University of Groningen
Convolution on homogeneous spacesCapelle, Johan
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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
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— 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 !
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