1–(–1)(n+k)‚ î

Ĕ(k

kî)

!

î(î0)

–î´

–

ìn

n

î++ ê ìk

k

‘.

So it is justified to denote Pf˚t–n by t–Ín

n.

Removing the poles by dividing by a suitable meromorphic function one obtains a

holomorphic family. For example, take (we write Ò≈~ for Ò(Ô≈

~)):

(VII.4.1.b) Ò1~–1:=

îÌ

î(î12îî

~)t1

~–1

ÒÍ~–1:=

îÌî(î2î1

~î+

î2)

tÍ~–1

.

Derivatives of ∂–distributions occur at integral points of this family, that is

(VII.4.1.c) Ò1–™m–1=(-1)mî(î™

mîm

!_)! ∂

(™m)

ÒÍ–™m=

(-1)mîî(î(™mîm

–î–¡î¡)_)!! ∂

(™m–¡)

These expressions follow from (VII.4.1.a) by using the residues of the Ì–function.

The theorem implies and requires (positive) inhomogeneity for the classical distributions

t+–n–1

, n a positive integer, i.e. the distributions usually defined by

207

— Chapter VII SL(2;Â) acting on ™* —

<t+–n–1,ƒ>=

´liè mº

“ ª´

°ƒ(t)t–n–1dt+n

k¶=

-

1

0îƒ

î(k

kî)

!î(î0)

–î´

–

ìn

n

î++ ìk

k+ î

Ĕ(n

nî)

!

î(î0)

log´‘.

The notation t+–n, though customary, is misleading. Yet, by use of this one defines

t≈–n:= t+

–n + ≈(–1).t––n ≈∑”1,Í’,

(positively) homogeneous if and only if ≈=Ín. For ≈=Ín+1 the behaviour of the distribution t≈–n is

best understood by considering it as constant term in the Laurent expansion of the family t≈~ around

the pole at ~=–n. For example, around ~=0,

(VII.4.1.d) t1~–1= ~_

¡(2∂)+t 1–1+ ~t1

–1log|t|+higher order terms,

where t1–1log|t| denotes the parti finie distribution Pf |t|¡log|t| (see [127]). In that way one derives

precise expressions such as

(VII.4.1.e) ·¨©–©≈–n‚t≈

–n=·(-1)n–1+≈(-1)‚ä(änä–

1ä1)ä! ©≈

–n.log|©|.∂(n–1) Å ©∑Â*.

describing the inhomogeneity of t≈n at ≈=Ín+1, n˘1. Though inhomogeneous when ≈=Ín+1, the

distributions t≈–n still have the property

(VII.4.1.f) ·¨©–©≈–n‚™t≈

–n=0 Å ©∑Â*.

Consider the unitarized right action of MA=Â* on the distributions on Â*™, as defined in Section

IV.13, so

v©T=T*|©|¡∂(©¡,º) T∑∂æ(Â*™), Â*ë© identified with ·©0

0©¡‚∑MA.

This is the continuous extension to ∂æ(Â*™) of the unitary representation in L™(Â*

™;dx) defined by

·v©(Ï)‚(x)=|©|Ï(x©) Ï∑ L™(Â*™;dx), x∑Â*

™, ©∑Â*.

This action is such that

(VII.4.1.g) v©^*|s|.S=^*|s|R©S S∑∂æ(Â*), ©∑Â*,

where R is the regular right action of Â* on itself. Some simple considerations also show that

(VII.4.1.h) v©(≈¤T)=≈(©)·≈¤|©|¨©T‚, T∑∂æ(Â), ©∑Â*, ≈∑”1,Í’,

with ¨ defined as at (VII.4.0.a).

Using (VII.4.1.g) and (VII.4.1.h), and Proposition VII.1.2 for decompositions in ∂æ(Â*™)N,

one can determine the space of zonal distributions homogeneous of a particular degree under the right

127 I.M.Gelfand and G.E. Shilow, Generalized Functions,Vol.1: Properties and Operations, Section I.4.2.

208

— VII.4 Homogeneous Zonal Distributions —

action of Â*, and these can be organized into holomorphic families, though some distributions cannot

be thus accommodated (see the theorem).

For ≈∑”1,Í’, ~∑Ç, let ∂æ(™*)~≈ denote the space of distributions on ™

* homogeneous of

degree ©≈~ under the unitarized right action of Â*, so those U satisfying

(VII.4.1.i) v©U=© ≈~U ©∑Â*.

Moreover, let ∂æ(™*)~≈,N denote the space of zonal elements in ∂æ(™

*)~≈.

Theorem VII.4.2 Homogeneous Zonal Distributions on å

™*

Let Ò≈~ be the holomorphic family defined by (VII.4.1.b).

Then ∂æ(™*)~≈,N is two-dimensional, and is spanned by s≈

~¤∂ and 1¤Ò≈~–¡ at all

except a discrete set of points (≈,~) in Mfl*aæÇ.

The space ∂æ(™*)01,N is also two-dimensional, but here s≈

~¤∂ and 1¤Ò≈~–¡ coincide.

The space ∂æ(™*)01,N is spanned by s1

º¤∂=1¤Ò1–¡=1¤∂ and 1¤Pf|t|¡–2log|s|¤∂.

The only other exceptional points are the (≈,~) with ~ a strictly negative integer, and

≈=Í~–¡. At those points ∂æ(™*)~≈,N is three-dimensional, and is spanned by s~

Í~–¡¤∂,

1¤Ò~Í–.~

¡–¡=1¤Pft~–¡ and ͤÒ~

Í– ~

¡.

Comment These results are not new, and can be found for example in [ 128] , where they are given

without proof. Our purpose is to show how these results eventually lead to a description of Hilbert

subspaces, see Theorem VII.8.2.

On general grounds it is to be expected that the dimension of ∂æ(™*)~≈,N at the exceptional

points does not drop, the families s≈~¤∂ and 1¤Ò≈

~ being holomorphic (compare (VII.5.4.a)). That the

dimension of ∂æ(™*)~≈,N should increase at certain isolated points is due to the fact that for ~ a strictly

negative integer, and ≈=Í~–¡, 1¤Ò~Í– ≈¡ is concentrated on the s–axis. The s–axis has two connected

components, giving an extra degree of freedom. This is indicated by the arrows in the table on the

next page. Note futhermore that equation (VII.4.1.e) for ≈=1, n=1 reads

¨©Pf|t|¡=|©|¡Pf|t|¡+2|©|¡.log|©|.∂ Å ©∑Â*.

This explains the occurrence at (≈,~)=(1,0) of the exceptional distribution 1¤Pf|t|¡–2log|s|¤∂.

128 Khélifa Harzallah,”Distributions Invariantes: Une Introduction,” Deux Courses d’Analyse Harmonique,

École d’Été d’Analyse Harmonique de Tunis, 1984 (Stuttgart: Birkhäuser, 1987), p. 250.

209

— Chapter VII SL(2;Â) acting on ™* —

Table VII.4.3 Zonal Distributions on å

™*

Homogeneous of Degree '˚ ≈~

for integers ~

~ -3 -2 -1 0 1 2 3

≈ Exceptional 1¤Pf|t|¡–2log|s|¤∂

1

™ s1~¤∂ 2|s|–£¤∂ 2s–™¤∂ 2|s|–¡¤∂

2¤∂2|s|¤∂ 2s™¤∂ 2|s|£¤∂

äÌä(ä™™ä~)1¤t1

~–18îä…√

£ ä_äπ¤Pft–¢ –4¤∂ææ –¢îä…√

1ä_äπ¤Pft–™

™îä…√1ä_äπ¤dt 2¤|t| ä…√

1ä_äπ¤t™

äÌä(ä™äî~™î+iî™)ͤt Í

~–1 î¡¡™iͤ∂æææ ⇓ –2iͤ∂æ ·⇓

‚

Í

äÌä(ä™™ä~)ͤt1

~–1 ⇑ –4ͤ∂ææ ⇑ ( )

äÌä(ä™ä

î~™î+iî™)1¤tÍ

~–1 î¡¡™i¤∂æææ –¢îä…√

iä_äπ¤Pft–£

–2i¤∂æ î™îä…√iä_äπ¤Pvt¡ 2i¤Í ä…√

1ä_äπ¤t 1¤t™

Í

™ sÍ~¤∂ 2s–£¤∂ 2sÍ

–™¤∂ 2s¡¤∂ 2ͤ∂ 2s¤∂ 2s™Í¤∂ 2s£¤∂

Recall that distributions of the form S¤∂ are of Weyl type 1, those of the form 1¤T+ͤTº are of

Weyl type -1 (Section VII.3). We shall say that a zonal distribution is of Weyl type ±1 when it is of

both types.

Corollary VII.4.4

The space of zonal distributions which are of Weyl type ±1 is spanned by 1¤∂, ͤ∂, and

1¤Pv˚_

1t˚˚. Therefore,

i) the type 1 zonal distributions are those of the form S¤∂+å1¤Pv˚_

1t˚ ˚ ˚, S∑∂æ(Â*), å∑Ç

ii) the type -1 zonal distributions are those of the form 1¤T+ͤTº, T∑∂æ(Â), Tû∑∂ûæ(Â).

Proof Let ∂æ(Â*™) ºN denote the space of zonal distributions that are right invariant for the unitarized

right action of the group Â>0. Then ∂æ(Â*™) ºN equals ∂æ(Â*

™) º1,N@∂æ(Â*™) ºÍ,N. Table (VII.4.3) shows

that ∂æ(Â*™) ºN is spanned by 1¤∂, ͤ∂, 1¤

Pv˚

˚_1t˚ , and 1¤

Pf

˚ä|_1tä|–2log|s|¤∂.

Let U be Weyl type ±1. Then U*E=U* E%, for E∑´æ(Â*)¤∂. Put E=r∂(r,0), r∑Â, r>0. Then

210

— VII.4 Homogeneous Zonal Distributions—

E%*E=∂p (compare IV.13.2.a), so that U=U* E%*E=U*E*E=U*r™∂(r™,0). This is saying that U∑∂æ(Â*™) ºN.

Conversely, if U∑∂æ(Â*™) ºN then if U is type 1 it follows that E*U=U*E=U=U*E% for E∑´æ(Â>0)¤∂.

Since the Dirac distribution ∂(-1,0) is central, one also has E*U=U*E=U*E% for E= ∂(-1,0), so that

E*U=U* E% for E∑´æ(Â*)¤∂, in other words , U is type -1. The same type of argument shows that U is

type 1 when it is type -1.

Recapitulating, a zonal distribution is both type 1 and type ---1 if and only if it belongs to

U∑∂æ(Â*™) ºN and if it is type 1 or type -1. So it is clear that 1¤∂, ͤ∂, 1¤

Pv˚

˚_1t˚ are type ±1. Finally,

1¤Pf

˚ä|_1tä| is type -1, and –2log|s|¤∂ is type 1, so for their sum to be type ±1 they would have to be

type ±1 each, so each would have to belong to ∂æ(Â*™) ºN, which is not the case !

VII.5 ‰-Homogeneous Zonal Distributions

As before, let å denote the algebra of convolution operators ∂æ(™*)îâ∂æ(™

*). Let Ω≤å denote the

algebra of central convolution operators, meaning that they commute with every convolution

operator ∂(™*)îâ∂æ(™

*). This makes sense because the space is weakly symmetric (Section IV.5),

so that every a∑å maps ∂(™*) into itself. Let Ó be a minimal SL(2;Â)–invariant Hilbert subspace of

∂æ(™*), with reproducing distribution U. According to Corollary IV.14.8 there exists a character of the

algebra Ω such that zU=∆(z)U, z∑Ω.

Definition VII.5.1

i) Let ∆ be a character of the centre Ω. A distribution U will be called Ω–homogeneous

of degree ∆ when for all z∑Ω

zU=∆(z)U. .

ii) A convolution operator a:∂(™*)îîâ∂æ(™

*) will be called symmetric when it equals its

transpose, so a=ta.

A convolution operators a:∂æ(™*)îîâ∂æ(™

*) will be called symmetric when its

restriction to ∂(™*)coincides with its transpose.

Proposition VII.5.2

A convolution operator a:∂æ(™*)îîâ∂æ(™

*) is central if and only if it is symmetric.

