Invariant Differential Operators and Spherical - Project Euclid

TOKYO J. MATH. VOL. 15, No. 1, 1992

Invariant Differential Operators and Spherical Sections on a Homogeneous Vector Bundle Katsuhiro MINEMURA Japan Women’s University (Communicated by Y. Shimizu)

Introduction. In this paper we study the structure of the algebra of invariant differential operators on a homogeneous vector bundle over a reductive coset space, define the eigensections and spherical sections, and determine the dimension of the space of the spherical sections in case of a vector bundle over a Riemannian symmetric space. Let $G$ be a connected Lie group with Lie algebra and $K$ be a closed subgroup with Lie algebra . We assume that $G/K$ is reductive, that is, there exists an $Ad(K)-$ (direct sum). Let and be the in such that invariant subspace be the universal enand complexifications of and respectively. Let the symmetric algebra of , and respectively, and veloping algebra of by 1 $=1,$ $x^{T}=-x,$ $(xy)^{T}=y^{T}x^{T}$ defined the canonical anti-automorphism of . Let be a representation of $K$ on a complex vector space $V$ of finite diin mension, and let $J$ be the kernel of be its differential representation of $J$ be the homogeneous vector bundle . denotes the image of under T. Let $G/K$ of G-invariant differential operalgebra over associated to and be the and K-invariants ators on . In \S 1 we establish a linear isomorphism between in $S(p_{c})\otimes EndV$ (Proposition 1.2). Then under a certain condition, we have an algebra (Theorem 1.3). isomorphism . In \S 2, we give a Let be a finite-dimensional representation of the algebra of type (Definition 2.1) and give a definition of a definition of an eigensection of (Definition 2.4) which is a generalization of the so-called (zonal) spherical section spherical functions ([5], [7]). In particular when $K$ is compact, we can give an upper bound of the dimension of the space of spherical sections (Theorem 2.5). In \S 3 we determine the dimension of the space of spherical sections by using the Poisson transform (Definition 3.2) when $G/K$ is a Riemannian symmetric space $\mathfrak{g}$

$\mathfrak{k}$

$p$

$\mathfrak{g}=f+p$

$\mathfrak{g}$

$\mathfrak{k}$

$U(\mathfrak{g}_{c})$

$p_{c}$

$\mathfrak{p}$

$\mathfrak{g},$

$S(p_{c})$

$\mathfrak{k}_{c}$

$\mathfrak{g}_{c}$

$T$

$p_{c}$


$(x, y\in \mathfrak{g}_{c})$

$\tau$

$ d\tau$

$U(\mathfrak{k}_{c})$

$\mathfrak{k}_{c}$

$\mathfrak{g}_{c},$


$ d\tau$


$J^{T}$

$E_{\tau}$

$\tau$

$D(E_{\tau})$

$D(E_{\tau})$

$E_{\tau}$

$D(E_{\tau})\cong U(\mathfrak{g}_{c})^{K}/U(\mathfrak{g}_{c})^{K}\cap U(\mathfrak{g}_{c})J^{T}$

$D(E_{\tau})$

$\chi$

$E_{\tau}$

(Theorem 3.5). Received May 13, 1991 Revised August 8, 1991

$\chi$

232

KATSUHIRO MINEMURA

We shall use $C$ for the field of complex numbers. If $M$ is an analytic manifold, 1 a differential operator and $f$ a differentiable function, the value of $Df$ at in $Mi$ often denoted by $f(x;D)$ . If $E$ is an analytic vector bundle over $M$, we denote by the space of hyperfunctions of type $E$ over $M$ (for details, see [13]). In this paper shall usually call a hyperfunction of type $E$ over $M$ a hyperfunctional section (or a hyperfunction) of $E$ over $M$ . For a group $G$ and G-modules $V$ and $W$, let denotI $V$ the set of G-invariants in and let $Hom_{G}(V, W)$ denote the set of homomorphism which commute with the action of $G$ . For a vector space $H$, let denote the dua spaoe of $H$ and let denote the canonical bilinear form on $H^{*}xH$. $x$

$\theta(E$

$w($

$simpl\urcorner$

$V^{G}$

$H^{*}$

$\langle, \rangle$

1.

The algebra of invariant differential operators.