211

— Chapter VII SL(2;Â) acting on ™* —

Remark Note that in view of (VII.3.0.c) symmetric is the same as Weyl group invariant.

Proof Let a∑å be symmetric. Then its propagator A (which belongs to ´æ(Â*)¤∂) satisfies A=A% .

The Weyl type decomposition (Theorem VII.3.4) then implies that U*A=A*U, U∑∂æ(™*)N, which

means that a is central (Definition IV.14.4).

Let E be the propagator of a convolution operator in the distributions, so E=S¤∂,

S∑´æ(Â*). Then if E has the property that E*U¡=0 for all Weyl type -1 distributions U¡, then E must

be 0. Indeed, take U¡=1¤Ï with Ï∑´(Â), then U¡*E% is smooth, with (U¡*E%)(gp)=<E,†g¡U¡>, g∑G,

so that (U¡*E%)(x)=<S(s),U¡(xs)>. The result is that (S¤∂)*(1¤Ï)=1¤Á, with Á(t)=<S(s),Ï(ts)>, in

particular ·(S¤∂)*(1¤Ï)‚· º¡‚=<S,Ï>. So <S,Ï>=0 for all Ï∑´(Â), so S=0.

Now let A be central. Then for all U¡ Weyl type -1 one has A*U¡=U¡*A%=A%*U¡. Then in

the preceding paragraph take E=A–A% !

Corollary VII.5.3 The centre Ω equals the algebra of convolution operators propagated by

symmetric compactly supported distributions concentrated on the s–axis.

Moreover, for every minimal SL(2;Â)–invariant Hilbert subspace Ó the associated

character ∆ of Ω is real, so

zT=∆(z)T,

∆(_z)= ä∆ä(äz) T∑Ó, z∑Ω.

Proof Theorem V.3.1 in the present context means that the convolution operators in the distributions

are those propagated by compactly supported distributions on the s–axis. The character ∆ is Hermitian

for general reasons (Corollary IV.14.8), so now that z is symmetric ∆ must also be real !

Example Take Z=|©¡|∂·©º

¡‚+|©|∂

·©º‚, ©∑Â*. The convolution operator z propagated by Z is the

continuous extension to the distributions of the map (z·Ï‚)(x)=|©|Ï(x©)+|©|¡Ï(x©¡), Ï∑´(™*), x∑™

*.

Since Z is symmetric, z is central. This means for instance that in every minimal SL(2;Â)–invariant

Hilbert subspace of ∂æ(™*) the operator z reduces to a real scalar.

Let ´ be the operator rîÿÿ êr+1, a G–invariant operator with propagator E=(∂æ¡–∂¡)¤∂º. Then t´=–´, so

that ø=´™ is symmetric and therefore central. The group M=”±I’ commutes with everything in sight.

212

— VII.5 Ω-Homogeneous Zonal Distributions —

In particular ß, defined by ß(T)=†–IT=T*∂(–¡,º), T∑∂æ(™*), is a central convolution operator.

For ≈∑”1,Í’, and ¬∑ let ∂æ(™*)≈

[¬] denote the space of distributions U on ™

* satisfying

(VII.5.3.a) øßU=≈(–1).¬.U.

Obviously, a Ω–homogeneous distributions satisfies at least (VII.5.3.a), for some ≈ and ¬, that is, for a

real–valued character on the subalgebra of Ω generated by ´™ and ß.

First assume that ¬≠0. Set ¬=~™ for a complex number ~. Then

Ker(ø–¬I)=Ker(´™–~™I)

=Ker(´–~I)@Ker(´+~I).

Involving also ß one gets

(VII.5.3.b) ∂æ(™*)≈

[~™]= ∂æ(™*)≈

~ @ ∂æ(™*)≈

–~, ~≠0, ≈∑”1,Í’.

while for ~=0

(VII.5.3.c) ∂æ(™*)≈

[º]= ∂æ(K)≈¤“dr‘ @ ∂æ(K)≈¤“log|r|dr‘ ≈∑”1,Í’.

Decomposition (VII.5.3.c) is not canonical, one rather has the uniquely determined G–invariant flag

0≤∂æ(™*)≈

º≤∂æ(™*)≈

[º], with ´(∂æ(™*)≈

[º])=∂æ(™*)≈

º, ´(∂æ(™*)≈

º=(0).

With the help of the decompositions (VII.5.3.b,c) one shows that every common eigendistribution for

´™ and ß is in fact an eigendistribution for every central convolution operator (see Corollary VII.5.3),

so (VII.5.3.b,c) describe all Ω–homogeneous distributions. It also follows that for ~≠0 every

Ω–homogeneous zonal distribution is the (necessarily unique) sum of two zonal distributions that are

homogeneous of opposite degrees, while for ~=0 the Ω–homogeneous zonal distributions have a

decomposition R¤dr+ L¤log|r|dr with ´(L¤log|r|dr)= L¤dr. Using these facts it is possible to

determine all Ω–homogeneous zonal distributions.

Let Ò≈~ be the holomorphic family defined by (VII.4.1.b). For ≈∑MÀ, ¬∑Ç let ∂æ(™

*)[≈¬,]N

denote the space of zonal distributions which are Ω–homogeneous of degree (≈,¬), compare

(VII.5.3.a).

213

— Chapter VII SL(2;Â) acting on ™* —

Theorem VII.5.4 ‰–Homogeneous Zonal Distributions on å

™*

∂æ(™*)[≈

¬,]N is 4-dimensional, except when (≈,¬)=(Ín–1,n™) for a non-zero integer n, in which

case its dimension is 5.

∂æ(™*)[≈

~,N™] is spanned by

s≈±~¤∂ and 1¤Ò≈

±~–¡ for ~≠º, (≈,~™)„”(Ín–1,n™)»n∑Û’

∂æ(™*)[1

º,]N is spanned by

1¤∂, 1¤Pf

˚ä|_1tä|, log|s|¤∂ and

˚Pf˚ä|_1tä|log|t|–log™|s|¤∂

∂æ(™*)[

ͺ], N

is spanned by

ͤ∂, 1¤tÍ¡, Í(s)log|s|¤∂, and 1¤tÍ

¡log|t|

∂æ(™*)[

Ín n

™–] ¡,N

is spanned by

s±Í

nn–¡¤∂, 1¤Ò±

Ínn––

¡¡, andͤҖ

Í|n n

|–¡

n∑Û, n≠º.

Outside ~=0 the theorem follows from (VII.5.3.b) and Theorem VII.4.2. For ~™=0 one can directly

solve the differential equation for the operator ´™, using the decomposition U=S¤∂+1¤T+ͤTû.

Note that since the solution space is 4-dimensional around ~=0 (both for ≈=1 and for ≈=Í) it is also

at least 4-dimensional at 0. To show this, first assume ≈=1. The meromorphic families solving the

differential equation ´™T=~™T near ~=0 are as follows:

(VII.5.4.a) s1~¤∂=1¤∂+ ~.“log|s|¤∂‘+~™ “2log™|s|¤∂‘ + higher order terms

1¤t1~–1= ~_

¡“1¤2∂‘+“1¤t1¡‘ + ~“1¤t1

¡log|t|‘+higher order terms.

and the same with –~ instead of ~ (compare (VII.4.1.d)). From this it follows for example that the

family –~–™·s1~¤∂+s1

–~¤∂)+2~¡·1¤t1~–1–1¤t1

–~–1‚ tends to a solution for the equation ´™T=0.

The limit is 1¤t1¡log|t|–log™|s|¤∂. By using the identities s1

~¤∂=(∂º+∂π)¤r~dr and 1¤t1~–1=

sin 1~–1∆d∆¤r~dr in this approach one can also show that

(VII.5.4.b) 1¤t1¡=sin 1

¡(∆)d∆¤dr+2(∂º+∂π)¤log(r)dr

1¤t1¡log|t|–log™|s|¤∂=·sin 1

¡(∆).log|sin∆|d∆‚¤dr+sin 1¡(∆).d∆¤log(r)dr

These identities confirm (VII.5.3.c).

214

— VII.5 Ω-Homogeneous Zonal Distributions —

For ≈=Í one has

sÍ~¤∂=ͤ∂+~ “log|s|.ͤ∂‘+ higher order terms

1¤tÍ~–1=“1¤tÍ

¡‘ + ~“1¤tÍ¡log|t|‘+higher order terms.

This is a much simpler situation, where four solutions can be immediately read off.

This yields the solutions given in Theorem VII.5.4. Solving the differential equation

´™T=0 one checks that these are all solutions !

VII.6 Co- ° vectors

In order to give a full description of all irreducible Hilbert subspaces of ∂æ(™*), and their

multiplicities, it is very useful to use some of the theory of (co)–ç°–vectors. In general, this theory

concerns abstract representations, and relates realizations of a representation in subspaces of

distributions to the behaviour of the co–ç°–vectors associated to it. We are here concerned with

representations that are already realized in Hilbert subspaces of ∂æ(≈), where ≈ is a homogeneous

space equipped with an invariant measure. In that case the imbeddings of the abstract Hilbert spaces

get caught up in the general convolution structure on ≈. This section explains how.

First, as discussed at (V.5.5.c), assume one has a continuous unitary representation of the

Lie group G on a Hilbert space Ó, with associated triple

(VII.6.0.a) Ó° ≤â Ó ≤â Ó–°.

We do not assume that † is irreducible.

If V∑Ó–°, the map TV:∂(G) îêâ Ó–°, TV(ƒ)=† ƒ–°V=ª

Gƒ(g)†gVdg, will take its

values in Ó, and will be continuous as map ∂(G)îêâÓ. Moreover, it is intertwining between † and

the left regular representation of G on ∂(G), so TV(Lgƒ)=†gTV(ƒ), g∑G, ƒ∑∂(G). In short, TVbelongs to LG(∂(G);Ó), the space of G–intertwining continuous linear maps ∂(G) îîâ Ó. An

Ó–valued distribution on G with this property is called a distribution vector for †. The important

result, due to Cartier [ 129] , is that all distribution vectors arise in the way described, so they are all

of the form TV.

129 Pierre Cartier, “Vecteurs Différentiables dans les Répresentations Unitaires des Groupes de Lie,”

Séminaire Bourbaki, 27th year, 1974/75, nº 454, pp. 20-34, Theorem 1.4.

215

— Chapter VII SL(2;Â) acting on ™* —

Furthermore, when T is a distribution vector for †, its adjoint j:=T* is a continuous linear map Ó

ììâ ∂æ(G) which is again intertwining, this time between † and the left regular representation of G in

∂æ(G), so j∑LG(Ó;∂æ(G)). This argument can be reversed, so that transposition sets up a bijective

correspondence between LG(Ó;∂æ(G)) and LG(∂(G);Ó). So, a consequence of Cartier’s result is that

the map V éìâ (TV)* sets up a bijective correspondence between Ó–° and LG(Ó;∂æ(G)).

When Ó is irreducible, the kernel of any j∑LG(Ó;∂æ(G)) is G–invariant, so that j is an

imbedding (unless j=0, of course). This means that in the irreducible case every j∑LG(Ó;∂æ(G))

constitutes a realization of Ó in the distributions on G. In any case (not assuming Ó is irreducible),

j(Ó) is a Hilbert subspace of ∂æ(G). Moreover, j has an extension j–°:Ó–°ììâ ∂æ(G). When

j=(TV)*, then j–°(V) is the reproducing distribution of j(Ó) as Hilbert subspace of ∂æ(G).

The next thing to remark is that for H a closed subgroup of G, the image j(Ó) will consist of

right H–invariant distributions if and only if the corresponding co–ç°–vector V∑Ó–° is

H–invariant (for the representation †–°, that is). Therefore, j can in that case be seen as mapping into

∂æ(G/H), and in the case where Ó is G–irreducible, j realizes Ó as subspace of ∂æ(G/H).

Set ≈=G/H. We now turn to the case where Ó is already imbedded in ∂æ(≈). Then Ó–° is

realized in ∂æ(≈), and can be characterized as follows:

(VII.6.0.b) Ó–°= ”V∑∂æ(≈)»·Åƒ∑∂(G)‚ ·†ƒV∑Ó‚’ [130]

This implies that Ó–° is contained in any closed subspace of ∂æ(≈) that contains Ó.

Of most interest are the H–fixed elements in Ó–°. It follows from (VII.6.0.b) that in terms

of the convolution product on the homogeneous space as defined in Chapter IV these can be

characterized by

(VII.6.0.c) ÓH–°=”V∑∂æ(≈)H»·ÅÏ∑∂(≈)‚ ·Ï*V∑Ó‚’.

There is always a privileged element in ÓH–°, because ÓH

–° contains the reproducing distribution of

Ó (as subspace of ∂æ(≈)).

We can now formulate what the general theory yields in this context, where Ó is an

imbedded space. We formulate it in the way we find convenient for the next sections.

130 As in Jacques Faraut, “Distributions Sphériques sur les Espaces Hyperboliques,” Journal de Mathéma-

tiques Pures et Appliquées 58 (1979), 369-444, see p. 373.This is an imbedded form of Cartier's result referred

to above. The point is that when V∑∂(≈) is such that †ƒV∑Ó for all ƒ∑∂(G), then ƒ éìↃV is a

distribution vector for Ó.

216

— VII.6 Co-ç° vectors —

Proposition VII.6.1

Let Ó be a Hilbert subspace of ∂æ(≈), with reproducing distribution U. Then

i) Every continuous linear G–equivariant map †:Ó ììâ∂æ(≈) has a continuous extension

†–°:Ó–° ììâ∂æ(≈). When equipped with the Hilbert structure that makes † into a

partial isometry, the image †(Ó) is an invariant Hilbert subspace of ∂æ(≈), with

·†(Ó)‚–°=†–°(Ó–°).

ii) For every V∑ÓH–°there exists a unique continuous linear G–equivariant map †V:Ó

ììâ∂æ(≈) such that † V–°(U)= VŸ.

iii) Every continuous linear G–equivariant map †:Ó ììâ∂æ(≈) arises in this way, that is,

there exists V∑ÓH–° such that †=†V. The reproducing distribution of †(Ó) as

Hilbert subspace of ∂æ(≈) is then † V–°(V).

Proof (i) is true in general, that is, (i) does not depend on any imbedding.

As to (ii), assume that V∑ÓH–°. First treat Ó as an abstract representation, that is, forget its

imbedding. The distribution vector associated to V is the map TV:∂(G) îîâÓ, TV(ƒ)=†ƒV.

Because V is H-invariant, TV is right H–invariant, in the sense that TV(Rhƒ)=TV(ƒ), h∑H, ƒ∑∂(G),

where R denotes the right regular representation of G in ∂æ(≈), restricted to H. So, V gives rise to a

distribution vector based on ≈, so a G–equivariant map T$V:∂(≈) îîâÓ. It follows from the

definitions that

T$V(Ï)=Ï*V Ï∑∂(≈).

Transposition of this map yields a continuous linear G–equivariant map †V:Ó ììâ∂æ(≈),

(VII.6.1.a) ·S»Ï*V‚Ó

=< †V(S)»Ï>∂æ(≈),∂(≈)

Ï∑∂(≈), S∑Ó.

The map †V has a continuous extension to Ó–°.

Now remember the original imbedding of Ó. Call this formal imbedding k, so the

reproducing operator of Ó is kk*:∂(≈) ììâ ∂æ(≈) (in this notation we suppress the identification of

Ó with its anti-dual). So, kk*(Á)=Á*U, Á∑∂(≈). In (VII.6.1.a) take S=k*Á. When one varies Á and

Ï, then (VII.6.1.a) becomes a separately continuous sesquilinear form on ∂(≈)*∂(≈). According to

Proposition IV.2.1 this means that there exists a zonal distribution W on ≈ such that

217

— Chapter VII SL(2;Â) acting on ™* —

(VII.6.1.b) <Á*W»Ï>∂æ(≈),∂(≈)

=< †V(k*(Á))»Ï>

∂æ(≈),∂(≈)

=·k*Á»Ï*V‚Ó Á,Ï∑∂(≈).

Slightly reworking this one obtains

<Ï*WŸ»Á>∂æ(≈),∂(≈)

=·Ï*V»k*Á‚Ó

=<Ï*V»Á>∂æ(≈),∂(≈)

So, Ï*V=Ï*WŸ, Ï∑∂(≈). (The anti-linear map ∂æ(≈)H êêâ ∂æ(≈)H mapping W to WŸ is defined as

before, that is by <Ï*W»Á>=<Ï»Á*WŸ>, Ï,Á∑∂(≈), so W=W…%, see Proposition IV.3.1).This means that

V equals WŸ, so W equals ŸV. Substitution in (VII.6.1.b) then yields †V(k*(Á))=Á* ŸV, so that

(VII.6.1.c) †V(Á*U)=Á* ŸV Á∑∂(≈).

This is the main point. It implies that †V–°(U)= VŸ. Conversely, † V

–° is the only continuous linear

G–equivariant operator Ó–° ììâ ∂æ(≈) satisfying † V–°(U)= VŸ, because this identity implies

(VII.6.1.c), so that it determines the operator on a dense subspace of Ó. This shows (ii).

As to (iii) in the proposition, if † is an operator with the properties described, its adjoint is

a distribution vector for the representation of G in Ó, based on ≈. Since Ó is continuously imbedded,

this distribution vector is necessarily the form Ï éêâ Ï*V for a V∑∂æ(≈)H, where V must be such that

Ï*V∑Ó, for all Ï. So,

·S»Ï*V‚Ó

=<†(S)»Ï>∂æ(≈),∂(≈)

, Ï∑∂(≈), S∑Ó.

But in view of (VII.6.1.a) this means that †V= †!

Notation The reproducing operator Ïéêâ Ï*U can be extended to ´æ(≈) (Theorem IV.2.2).

Moreover, from criterion (VII.6.0.b) one sees that this extension maps into Ó–°, and that the same is

true for convolution operators propagated by V∑Ó–° other than U. It follows that equation (VII.6.1.c)

can be extended to yield

(VII.6.1.d) † V–°

(S*U)=S* ŸV S∑´æ(≈).

Let U∑∂æ(≈)H be of positive type, and Ó the Hilbert subspace of ∂æ(≈) reproduced by U.

For V∑ÓH–° and any W∑Ó–°, the notation

WU*ŸV

218

— VII.6 Co-ç° vectors —

will be used for † V–°

(W). This convenient notation is motivated by (VII.6.1.d), which now reads

(VII.6.1.e) (S*U)U*ŸV =S* ŸV S∑´æ(≈)

From the definition one also has VU* U=V, obviously (since †U=identity). These notations are useful

in the calculations in Section VII.8.

Rather than Proposition VII.6.1 we will use its following consequence.

Theorem VII.6.2

(i) Let Ó be a Hilbert subspace of ∂æ(≈). Then Ó is irreducible if and only if

ÓH–°§ Ó ŸH

–° is one-dimensional.

(ii) Let Ó and ˚ be Hilbert subspaces of ∂æ(≈). Let Ó and ˚ both be irreducible.

Then Ó and ˚ are equivalent if and only if ÓH–°§˚ ŸH

–°≠(0).

In that case, if U is the reproducing distribution of Ó, and 0≠V∑ÓH–°§˚ ŸH

–°, then the

reproducing operator of is a positive real multiple of VU*ŸV .

Proof Let L:Ó ììâ Ó be a continuous linear G–intertwining operator. Composition with the

imbedding of Ó yields a continuous linear G–intertwining operator L¡:Ó ììâ ∂æ(≈), which according

to Proposition VII.6.1 is associated with a V∑Ó–° such that L¡(f)=fU*ŸV , f∑Ó. Moreover, L¡ has an

extension L¡–° to Ó–°, with L¡

–°(U)=UU*ŸV = ŸV , so that ŸV ∑L ¡

–°(Ó–°)=(L(Ó))–°=Ó–°, and, of

course, ŸV is zonal. If ÓH–°§ Ó ŸH

–° is one-dimensional, the result is that V is a multiple of U, say

V=åU, so that L¡(f)=fU*_å ŸU = _åf

U*U =_åf, f∑Ó, that is, L is a multiple of the identity. So every bounded

G–intertwining operator is a multiple of the identity, so that Ó is irreducible.

On the other hand, let Ó be irreducible, and let both V and ŸV belong to ÓH–°. Let †V:Ó

ììâ ∂æ(≈) be the operator associated with V, so †V(f)=fU*ŸV , f∑Ó. Let †û

V be the restriction of †V to

the dense subspace Óû, where Óû is the range of the reproducing operator of Ó, so Óû=∂(≈)*U. Then

†ûV·Óû‚=∂(≈)*U

U*ŸV =∂(≈)*ŸV, which is contained in Ó (according to (VII.6.0.c)). This means that

†ûV is a densely defined linear G–equivariant operator in Ó. Moreover, †û

V is closeable, its closure

being the intersection of the graph of †V with Ó*.Ó. Now Schur’s Lemma for closeable operators

(Lemma IV.14.11) implies that for some number å∑Ç one has †ûV(fû)=åfû, fû∑Óû, so that actually

†V(f)=åf, f∑Ó. This means Ï*ŸV =Ï*UU*ŸV =åÏ*U, Ï∑∂(≈), so that ŸV=åU, V=å_U. Apparently this

219

— Chapter VII SL(2;Â) acting on ™* —

holds whenever both V and ŸV belong to ÓH–°, so ÓH

–°§ Ó ŸH–° is one-dimensional, more precisely,

it equals ÇU.

Part (ii) of the theorem is shown along the same lines. Instead of Schur’s Lemma for

closeable operators one needs a lemma to the effect that a densely defined closeable operator

intertwining between two distinct irreducible Hilbert space representations is essentially a complex

multiple of a unitary operator, which can be proved in the same way as Lemma IV.14.11. Finally, VU*ŸV

then becomes the reproducing distribution of a Hilbert subspace of ˚–°. According to Lemma V.5.7

the irreducibility of ˚ implies that the only invariant Hilbert subspaces of ˚–° are positive

multiples of ˚, so that the reproducing operator of ˚ has to be a positive real multiple of VU*ŸV !

Corollary VII.6.3 Assume ≈=G/H has a G–invariant measure. Let ÓH–°= ÓŸ H

–°, for every

Hilbert subspace of ∂æ(≈). Then (G,H) is a Generalized Gelfand Pair.

This result is closely related to Proposition IV.7.6.

Proof Assume that ÓH–°= ÓŸ H

–°, for every Hilbert subspace of ∂æ(≈). Let Ó be irreducible. Then

Theorem VII.6.2.(i) yields that ÓH–° is one-dimensional, so Ó has multiplicity one. So every

irreducible subspace Ó has multiplicity one, meaning that (G,H) is a Generalized Gelfand Pair !

VII.7 Partial Symmetries and Subcones of Hilbert Subspaces

In Section IV.7 we indicated how symmetries in a homogeneous space ≈ can sometimes be used to

prove that the representation in the distributions on ≈ is multiplicity free (see Proposition IV.7.6). In

spite of the fact that the representation of SL(2;Â) in ∂æ(Â*™) is not multiplicity free, there are (partial)

symmetries that still have a role to play. In this section we first discuss two partial symmetries which

are adapted to the two cones of Hilbert subspaces that were introduced earlier (Definition VII.3.1).

We then go on to discuss minimal spaces within these cones.

In the first place, let ß be the involutive diffeomorphism defined by ß(kap)=(k¡a¡p). We will call

this a partial symmetry because it maps every singleton N–orbit F (every point on the s–axis) to its

220

— VII.7 Partial Symmetries; Subcones of Hilbert Subspaces —

inverse F%, so that it partially satisfies condition (i) in Proposition IV.7.2. If one identifies ™* with Ç*

in the usual manner, then ß is the map zéâz¡. Let ß*:∂æ(™* ) îâ∂æ(™

* ) denote the push-forward by

ß, and let ßû*:∂æ(™* ) îâ∂æ(™

* ) denote the map ßû*V=r™ß*V (that is, ß*V multiplied by the smooth

function r™=|x|™). Then ßû* is the unitarized version of ß*, in the sense that its transpose coincides

with its restriction to ∂(™* ). This implies that ßû* operates unitarily in L™(™

* ;dx). Explicitly,

(ßû*Ï)(z)=|z|–™Ï(z–¡), Ï∑L™(™* ), z∑™

* =Ç*. Now define

^V= ßû*îV , V∑∂æ(™

* ).

Then ^ is an anti-linear involution in ∂æ(™* ) with the following properties.

Proposition VII.7.1

i) ^= ^* (more precisely, ^*:∂(™*)îâ∂(™

*) is the restriction of )

ii) ^(Ï*U)=(^Ï)*(^U), when Ï∑∂(™*), and U is Weyl type 1 zonal.

iii) ^U= UŸ when U is Weyl type 1 zonal.

iv) ^Ó=Ó for all Weyl type 1 Hilbert subspaces of ∂æ(™* ).

Statement iv) means that ^ operates as an anti-linear isometry in every Weyl type 1 Hilbert subspace

of ∂æ(Â*™). One can compare the proof of Proposition IV.7.6. .

Proof i) is valid precisely because ßû* is unitary.

Let U be zonal, and of Weyl type 1. Then †kaU=†k(∂ap*U)=†kU*(∂ap), k∑K, a∑A. So, for

Á∑∂(Â*™) one has <†kaU»Á>=<U»Áka>, with Áka=†k¡Á*(∂ap)%, so that Áka(y)=Á(kya), y∑ ™

*.

Therefore, the kernel associated to U is of the form <Ï*U»Á>=ªKdkªAdaa™®Ï(kap)<U»Áka>=

<U»ªKdkªAdaa™® Ïä(kap)Áka>. The integral ªKdkªAdaa™® Ïä(kap)Áka evaluated at the point ñbp, ñ∑K,

b∑A, equals ªKdkªAdaa™®(^Ï)(kap)Á(k¡ña¡bp). So, under the identification ™*=K*A, kap=(k,a),

one has

(VII.7.1.a) <Ï*U»Á>=<U»(^Ï)K*A

Á> Ï,Á∑∂(™*), U∑∂æ(™

*)N, U type 1.

where K*A

denotes the standard convolution product on the direct product group K*A. Since

^(ÏK*A

Á)=(^Á)K*A

(^Á), Ï,Á∑∂(™*), one finds that <UŸ»Ï

K*A

Á>=<(^Á)*UŸ»Ï>=…<…Ï…*…U…|……^ ……Á>=

…<…U…|………(…^ ……Ï…)…K*…A

…(…^ ………Á)>=…<…U…|…^ ……(…Ï…K*…A

…………Á…)>=<^U»ÏK*A

Á>. Since ∂(KA)*∂(KA) is dense in ∂(KA), one gets

^U= UŸ. This settles iii).

221

— Chapter VII SL(2;Â) acting on ™* —

Since <^(Ï*U)»Á>=…<…Ï…*…U…»……^ ……Á>, and since the latter expression according to the identities just given

also equals <^U»ÏK*A

Á>=<(^Ï)*(^U)»Á>, one finds moreover that

(VII.7.1.b) ^(Ï*U)=(^Ï)*(^U) Ï∑∂(™*), U∑∂æ(™

*)N, U type 1.

This proves ii).

Let Ó be an SL(2;Â)–invariant Hilbert subspace of type 1. Let Ó be its reproducing

operator. Then Ó is of the form ÓÏ=Ï*U for a type 1 zonal distribution of positive type. Being of

positive type, U has the Hermitian symmetry U=UŸ, so that ^(Ï*U)=(^Ï)*(^U)=(^Ï)*UŸ. This

means that Ó commutes with ^. But then the reproducing operator of Ó equals ^Ó^*=^Ó^=

^™Ó=Ó, so that ^Ó=Ó !

To strongly zonal distributions, so those of Weyl type 1, the theory in Section V.5. applies. Recall

that a Hilbert subspace is called bilaterally invariant when it is invariant for the left action of the

group as well as for the unitarized right action of the group of G–invariant diffeomorphisms, so in this

case, for SL(2;Â) and the unitarized right action v of Â* in ∂æ(Â*™). The latter is such that for an

L™–function Ï one has (v©Ï)(x)=|©|Ï(x©), and ||v©Ï||L™=||Ï||L™, for ©∑Â*, x∑Â*™.

Proposition VII.7.2

Let U be a zonal distribution of Weyl type 1. Then U is of positive type with respect to the

action of SL(2;Â) if and only if r¡U is of positive type as distribution on ™* when ™

* is

interpreted as the group K*A with measure dkda=d∆r¡dr.

Moreover, a zonal distribution U of positive type is of Weyl type 1 if and only if the

Hilbert subspace reproduced Ó by U is a bilaterally invariant Hilbert subspace of ∂æ(™*).

Finally, a bilaterally invariant Hilbert subspace is irreducible for the bilateral action of

SL(2;Â)*Â* if and only if it is irreducible for the left action of SL(2;Â) alone.

Proof Formula (VII.7.1.a) can also be written as <Ï*U»Á>=<r™ÏK*A

U»Á>=<(rÏ)K*A

(r¡U)»rÁ>. This is

formula (V.5.0.h) for this particular case, and the first statement in the proposition repeats Proposition

V.5.1. The second statement is Theorem IV.13.5. The final statement is an easy consequence of

Schur’s Lemma. Indeed, the bilateral representation SL(2;Â)*Â* and the right representation of Â*

commute because Â* is abelian. So when Ó is minimal bilaterally invariant then Â* operates by

scalars in Ó, so any SL(2;Â)–invariant Hilbert subspace of Ó will automatically be bilaterally

222

— VII.7 Partial Symmetries; Subcones of Hilbert Subspaces —

invariant as well, so every SL(2;Â)–invariant Hilbert subspace of Ó is a multiple (possibly 0) of Ó !

Recall that for w∑W=”±1’ we use the notation Hilb

wSL(2;Â)∂æ(™

*) for the (closed, convex) cone of

Hilbert subspaces of type w (Definition VII.3.1). For w=1 this is the cone of bilaterally invariant

Hilbert subspaces. The final statement in Proposition IV.7.2 is equivalent to saying that the extremal

rays in the subcone Hilb

1SL(2;Â)∂æ(™

*) are precisely those extremal rays in the larger cone that belong

to the subcone.

Recall that for that ≈∑”1,Í’ the notation L™(K;ä™däπ∆)≈

signifies the space of L™–functions on

the circle K that have the property that f(∆+π)=≈(-1)f(∆), d∆–almost everywhere, equipped with

Hilbert norm ||f||=·ªº™π

|f(∆)|™ä™däπ∆‚

™. Let L™(K;ä™däπ∆)

+Í

denote the space of functions in L™(K; ä™däπ∆)Í

with all

Fourier coefficients of negative order equal to 0, so those f∑L™(K;ä™däπ∆)Í for which <f»ein∆>=0, n<0.

Let L™(K;ä™däπ∆)

–Í

denotes the space of functions in L™(K; ä™däπ∆)Í

with all Fourier coefficients of positive

order equal to 0. Obviously,

(IV.7.2.a) L™(K;ä™däπ∆)Í

¤“™πdr‘=L™(K;ä™däπ∆)

+Í

¤“™πdr‘@L™(K;ä™däπ∆)

–Í

¤“™πdr‘

Theorem VII.7.3

The extremal rays in the cone Hilb

1SL(2;Â)∂æ(™

*) of bilaterally invariant, or Weyl type 1, Hilbert

subspaces of ∂æ(™*) are generated by

L™(K;ä™däπ∆)≈

¤“™πriµdr‘ for ≈∑”1,Í’, µ∑Â, (≈,µ)≠(Í,0),

L™(K;ä™däπ∆)

+Í

¤“™πdr‘

L™(K;ä™däπ∆)

–Í

¤“™πdr‘ .

The bilateral action of SL(2;Â)*Â* on ™* is multiplicity free, that is, Hilb

1SL(2;Â)∂æ(™

*) is a

lattice cone. Therefore, every bilaterally invariant Hilbert subspace has a unique integral

decomposition in extremal rays of Hilb

1SL(2;Â)∂æ(™

*). In particular,

(VII.7.3.a) L™(Â*™;dx) = ·î™

¡ìπ‚™

Ŧ=

@

1,Í ª

@

Âdµ L™(K;

ä™däπ∆)≈

¤“™πriµdr‘

is a Plancherel decomposition of L™(Â*™;dx) which is unique within Hilb

1SL(2;Â)∂æ(™

*).

223

— Chapter VII SL(2;Â) acting on ™* —

Comment The factors ™π here play no essential part —one could also write L™(K;ä™däπ∆)¤“™πriµdr‘=

™πL™(K;d∆)¤“riµdr‘ etcetera—they are there merely to keep in line with the normalization of the

measures on K and A, so that L™(K;ä™däπ∆)

¤“™πdr‘ reads as L™(K;dk)¤“a®da‘ (compare (VII.0.1.a)).

Formula (VII.7.3.a) shows that setting n=2 in the Plancherel formula (VI.4.4.a) for SL(n;Â), n>2, also

yields a Plancherel formula for n=2, one difference being that (VII.7.3.a) is not unique when one

considers only SL(2;Â) acting from the left, another difference is that not all the spaces that occur in

the integral are irreducible.

Proof The last two statements follow from Theorem V.5.3 and Proposition V.5.4. A more direct way

to see that the bilateral action of SL(2;Â)*Â* on ™* is multiplicity free is to use the fact that according

to Theorem VII.7.1 one has ^Ó=Ó for all bilaterally invariant Hilbert subspaces (we used this type

of argument before in the proof of Proposition IV.7.6).

That the L™(K;ä™däπ∆)≈

¤“™πriµdr‘ are irreducible except at (≈,~)=(Í,0) can be seen using the

theory in Section VII.6 (note that Bruhat theory — see the paragraph featuring (V.5.1.g)) — yields

irreducibility except at the two points (1,0) and (Í,0), but is not decisive at those points). First assume

µ≠0. It follows from from (V.5.1.b) that the space reproduced by the distribution πs≈iµ¤∂ is

Ó≈iµ:=L™(K;

ä™däπ∆)≈¤“™πriµdr‘.

(because in KA–coordinates πs≈iµds¤∂=^*(≈dm¤aiµa®da)). Then ·Ó≈

iµ‚–° equals ∂æ(™*)≈

iµ (the

essential point about this is that K is compact, see (V.5.5.d) and comment). According to Theorem

VII.4.2 this means that space ·Ó≈iµ‚N

–° is two-dimensional, and is spanned by s≈iµ¤∂ and 1¤t≈

iµ–1.

Since in view of Proposition VII.1.4 one has s≈iµŸ¤∂=s≈

iµ¤∂ and 1¤t≈iµ–1Ÿ=≈(-1)t≈

–iµ–1 it follows

that ·Ó ≈iµ‚N

–°§··Ó ≈iµ‚N

–°‚Ÿ is one-dimensional (as long as µ≠0). Then Theorem VII.6.2.(i) yields

that Ó≈iµ is SL(2;Â)–irreducible.

The same type of argument settles the case ≈=1, ~=0, though the distributions involved are

somewhat different. More precisely, ·Ó01‚N

–°=∂æ(™*)01,N, and is spanned by 1¤∂ and

1¤Pf|t|¡–2log|s|¤∂ (see Theorem V.5.2). Since (1¤∂)Ÿ=1¤∂ and ·1¤Pf|t|¡-2log|s|¤∂‚Ÿ=

1¤Pf|t|¡+2log|s|¤∂ it follows again that ·Ó 01‚N

–°§··Ó 01‚N

–°‚Ÿ=1.

Finally, ·Ó0Í‚N

–°=∂æ(™*)0Í,N is spanned by ͤ∂ and 1¤Pv

˚t¡, and since (ͤ∂)Ÿ=ͤ∂

and (1¤Pv ˚t¡)Ÿ=–1¤Pv˚t¡ it follows that ·Ó0Í‚N

–°§··Ó0Í‚N

–°‚Ÿ is two-dimensional, so Ó0Í is

reducible. Decomposition (IV.7.2.a) is then a decomposition into minimal spaces. It is in fact a

224

— VII.7 Partial Symmetries; Subcones of Hilbert Subspaces —

decomposition both within Hilb

¡SL(2;Â)∂æ(™

*) and within Hilb

¡SL(2;Â)∂æ(™

*), and that it is a

minimal decomposition will be shown in the proof of Theorem VII.7.8. At this point we note that in

view of the second statement in the theorem the two components occurring in the decomposition are

necessarily inequivalent for the bilateral action of SL(2;Â). Since the right action of Â* is trivial in

both components, it follows that they are also inequivalent for SL(2;Â) acting alone !

We now turn to the type -1 Hilbert subspaces, and an associated partial symmetry.

Let ¨:Â*™ îâÂ*

™ be the reflection defined by ¨(kap)=k¡ap, so in (s,t)–coordinates this is

simply ¨(s,t)=(s,–t). We call ¨ a partial symmetry because it has the property that outside the

s–axis it maps every N–orbit F to its inverse F%, so that it partially satisfies conditions (i) and (ii) in

Proposition IV.7.2. Define ∆V=¨*Vä. Then ∆ is an anti-linear involution in ∂æ(™* ) with the

following properties.

Proposition VII.7.4

i) ∆= ∆* (more precisely, ∆*:∂(™*)îâ∂(™

*) is the restriction of ∆)

ii) ∆(Ï*U)=(∆Ï)*(∆U), Ï∑∂(™*), when U is Weyl type -1 zonal.

iii) ∆U= UŸ when U is Weyl type -1 zonal.

iv) ∆Ó=Ó for all Weyl type –1 Hilbert subspaces of ∂æ(™* ).

Statement (iv) means that ∆ operates as an anti-linear isometry in every Ó∑Hilb

¡SL(2;Â)∂æ(™

*).

Proof Since the push-forward ¨* operates unitarily in L™(™*, dx), the restriction of ¨* to ∂(™

*)

equals t(¨*), implying i).

Let U be zonal, and of Weyl type -1. Then †kaU=†k(∂ap*U)=†kU*(∂ap)%, k∑K, a∑A. So,

for Á∑∂(™*) one has <†kaU»Á>=<U»a–™®Áka>, with Áka(y)=Á(kya¡), y∑™

*. Therefore, the kernel

associated to U is of the form <Ï*U»Á>=ªKdkªAdaÏ(kap)<U»Áka>=<U»ªKdkªAda Ïä(kap)Áka>. The

integral ªKdkªAda Ïä(kap)Áka evaluated at the point ñbp, ñ∑K, b∑A, equals ªKªAdkda(∆Ï)(kap)

Á(k¡ña¡bp). This means that under the identification ™*=K*A, kap=(k,a), one has

(VII.7.4.a) <Ï*U»Á>=<U»(∆Ï)K*A

Á> Ï,Á∑∂(™*), U∑∂æ(™

*)N, U type -1.

where K*A

denotes the standard convolution product on the direct product KA=K*A.

225

— Chapter VII SL(2;Â) acting on ™* —

The proof is now completed by imitating the proof of Proposition VII.7.1: just replace ^ by ∆ and 1

by -1 !

The type -1 Hilbert subspaces (so those with Weyl type -1 reproducing distributions) are not as easily

characterized as the bilaterally invariant Hilbert subspaces. We give a very general description, and

then we determine all the minimally SL(2;Â)–invariant elements within Hilb

¡SL(2;Â)∂æ(™

*).

Let Ó:∂(™*)îâ∂æ(™

*) be the reproducing operator of a Weyl type –1, SL(2;Â)–invariant

Hilbert subspace of ∂æ(™*). Let a belong to å, so let a be an SL(2;Â)–invariant operator in the

distributions. Then Definition VII.3.1 implies

(VII.7.4.b) aÓ=Ó(ta) (ta is the transpose of a).

This implies that a maps Óû:=u(∂) into itself. Assume a is real, that is, it commutes with the

conjugation TéâTä, T∑∂æ(™*). Let aû denote the restriction of a to Óû. Then (VII.7.4.b) implies that

aû is symmetric in the Hilbert space Ó, on the domain Óû. Indeed, ·aûÓÏ»ÓÁ‚Óû=·Ó(taÏ)»ÓÁ‚Óû

=

<Ó(taÏ)»Á>∂æ,∂=<aÓÏ»Á>∂æ,∂=<ÓÏ»taÁ>∂æ,∂=·ÓÏ»ÓtaÁ‚Óû=·ÓÏ»aûÓtaÁ‚Óû

, Ï,Á∑∂(™*).

For the operator a one can take, for example, v©, ©∑Â*, with v the unitarized right

representation of MA=Â* in ∂æ(™*). What one then obtains is a representation of Â* in the symmetric

operators in Ó, with common domain Óû. The argument can be reversed to show that this property

characterizes the type -1 spaces. In general, this is about all one can say, since there is no reason for Ó

to be v–invariant.

When Ó is minimal SL(2;Â)–invariant, and of Weyl type -1, much more can be said.

Proposition VII.7.5

The minimal SL(2;Â)–invariant elements in the cone Hilb

¡SL(2;Â)(∂æ(Â*

™)) are of the form

˚≈~¤[™πr~dr], with ~ a real number, and ˚≈

~ a K–invariant Hilbert subspace of ∂æ(K)≈.

They are also minimal in the cone Hilb

¡SL(2;Â)(∂æ(Â*

™)).

The proposition does not exclude the possibility that there exist elements in Hilb

¡SL(2;Â)(∂æ(Â*

™)) that

are extremal in Hilb˚SL(2;Â)(∂æ(Â*™)) but not in Hilb

¡SL(2;Â)(∂æ(Â*

™)), so there might be a difference

in this respect with the case w=1 (see Proposition VII.7.2).

226

— VII.7 Partial Symmetries; Subcones of Hilbert Subspaces —

Proof Let Ó be SL(2;Â)–minimal. According to the remarks just made, the restriction to the domain

Óº of the unitarized right representation v of Â* is a representation in the densely defined symmetric

(and therefore closeable) operators in Ó, commuting with the representation of SL(2;Â) in Ó. One can

now use Schur’s Lemma for closeable densely defined operators (Lemma IV.14.11) to see that

SL(2;Â) operates by scalars. So, there is a character ©≈~ of Â*, ≈∑”1,Í’, ~∑Ç, with the property that

v©S=©≈~S, S∑Óº, ©∑Â*. Since v© is symmetric, the character ©≈

~ must be real, so ~∑Â. Since Ó is

continuously imbedded in ∂æ(Â*™), one has in fact v©S=©≈

~S for all S∑Ó. This means that Ó consists

entirely of distributions homogeneous of degree ©≈~, so Ó is a Hilbert subspace of ∂æ(™

*)≈~=

∂æ(K)≈¤[™πr~dr]. Therefore, Ó is of the form ˚≈~¤[™πr~dr], for some Hilbert subspace ˚≈

~ of

∂æ(K)≈. The action of K is simply in the first variable, so, †k(S¤™πr~dr)=(LkS)¤™πr~dr, k∑K,

S∑∂æ(K), L the left regular representation of K in ∂æ(K). So, ˚≈~ must be K–invariant.

The final statement in the proposition is trivial. An extremal ray in a cone is extremal in

any subcone to which it belongs !

A distribution homogeneous of degree ©~≈ is of the form U=T¤™πr~dr, for some T∑∂æ(K)≈.

Definition VII.7.6 The distribution T∑∂æ(K) will be called the K–trace of U.

The definition is such that the K–trace of the Lebesgue measure dx is the normalized Haar measure

ä™däπ∆ on K.

Since ∂æ(™*)≈

~ is a closed subspace of ∂æ(Â*™), it contains Ó–° for every Hilbert subspace

Ó that it contains. In particular contains the reproducing distribution U of Ó. Let T be the K–trace of

U, and assume U is type -1. In (VII.7.4.a) take Ï=˚¤å, Á=¬¤∫, ˚,¬∑∂(K), å,∫∑∂(Â+). Then,

reworking the expression, one gets

(VII.7.6.a) <·(˚¤å)*(T¤™πr~dr)»(¬¤∫)>=<˚K*T»¬> ªº

°å(r)r~™πdrªº

°r~∫_(r)™πdr.

In general, the repoducing kernel associated to the one-dimensional Hilbert subspace “S‘ of

∂æ(≈), ≈ a manifold, is given by ˚(å¤∫_)=ä<äS…ä|äå><S|∫>. So, since ~ is real, in the expression

ªº

°r~å(r)™πdrªº

°r~∫_(r)™πdr one recognizes the reproducing kernel of the one-dimensional Hilbert

subspace “™πr~dr‘ of ∂æ(Â+). So it follows from (VII.7.6.a) that if T is of positive type, reproducing

the Hilbert subspace ˚ of ∂æ(K), then U reproduces ˚¤“™πr~dr‘. Reversal of this argument shows

227

— Chapter VII SL(2;Â) acting on ™* —

Proposition VII.7.7

A Weyl type -1 and zonal distribution homogeneous of degree ©~≈ is of positive type if and only if

its degree of homogeneity is real, and if its K–trace is a distribution of positive type on K. The

SL(2;Â)–invariant Hilbert subspace Ó of ∂æ(™*) reproduced by U is then of the form

˚¤“™πr~dr‘

with ˚ equal to the Hilbert subspace of ∂æ(K)≈ reproduced by the trace of U.

Propositions IV.7.5 and IV.7.7 make it possible to determine explicitly all the minimal SL(2;Â)-

invariant elements in Hilb

¡SL(2;Â)(∂æ(Â*

™)). To prepare for this, we first present a short treatment of

rotation invariant subspaces of ∂æ(K), with K the one dimensional group of rotations. In particular we

deal with Sobolev spaces.

In general, let T∑∂æ(K) be of positive type, so with its Fouriertransform Tfl, defined by

Tfl(n)=<T»ein∆>, being nonnegative. Then T reproduces the K–invariant Hilbert subspace

Ó= n¶∑Û

Tfl(n) “ein∆‘.

The spectrum of T, denoted ÛT, is defined as the support of Tfl, so ÛT=”n∑Û» Tfl(n)≠0’. Then Ó

equals the space of distributions S∑∂æ(K) whose spectra ÛS are contained in ÛT, and which satisfy

n∑¶ÛT

(Tfl(n))¡|Sfl(n)|™<°. The Hilbert structure on this space is given by

(VII.7.7.a) (S»R)Ó

=n∑

¶ÛT

(Tfl(n))¡Sfl(n) …Rfl…(…n), S,R∑Ó.

One shows that Ó is complete for the Hilbert norm. From (VII.7.7.a) it is clear that T belongs to Ó if

and only if its Fourier transform belongs to ñ1(Û).

All K–invariant Hilbert subspaces of ∂æ(K) are of the form described. This can be shown

using the fact that the K–invariant operators ∂(K)îâ∂æ(K) are the convolution operators by arbitrary

distributions (this follows from the theory in Chapter IV).

Standard translation invariant Hilbert subspaces of ∂æ(K) are the Sobolev spaces ßs, s∑Â

[131] . They can be defined by

ßs= n¶∑Û

(1+n™)–s“ein∆‘.

131 François Trèves, Linear Partial Differential Equations with Constant Coefficients: Existence,

Approximation and Regularity of Solutions (New York: Gordon and Breach, 1966).

228

— VII.7 Partial Symmetries; Subcones of Hilbert Subspaces —

For m∑ˆ the Sobolev space ß–m is the Hilbert subspace of ∂æ(K) reproduced by (1-îddî∆™-™)m∂º.

Standard properties are ߺ=L™(K;d∆), ßs≥ßt for s<t, and s§á°ßs=∂(K), sá

∞–°ßs=∂æ(K), so the

Sobolev spaces form a decreasing continuum of Hilbert spaces of increasingly regular distributions,

interpolating between ∂æ(K) and ∂(K). An m–th order constant coefficient differential operator maps

ßs into ßs–m.

We denote by ßs1 the closed subspace of ßs consisting of distributions which are even

with respect to rotations over an angle π. They are characterized by having their spectra contained in

2Û. By ßsÍ we denote the closed subspace of ßs consisting of those distributions that are odd with

respect to rotations over an angle π, and that have spectra contained in 2Û+1. One has ßs=ßs1@ßs

Í,

which is the isotypical decomposition of the representation of M in ßs.

The choice of the weight function (1+n™)–s is rather arbitrary:what matters mainly are its

asymptotic behaviour (which is of the order |n|-™s) and its support (which is all of Û). We make this

more precise.

Let Ó¡, Ó™≤∂æ(K) be two Hilbert subspaces of positive type. It can be shown that Ó¡≤Ó™ if

and only if there exists a positive real number å such that Ó¡¯åÓ™ [132] . Here Ó¡≤Ó™ is simply an

inclusion of vector spaces, while Ó¡¯åÓ™ has its usual meaning, that is, the reproducing operators Ó¡

and Ó™ of Ó¡ and Ó™ have the property that åÓ™–Ó¡ is of positive type. This is shown by using the

continuity of the inclusion Ó¡≤Ó™ as guaranteed by the Closed Graph Theorem, and by setting the

norm of this inclusion equal to √_å. Let T¡ and T™ be the reproducing distributions of Ó¡ and Ó™. Then

Ó¡≤Ó™ if and only if Tfl¡(n)¯å Tfl™(n) for all n∑Û. In particular, the spectrum of T¡ will then be

contained in the spectrum of T™, so ÛT¡≤ÛT™. Write Ó¡ŸÓ™ if Ó¡ and Ó™ are identical as vector

spaces. In view of the preceding remarks this is equivalent to the existence of a positive constant å

such that simultaneously Ó¡¯åÓ™ and Ó™¯åÓ¡, which is the same as saying that Tfl¡(n)¯å Tfl™(n) and

Tfl™(n)¯å Tfl¡(n). In particular, ÛT¡=ÛT™. So, in that case Ó¡ and Ó™ can be seen as one and the same

topological vector space, equipped with two different, yet equivalent norms.

132 Laurent Schwartz, "Sous-espaces Hilbertiens d'Espaces Vectoriels Topologiques et Noyaux Associés

(Noyaux Reproduisants)" Jour. Anal. Math. 13 (1964), pp. 115-256, Proposition 2 on page 137.

229

— Chapter VII SL(2;Â) acting on ™* —

Theorem VII.7.8 Complementary and Discrete Series

A) Weyl type -1 minimal zonal distributions of positive type are positive multiples of the

following:

i) äÌä(ä™™ä~

-)

1¤t1~–1 for -1<~¯1

ii) U–k:=·–i îÿÿ_t‚

k·2ͤ∂-_™πä1_i1¤Pv˚t–1‚

äU–k:=·i îÿÿ_t‚

k·2ͤ∂+_™πä1_i1¤Pv˚t–1‚ k∑ˆ, k˘0.

B) The minimal SL(2;Â)–invariant elements in the cone Hilb

¡SL(2;Â)(∂æ(Â*

™)) reproduced

by the distributions in A) are:

i.a) ˚1~¤[™πr~dr], for -1<~<1

with ˚1~ equal to the Sobolev space ß1

™~, but equipped with the equivalent Hilbert

structure

·S»R‚˚1

~=™√_πn

¶∑2Û

Ì(™–™~)ÌäÌ

(ä(™ä™nän+…–

™ä™~ä~…++

™ä™))Sfl(n) Räflä(än).

i.b) For ~=1 one has the one-dimensional space

î™î√¡__π“dx‘ ,

where dx is Lebesgue measure on Â*™.

ii) ˚Í–k

k,++¡¤“™πr–kdr‘

˚Í–k

k,+–¡¤“™πr–kdr‘ , k∑ˆ, k˘0.

with ˚Í–k

k,++¡ equal to ßÍ

–™kk+,

¡+, the space of the elements in the Sobolev space of order

–™k whose spectra are contained in ·2Û+2(1+(-1)k))‚§(k,°), and with ˚Í–k

k,+–¡ equal

to ßÍ–™

kk+,

¡–, the space of the elements in the Sobolev space of order –™k whose spectra

are contained in “2Û+2(1+(-1)k)‘§(–°,–k).

The inner products in these space are given by

·S»R‚˚Í

–k,±=

n|n

¶

∑|>™kˆ+1

…(…n…™…–…(…k…-…1…)…™…)…(…n

π…™…–…(…k…-…3…)…™…)….….….…(…n…™…-…1)

Sfl(n) Räflä(än) k even

·S»R‚˚1

–k,±=

n|n

¶

∑|>™kˆ

…(…n…™…–…(…k…-…1…)…™…)…(……n…™…–

π…(…k…-…3…)……™…)….….….…(…n…™…-…2…™…)|…n|Sfl(n) Räflä(än) k odd

(For k=0 the denominator in the first of these two expressions should be interpreted as 1)

230

— VII.7 Partial Symmetries; Subcones of Hilbert Subspaces —

Comment We have not been able to show uniqueness for the integral decompositions within the

cone Hilb ¡SL(2;Â)(∂æ(™

*)), in spite of the presence of the symmetry ∆.

The series of representations of SL(2;Â) in the subspaces under B.i.a is known as the

complementary series. In the next section we shall see that the representations for ~ and –~, 0<~<1,

are equivalent, but considering the Sobolev spaces ˚1~ one sees that these two realizations of one and

the same representation are very different. More precisely, Ó~1:=˚1

~¤[™πr~dr] for ~>0 consists of

functions that are locally L™, which is not true for Ó1–~.

The series under ii) is known as the discrete series, the case k=0 is known as the limit of

the discrete series [133] . On the right and left half-planes s>0 and s<0 the distributions U–k and Uì–k

coincide with multiples of the distributions 1¤(t+iº)–n–1:=uliè mº

1¤(t+iu)–n–1 and

1¤(t–iº)–n–1:=ulià mº

1¤(t+iu)–n–1 (as defined, e.g., in [134] ). Globally, however, they do not

have such a form. The distributions U–k and Uì–k for k>0 cannot be approached continuously by other

zonal distributions.

As a limit of the complementary series for ~è–1 one has the space

˚1–¡¤[r–¡dr],

with reproducing distribution of –¢îä…√1ä_äπ¤Pft–™

. Here ˚1–¡ is equal to ß1

–™,*, the space of the

elements in the Sobolev space on K of order –™ whose spectra are contained in 2Û-0. The Hilbert

structure on ˚1–¡ is defined by

·S»R‚˚1

–¡ = 4√_πnn≠

¶∑2

0Û

|n1_| Sfl(n) Räflä(än), S,R∑˚1

–¡

The space ˚1–¡ is reducible, its minimal decomposition is

˚1–¡=4√_π˚1

–¡,+@ 4√_π˚1–¡,–,

with ˚1–¡,+ and ˚1

–¡,– the two conjugate spaces obtained by taking k=1 in B.ii). This follows

from the fact that 4√_πU¡@4√_πîU¡=–¢îä…√1ä_äπ1¤Pft–™.

Proof

The Fourier coefficients of the traces of the holomorphic families äÌä(ä™™ä~

-)1¤t1

~–1 and äÌä(ä™äî~™î+iî™)

1¤tÍ~–1

are as follows

133 Mitsuo Sugiura, Unitary Representations and Harmonic Analysis: An Introduction (Amsterdam: North

Holland, Tokyo: Kodansha, 1975, 2nd Ed. 1990), Chapter V, §3.134 I.M.Gelfand and G.E. Shilow, Generalized Functions,Vol.1: Properties and Operations,Section I.4.4.

231

— Chapter VII SL(2;Â) acting on ™* —

(VII.7.8.a) <äÌä(ä™™ä~

-)

sin1~–1∆ ä™

däπ∆»e

in∆>=î™î√¡__π äÌä(ä™ä…–

1ä™…~)

ÌäÌ

(ä(™ä™nän–ä…+

™ä™~ä~…++

™ä™))È2Û(n) ,

(which is even with respect to néâ-n)

(VII.7.8.b) < äÌä(ä™äî~™î+iî

™)sin Í

~–1∆ä™däπ∆»e

in∆>= î™î√¡__π äÌä(ä1ä…–

-1ä™…~)

ÌäÌ

(ä(™ä™nän–ä…+

™ä™~ä~…++

™ä™))È2Û+1(n),

(which is odd with respect to néâ-n).

These formulas can be derived, e.g., from the formulas in [135] . These Fourier coefficients should be

interpreted as holomorphic functions in ~∑Ç, the normalization being such that the poles of nominator

and denominator cancel each other out (for any fixed n). The coefficients are real–valued for ~∑Â, and

positive for large positive n. The first question is for which values of ~ the Fourier transforms are

nonnegative for all n.

For ≈∑Mfl=1,Í, ~∑Ç, w∑W let ∂æ(Â*™)≈

~,N,w denote the space of zonal distributions

homogeneous of degree ©≈~, and of Weyl type w. Theorem VII.4.2 and Corollary VII.4.4 imply that

∂æ(Â*™)≈

~,N,-1 is one-dimensional except at the points (≈,~)=(Í~–¡,~) with ~ a negative integer. The

exceptions include the case (≈,~)=(Í,0). The reason for this is that according to Theorem VII.4.2 the

space ∂æ(™*)ºÍ,N is spanned by 1¤

Pft¡ and ͤ∂, and both of these are type -1 according to

Corollary VII.4.4. The fact that (1,0) is in this respect not exceptional is that 1¤Pf|t¡|–2log|s|¤∂ is

not type -1.

For the non-exceptional points, so those where dim∂æ(Â*™)≈

~,N,w=1, there is little difficulty.

From the oddness of the Fourier transform in (VII.7.8.b)

it is

immediate that for ≈=Í there are no

candidates for positivity. Assume ≈=1, and the pair (≈,~) not exceptional, so ~„(2Û+1)<0.

Equivalently, (2+2~)„Û¯0. Denote the right hand side in (VII.7.8.a) by F~(n), and, the

Fouriertransform being even, consider only n∑2Û, n˘0. One has the recurrence relation

(VII.7.8.c) F~(n+2)= …nn…–+

1…1+…+

~_~F~(n) n∑2Û, n˘0

Moreover, as long as (2+2~)„Û¯0 one has F~(0)=î™î√¡__π äÌä(ä™ä…+

1ä™…~)≠0. In view of (VII.7.8.c) this implies

that for |~|>1 the values of F(2) and F(0) are non-zero and have opposite signs, so there are no

candidates for positivity there. For |~|<1 the fact that F~ is even, the fact that F~(0)=î™î√¡__π äÌä(ä™ä…+

1ä™…~)>0

135 Niels Nielsen, Handbuch der Gammafunction (Leipzig: Teubner, 1906), pp. 158-59. The formulas can be

represented in different shapes, see Tom H. Koornwinder, “The Representation Theory of SL(2;Â); a Non-

Infinitesimal Approach," L’Enseignement Mathématiques 2nd Series 28 (1982), pp. 53-87, on p. 71.

232

— VII.7 Partial Symmetries; Subcones of Hilbert Subspaces —

and formula (VII.7.8.c) together imply that F~ is strictly positive for all n.

Applying Proposition VII.7.7 one obtains a series of SL(2;Â)–invariant Weyl type -1

Hilbert subspaces Ó~1:=˚~

1¤[™πr~dr], where ˚~1 denotes the K–invariant Hilbert subspace of ∂æ(K)

reproduced by äÌä(ä™™ä~

-)

sin1~–1∆ ä™

däπ∆. The spaces ˚~

1 can be determined as follows.

Consider the expression ÌäÌ

(ä(™ä™nän–ä…+

™ä™~ä~…++

™ä™)), strictly positive for |~|<1 and n∑2Û. From the general

asymptotic behaviour

Ì(m+¬)mŸá° m¬Ì(m) ¬∑Ç (see [136] )

it follows that

(VII.7.8.d) ÌäÌ

(ä(™ä™nän–ä…+

™ä™~ä~…++

™ä™))nn∑Ÿá2°Û 2

~n–~.

Since ÌäÌ

(ä(™ä™nän–ä…+

™ä™~ä~…++

™ä™)) is even, the asymptotic behaviour for |n|á°, n∑2Û, is of the order 2~|n|–~. Let ˚1

~

be the Hilbert subspace of ∂æ(K) reproduced by äÌä(ä™™ä~

-)

sin1~–1∆ ä™

däπ∆. Then (VII.7.8.d) proves that, in

view of the considerations leading up to the theorem, one has

(VII.7.8.e) ˚1~

Ÿ ß1™~

–1<~<1.

with ß1™~

the even Sobolev space of order ™~. So, ˚1~

and ß1™~

denote one and the same Banach space,

equipped with two equivalent Hilbert norms.

One particular consequence of (VII.7.8.e) is that

(VII.7.8.f) (Ó1~)–°

=∂æ(™*) 1

~ –1<~<1.

The point is that ß1™~

contains all of ∂(K)1. Indeed, ∂æ(™*) 1

~ is closed and SL(2;Â)–invariant, so for

any ƒ∑∂(SL(2;Â)), U∑∂æ(™*) 1

~, the distribution ªSL(2;Â)ƒ(g)†gUdg belongs to ∂æ(™*)1

~=

∂æ(K)1¤™πr~dr, and, being smooth, it belongs to ∂(K)1¤™πr~dr, and therefore to ß1™~

¤r~dr=Ó1~.

Then use criterion (VII.6.0.b).

Equation (VII.7.8.f) gives a means to show that the Ó1~ are SL(2;Â)–irreducible. Indeed, for

~≠0 the space (Ó~1)N

–° of N–fixed co–ç°–vectors associated to Ó~1 equals ∂æ(™

*) 1~,N, and the latter

equals Ç(1¤t1~–¡)@Ç(s 1

~¤∂) according to Theorem VII.4.2. Since for ~ real ·1¤t1~–¡‚Ÿ=1¤t1

~–¡ and

(s1~¤∂)Ÿ=(s1

–~¤∂) one has (Ó~1)N

–°§ “(Ó~1)N

–°‘Ÿ=Ç(1¤t1~–¡). This space being one-dimensional,

the representation space Ó~1 is irreducible, according to Theorem VII.6.2.(i). For (≈,~)=(1,0) we can

refer back to Theorem VII.7.3.

136 Niels Nielsen, Handbuch der Gammafunction, Chapter 1.

233

— Chapter VII SL(2;Â) acting on ™* —

For ~=1 there is little need to use (VII.7.8.a). The space ∂æ(Â*™)≈

~,N,–1 equals Çdx, and the space

reproduced by an invariant measure dx equals “dx‘. This completes the proof of A(i) and B(i) in the

theorem.

The family äÌä(ä™™ä~

-)

sin1~–1∆d∆ reduces to a distribution supported by ”0,π’ for ~∑2Û, ~¯0. This

distribution being even with respect to translations through π, and also with respect to the reflection

∆éêâ–∆, its Fourier transform is an even polynomial P, restricted to 2Û. More precisely, at ~=-2m,

m∑ˆ, m˘0, formula (VII.7.7.a) takes the form:

2(-1)mä(ä2äm…mä)

!!<∂(2m)û(sin∆

ä™däπ∆)»e

in∆>= 䙡äπ(ä2ä

m…mä)

!!(n™–(2m-1)™)(n™–(2m-3)™)...(n™-1)È2Û(n).

(the pull-back is taken with respect to the measure ä™däπ∆, for the factor (-1)mä(ä2ä

m…mä)

!! see (VII.4.1.c)). For

m fixed, this expression is of indefinite sign in n, except when m=0. This merely confirms the results

above. However, the “twisted” distribution äÌä(ä™™ä~)

Í(cos∆)sin1~–1∆

ä™däπ∆ for ~=–2m (the trace on the

circle of the homogeneous distribution äÌä(ä™™ä~

)ͤt1

~–1 for ~=–2m), is the same even polynomial, but

now evaluated at odd integers. So

2(¡)mä(ä2äm…mä)

!!<Í(cos∆)∂(2m)û(sin∆

ä™däπ∆)»e

in∆> = 䙡äπ( ä2ä

m…mä)

!!(n™–(2m-1)™)(n™–(2m-3)™)...(n™-1)È2Û+1(n).

The support of this polynomial is (2Û+1)§”n∑Û»|n|>™m’. Up to a constant this transform coincides

with the transform of äÌä(ä™äî~™î+iî™)

sin Í~–1∆

ä™däπ∆, except that the latter is odd with respect to the reflection n

éêâ –n. The two transforms agree for positive n (see formulas (VII.7.8.a,b)). It follows from this that

within the 2–dimensional space ∂æ(Â*™)Í

–,™mN,–1 the elements of positive type are positive sums of two

extremals obtained by adding and subtracting, so take

2(-1)m(ä2äm…mä)

!!ͤ∂(2m)±2πä

i(-1)m(m!)1¤Pf˚t–2m–1

with Fourier transforms given by

äπ¡(ä2ä

m…mä)

!!(n™–(2m-1)™)(n™–(2m-3)™)...(n™-1)È

(ˆ§(2Û+1))(±n) .

Multiplying by (2m)!.(m!)¡ one obtains the distributions

U–™m= (-1)m

“2ͤ∂(2m)–_™πä

1_i(2m)!1¤Pf˚t–2m–1‘

and its complex conjugate. These allow the convenient expression given in the theorem.

234

— VII.7 Partial Symmetries ; Subcones of Hilbert Subspaces —

Let ßÍ–m,+ denote the closed subspace of the Sobolev spaces ßÍ

–m consisting of distributions with

spectra contained in (2m,°), and let ßÍ–m,– denote the closed subspace of the Sobolev space ßÍ

–m

consisting of distributions with spectra contained in (–°,–™m). Considering the asymptotic behaviour

(which is described by VII.7.8.d) as well as the spectra of the distributions U–™m and Uì–™m

one finds

that the Hilbert subspaces they reproduce are of the form ßÍm,±¤“r–™mdr‘, with the equivalent inner

product as given in the theorem. This settles the matter for the exceptional points (~,Í) with ~∑2Û,

~¯0.

For the other exceptional points the argument is similar. The result is that for m∑Û>0 within

the 2–dimensional space ∂æ(Â*™)1

–™,mN,

+-

¡1 the elements of positive type are generated by the two

extremals

U–™m+1=(-1)m

“2iͤ∂(2m–1)+

䙡äπ(2m–1)!1¤Pf˚t–2m‘

and its conjugate.

Let ß1–m+™,+ denote the closed subspace of the Sobolev space ß1

–m+™ consisting of

distributions with spectrum contained in (2m,°), and let ß1–m+™,– denote the closed subspace of the

Sobolev space ß1–m+™ consisting of distributions with spectra contained in (–°,–™m). The subspaces

that the U1–™m+1,± reproduce are of the form ß1

–m+™,±¤“r–™mdr‘, with the inner product as in the

theorem.

The irreducibility of the spaces ˚Í–k

k,+±¡¤“™πr–kdr‘ is shown by using the same type of argument as

for the series for -1<~<1. One shows that ·˚Í–k

k,+±¡¤“™πr–kdr‘‚–° equals ∂æ(™

*)Í–k

k,+±¡, with

∂æ(™*)Í

–kk,+±¡ denoting the space of distributions homogeneous of degree (Ík+1,–k) whose K–traces

satisfy the same type of condition on their spectra as the distributions in ˚Í–k

k,+±¡ (so, without the

Sobolev growth condition). For n>0 the space of zonal elements in ∂æ(™*)–Í

kk+¡ is three-dimensional

according to Theorem VII.4.2. Slightly changing the basis described in Theorem VII.4.2, one sees

that ∂æ(™*)

–Í

kk+¡,N is spanned by U–k, U_–k, and by sÍ

–kk+¡¤∂, whose K–trace is ä™

¡äπ(∂º+(¡)k+¡∂π).

The spectrum of 䙡äπ

(∂º+(¡)k+¡∂π) is all of ·2Û+2(1+(-1)k))‚, while the spectra of the K–traces of

U–k and U_–k are ·2Û+2(1+(-1)k))‚§±(k,°) respectively. It follows from this that ·˚Í–k

k,+±¡¤

“™πr–kdr‘‚N–°=∂æ(™

*)Í–k

k,+±¡,N is one-dimensional, in other words, the only N–fixed co–ç° vectors

associated to the Hilbert spaces ˚Í–k

k,+±¡¤“™πr–kdr‘ are multiples of their reproducing distributions.

According to Theorem VII.6.2 this proves at once that they are irreducible and of multiplicity one !

235

VII.8 Minimal Invariant Hilbert Subspaces

Traditionally, one considers the function spaces ∂(™*)≈

~ (sometimes completed) and tries to see

whether there exists an invariant bilinear (or rather: sesquilinear form), making yet other completions.

We approach this subject from another direction: we set out to describe all minimal invariant Hilbert

subspaces of the distributions on ≈=™* in terms of their reproducing distributions. We show that for

every Hilbert subspace there exists at least one equivalent Hilbert subspace consisting of

homogeneous distributions. We then derive explicit formulas for the realizations, in terms of the

corresponding minimal zonal distributions of positive type.

For the definition of Ω–homogeneous distributions see Section VII.5.

Theorem VII.8.1 Subrepresentation Theorem for ¶¾(˚å

™*

)

Every minimal SL(2;Â)–invariant Hilbert subspace of ∂æ(™*) consists of Ω–homogeneous

distributions. Moreover, for every minimal Hilbert subspace of ∂æ(™*) there exists at least one

equivalent Hilbert subspace of ∂æ(™*) consisting of homogeneous distributions.

More precisely, for every minimal SL(2;Â)–invariant Hilbert subspace Ó of ∂æ(™*)

there exist a real number ¬ and a character ≈∑MÀ=”1,Í’, such that Ó is contained in ∂æ(™*)≈

[¬].

Moreover, ∂æ(™*)≈

[¬] contains at least one Hilbert subspace Ó¡ equivalent to Ó with

Ó¡ ≤ ∂æ(™*)≈

~ ≤ ∂æ(™*)≈

[¬], and ~™=¬.

Proof Being minimal, Ó is realized in Ω–homogeneous distributions. That ¬ must be real was part of

Corollary VII.5.3. Moreover, ∂æ(™*)≈

[¬] being closed, (VII.6.0.b) implies that Ó–°≤ ∂æ(™*)≈

[¬]. In

particular, ÓN–°≤∂æ(™

*)≈[¬,N] . Furthermore, according to Theorem VII.5.4 the space ∂æ(™

*)≈[¬,N] of

zonal distributions which are Ω–homogeneous of fixed degree is 5-dimensional at the most. So, ÓN–°

is finite dimensional.

Since MA=Â* normalizes N it operates in ÓN–° (through the restriction of the

quasi–regular representation †). But ÓN–° is a finite dimensional complex vector space. The group

MA is abelian, and an abelian group of linear transformations of a finite-dimensional complex vector

space always possesses a common eigenvector (just take a non-zero vector in a minimal subspace).

So, ÓN–° contains at least one vector V≠0 that satisfies

236

— VII.8 Minimal Invariant Hilbert Subspaces —

†©V=© ≈åV ©∑Â*

for some character ©≈å of Â*, with ≈∑”1,Í’, å∑Ç. Rewrite this as ∂p©*V=©≈

åV, so that ŸV*Ÿ∂p©=©≈åäŸV,

that is, ŸV*|©|¡∂p©¡=©≈åä+1ŸV, which is saying that ŸV is homogeneous of degree ©≈

åä+1 with respect to

the unitarized right action of Â*. Obviously, ∂(™*)*ŸV≤∂æ(™

*) ≈åä+1. Now ∂(™

*)*U is dense in Ó, so

that ∂(™*)*ŸV=∂(™

*)*UU*ŸV= †V(∂(™

*)*U) is dense in Ó¡:=†V(Ó). Since ∂æ(™*) ≈

åä+1 is closed, it

follows that Ó¡≤∂æ(™*) ≈

åä+1. Set ~=åä+1. Then ~™ equals ¬ (see Section VII.5). Finally, since Ó is

irreducible, †V:Óììâ∂æ(™*) is an imbedding, so Ó¡ is indeed equivalent to Ó !

All minimal zonal distributions can now be calculated, in terms of the Ω–homogeneous distributions

already determined in Theorem VII.5.4. For notations see Section VII.2.

Theorem VII.8.2 Minimal Zonal Distributions of Positive Type

The following is a complete list of all minimal zonal distributions of positive type on ™* under the

action of SL(2;Â). In this list Ô signifies the Ω-character of the distributions involved.

πAs1–~¤∂ –™π~B.1¤t1

–~–1+™π~C.1¤t1~–1+ πDs 1

~¤∂,

Ô=(1,~™), ~™<0;multiplicity 2 with A,D˘0, B= Cä, BC= îtîa™înπîî™îπ~î~

AD

πA.1¤∂–B·2log|s|¤∂-1¤Pfä|_1 tä|‚+C·2log|s|¤∂+1¤Pfä|_

1tä|‚+D· π_

4(1¤Pfä|_1tä|log|t|–log™|s|¤∂)+3_

1π.1¤∂‚

Ô=(1,0); multiplicity 2 with A,D˘0, B= Cä, BC=AD

πAs1–~¤∂ –™π~B.1¤t1

–~–1+™π~C.1¤t1~–1+ πDs 1

~¤∂

Ô=(1,~™), 0<~™<1; multiplicity 2 with B,C˘0, A= D_, BC= îtîa™înπîî™îπ~î~

AD

πAsÍ–~¤∂ –B.1¤tÍ

–~–1+C.1¤t Í~–1+ πDs Í

~¤∂

Ô=(Í,~™), ~™<0; multiplicity 2 with A,D˘0, B= Cä, AD=îtîa™înπîî™îπ~î~

BC

A·–iîÿÿ_t‚

k·πͤ∂+i.1¤Pv˚t–1‚ and D·i îÿÿ_t‚

k·πͤ∂–i.1¤Pv˚t–1‚

Ô=(Ík–1, k™), k∑Û, k™≠1; multiplicity 1 with A, D > 0

A·–πiͤ∂æ–1¤Pv˚t–™‚ and D ·πiͤ∂æ–1¤Pv˚t–™‚ and E dx

Ô=(1,1); multiplicity 1 with A, D, E > 0.

For two distributions in this list to reproduce equivalent Hilbert subspaces their Ω-characters must

be equal. When two distributions in this list indeed have the same Ω-character Ô then they

reproduce equivalent Hilbert subspaces when Ô=(1,~™), ~™<1, or when Ô=(Í,~™), ~™<0. Otherwise

they reproduce equivalent Hilbert subspaces only if they are proportional.

237

— Chapter VII SL(2;Â) acting on ™* —

Proof Assume ~≠0. Set C≈~=√_π îî

ÌÌî( (î™ ™î–~î™)î~) for ≈=1, and C≈

~=√_πiäÌÌ

(ä(1ä™äî+–î™™î™î~~)) for ≈=Í. Define the

meromorphic families

(VII.8.2.a) S≈~:=πs ≈

~¤∂ ,

T≈~:=C ≈

~.1¤t≈~–1 , ≈=1,Í, ~„Û.

Then S Ÿ≈~=S≈

–~_, T Ÿ≈~=T≈

~_.

First take ~=iµ, µ∑Â. Let Ó≈–iµ and Ó≈

iµ be the two Hilbert subspaces reproduced by S≈–iµ

and S≈iµ respectively. According to Theorem VII.7.3 both spaces are irreducible. Moreover, since

T≈–iµ∑·Ó≈

–iµ‚N–°, and ·T≈

–iµ‚Ÿ=T≈iµ∑·Ó≈

iµ‚N°, it follows by Theorem VII.6.2.(ii) that Ó≈

–iµ and Ó≈iµ

are equivalent [137] . It follows by the same theorem that there exists a constant A≈iµ such that

(VII.8.2.b) T≈–iµ

S≈*–iµ T≈

iµ=A ≈iµ

S≈iµ

Since 1¤t≈–iµ–1=·sin ≈

–iµ–1(∆)ä™däπ∆‚¤·™πr–iµdr‚=·sin ≈

–iµ–1(∆)ä™däπ∆¤∂¡‚*

·2(∂º+≈(¡)∂π)¤™πr–iµdr‚=·sin ≈–iµ–1(∆)

ä™däπ∆¤∂¡‚*S ≈

–iµ, one obtains from (VII.8.2.b) and

(VII.6.1.e) that

A≈iµ

2(∂º+≈(¡)∂π)¤™πriµdr=·C≈–iµsin≈

–iµ–1(∆)ä™däπ∆¤∂¡‚*·T ≈

iµ‚

=·C ≈–iµsin≈

–iµ–1(∆)ä™däπ∆¤∂¡‚*·C ≈

iµsin≈iµ–1(∆)

ä™däπ∆¤™πriµdr‚

=·C≈

–iµsin≈–iµ–1(∆)

ä™däπ∆*C ≈

iµsin ≈iµ–1(∆)

ä™däπ∆‚¤™πriµdr,

so that A≈iµ is determined by

A≈iµ 2(∂º+≈(¡)∂π)=

C≈–iµsin≈

–iµ–1(∆)ä™däπ∆*C ≈

iµsin ≈iµ–1(∆)

ä™däπ∆.

Here 2(∂º+≈(¡)∂π) is the unit element in the convolution algebra of distributions on the circle that are

even (respectively odd) with respect to ∆ éêâ∆+π, so its Fourier transform is È2Û+2(1+≈(-1)). Use

the Fourier transforms in (VII.7.8.a,b) to show that A≈iµ=1. So we have

(VII.8.2.c) T≈–iµ

S≈*–iµ T≈

iµ=S ≈iµ ≈=1,Í, µ∑Â.

According to Theorem VII.6.2 the Hilbert subspaces equivalent to Ó≈–iµ are generated by

137 It will be seen that these arguments are similar to arguments used by F. Bruhat in his “Sur les

Représentations Induites des Groupes de Lie,” Bull. Soc. Math. France 84 (1956), pp. 97-205.

238

— VII.8 Minimal Invariant Hilbert Subspaces —

·Ó≈–iµ‚N

–°. Set V=å.S≈–iµ+∫.T≈

–iµ. Then according to Theorem VII.6.2 the reproducing distribution

of †V(Ó≈–iµ) is

·å.S ≈–iµ+∫.T≈

–iµ‚S≈

*–iµ

·_å.S ≈–iµ+∫_.T≈

iµ‚=|å|™S≈–iµ+_å∫.T≈

–iµ+ å∫_.T≈iµ+ |∫|™ S≈

iµ.

It follows that every distribution of the form

(VII.8.2.d) a.S≈–iµ+b.T ≈

–iµ+c.T≈iµ+ d.S ≈

iµ a,d˘0, b= cä, ad=bc≠0

is a minimal distribution of positive type, and that the Hilbert subspaces reproduced by distributions

of this form are equivalent to Ó≈–iµ (for fixed µ and ≈).

It follows from Theorem VII.8.1 that if Ó is a minimal Hilbert subspace of ∂æ(™*), and if

its Ω–character is ©≈¬ with ¬=(iµ)™, then Ó is equivalent to either Ó≈

–iµ or Ó≈iµ, and since these two

are equivalent Ó is equivalent to both. So, the reproducing distribution of Ó is of the form (VII.8.2.d).

Using (VII.8.2.a) the form (VII.8.2.d) is easily translated into the forms for Ô=(≈,~™), ≈=1,Í, ~™<0 in

the theorem, using the identy Ì(z)Ì(1-z)= îîsîiînπî(îπîz).

Next, consider the case 0<~™<1. Let Ó1–~ be the space reproduced by T1

–~. Then according

to (VII.7.8.f) one has (Ó1–~

)–°=∂æ(™*) 1

–~, so that the space of N–fixed co–ç°–vectors associated

to Ó1–~

is spanned by S1–~ and T1

–~. It follows that S1–~ gives rise to another realization of Ó1

–~. One

then shows that

(VII.8.2.e) S1–~

T1*–~

S1~=T 1

~

Informally, (VII.8.2.c) with ≈=1, ~=iµ yields

S1–~

T1*–~

S 1~=S1

–~

T1*–~

T1–~

S1*–~ T1

~=T 1~.

A correct proof can be given imitating the argument following (VII.8.2.b). The rest of the argument

for 0<~™<1 is as above, yielding that

(VII.8.2.f) a.S≈–iµ+b.T ≈

–iµ+c.T≈iµ+ d.S ≈

iµ, b,c˘0, a= dä, ad=bc≠0

is a minimal distribution of positive type, and that the Hilbert subspaces reproduced by distributions

of this form are equivalent to Ó≈–iµ (for fixed µ and ≈). This gives the distributions for Ô=(1,~™),

0<~™<1.

For ~=k∑Û, ≈=Ík+1 all minimal homogeneous Hilbert subspaces of ∂æ(™*)≈

[~™] have a

one-dimensional space of N–fixed co-ç°–vectors (according to the proof of Theorem VII.7.8). This

implies that the only Hilbert subspaces of ∂æ(™*) equivalent to a homogeneous Hilbert subspaces Ó

239

— Chapter VII SL(2;Â) acting on ™* —

of ∂æ(™*)≈

[~™] are the positive multiples of Ó. So ~=k∑Û, ≈=Ík+1 we can refer back to Theorem

VII.7.8.

What remains is the case ≈=1, ~=0 (induction of the trivial character). We sketch a proof.

According to Theorem VII.4.3 the space ∂æ(™*)º1,N

of zonal distributions homogeneous of degree ©º1

is spanned by π.1¤∂ and 1¤Pfä|_1tä|–2log|s|¤∂. However, since (π.1¤∂)Ÿ=π.1¤∂ and ·1¤Pfä|_

1tä|-

2log|s|¤∂‚Ÿ=1¤Pfä|_1tä|+2log|s|¤∂ it follows that within ∂æ(™

*)º1,N it is only the real multiples of

π.1¤∂ that have the Hermitian symmetry. This means that there can only be a single ray of Hilbert

subspaces within ∂æ(™*)1

º (the space of distributions homogeneous of degree ©º1), generated by the

Hilbert subspace Ó reproduced by π.1¤∂ (Ó equals L™(K;ä™däπ∆)1¤“™πdr‘, according to the proof of

Theorem VII.7.3). Then Theorem VII.8.1 implies that every minimal Hilbert subspace of ∂æ(™*)≈

[º]

(the space of distributions which are Ω–homogeneous of degree ©º1) is equivalent to Ó. So it will do to

calculate all Hilbert subspaces equivalent to Ó. Since (VII.7.8.f) yields (Ó1º)–°=∂æ(™

*) 1~, Theorem

VII.4.2 yields (Ó1º)N–°=Ç(1¤∂)@Ç(1¤Pfä|_

1tä|-2log|s|¤∂). The remaining difficulty is in determining

(1¤Pfä|_1tä|–2log|s|¤∂)

π.1*¤∂

(1¤Pfä|_1tä|+2log|s|¤∂).

To do this, use the fact that for ~ tending to zero (1¤t1~–¡–~_

2s~1¤∂) tends to 1¤Pfä|_

1tä|–2log|s|¤∂ (see

(VII.5.4.a)). Take ~ real, 0<~™<1. Since 1¤t1~–¡–~_

2s~1¤∂ is a distribution vector belonging to the

Hilbert space reproduced by 2π~.1¤t1~–¡, the reproducing distribution of the equivalent Hilbert

subspace associated to1¤t1~–¡–~_

2s~1¤∂ equals

(VII.8.2.g) (1¤t1~–¡–~_

2s~1¤∂)

™π~1*¤t1

~–¡(1¤t1~–¡–~_

2s1–~¤∂)

= π_1 ~_

2(1¤t1~–¡–î

tîa™înπîî™îπ~î~

1¤t1–~–¡)– π_

1~î4-™

(s~1¤∂+s1

–~¤∂)

(this is based on (VII.8.2.e)). Using the series at (VII.5.4.a) and the Laurent series for îtîa™înπîî™îπ~î~ one

shows that taking the limit of (VII.8.2.g) for ~=0 leads to the identity

(1¤Pfä|_1tä|–2log|s|¤∂)

π.1*¤∂

(1¤Pfä|_1tä|+2log|s|¤∂)= π_

4(1¤t1¡log|t|–log™|s|¤∂) + 3_

1π(1¤∂).

Which is enough !

240