In this section we assume that $G$ is a connected Lie group with Lie algebra is a closed subgroup of $G$ with Lie algebra such that $G/K$ is reductive, that is, therI exists an $Ad(K)$ -invariant subspace complementary to in . We fix such once an for all. We keep to the notation in the introduction. Let be a representation of $K$ on a complex vector space of finite dimension an let be the vector bundle over $G/K$ associated to . Let be the space $G$ functions on with values in and let be the space of the elements in such that $f(gk)=\tau(k^{-1})f(g)$ for all in $G$ and all in $K$. We identify th $\mathfrak{g}an($

$K$

$f$

$f$

$p$

$V_{f}$

$\tau$

$E_{f}$

$C^{\infty}(G, V_{\tau})$

$u^{\sim}$

$E_{f}$

at

$j$

$k$

$g$

$eK$

with

$V_{\tau}$

and define

lies in and . Hereafter we identify by Let act on $C^{\infty}(G, \tau)$

$C^{\infty}(E_{\tau})$

$u\mapsto u^{\sim}$

$0$

$C^{\infty}(G,\tau)$

$V_{\tau}$

$u^{\sim}(g)=g^{-1}u(gK)$

is isomorphic to with

$C^{\infty}(G, \tau)$


$($

$($

$C^{\infty}(G, V_{f})$

$\tau$

$C^{\infty}$

fiber of

$\mathfrak{p}$

$\mathfrak{g}$

for

$g$

in

. and in by the correspondenc

$G$


$($

$u$


$The\iota$

$($

.

$C^{\infty}(G, V_{\tau})$

$\mathfrak{g}$

$Xf(g)=(\frac{d}{dt}f$

(

$g$

exp $tX$)

$)_{t=0}$

$(X\in \mathfrak{g})$

.

Then this action induces a representation of on . Let be anothe and let denote the space finite-dimensional representation of $K$ on $T$ $X$ , set $(X\otimes T)f=$ in . For in and all linear maps from to $T\cdot Xf(f\in C^{\infty}(G, V_{\tau}))$ . This assigns a differential operater from to to eacl ([15, 5.4.5]). Let denote the space of al element in (for the definition see [15 to homogeneous differential operators from simply by $D(E_{f})$ and call its elements invari 5.4.1]). If equals , we denote ant differential operators of . canonically. Let $K$ act and Since and are K-modules, $K$ acts on by $C^{\infty}(G, V_{\tau})$


$L(V_{\tau}, V_{\sigma})$

$V_{\sigma}$

$G\times V_{\tau}$

$GxV_{\sigma}$

$D(E_{f}, E_{\sigma})$

$U(\mathfrak{g}_{c})\otimes L(V_{\tau}, V_{\sigma})$

$E_{\sigma}$

$E_{\tau}$

$D(E_{\tau}, E_{\sigma})$

$\sigma$

$\tau$

$0$



$V_{\sigma}$

$V_{\tau}$

$\sigma$

$E_{\tau}$


$\mathfrak{g}$

$\mathfrak{p}$

$S(p_{c})$

$01$


$kT=\sigma(k)\cdot T\cdot\tau(k^{-1})$

and and let $K$ act on Proposition 5.4.11], for any element $U(\mathfrak{g}_{c})\otimes L(V_{\tau}, V_{\sigma})$

$(T\in L(V_{\tau}, V_{\sigma}))$

$S(p_{c})\otimes L(V_{f}, V_{\sigma})$

by tensor product. By [15

$D\in(U(\mathfrak{g}_{c})\otimes L(V_{\tau}, V_{\sigma}))^{K}$

233

INVARIANT DIFFERENTIAL OPERATORS

$D(C^{\infty}(G, \tau))\subset C^{\infty}(G, \sigma)$

through the isomorphism from . We define a map and $D$ to the above . Thus, for by sending $\Delta=\mu_{0}(D)$ , putting we have

Therefore

$D$

defines an element

$\Delta\in D(E_{\tau}, E_{\sigma})$

$D\in(U(\mathfrak{g}_{c})\otimes L(V_{\tau}, V_{\sigma}))^{K}$

$(\Delta u)^{\sim}=D(u^{\sim})$

be the symmetrization from

$\Lambda$

to We regard

$L(V_{\tau}, V_{\sigma}))$

.

$S(\mathfrak{g}_{c})$

to

$S(p_{c})$

and

$ u\in$

.


and let

defined by as the subalgebra of . Since

$v$

be the linear map from

$v(p\otimes T)=\Lambda(p)\otimes T(p\in S(\mathfrak{g}_{c}),$


$S(\mathfrak{g}_{c})\otimes L(V_{\tau}, V_{\sigma})$

to

$\mu_{0}$

$\Delta$


Let

$ C^{\infty}(E_{\tau})\cong$

$(U(\mathfrak{g}_{c})\otimes L(V_{\tau}, V_{\sigma}))^{K}$

$C^{\infty}(E_{\sigma})\cong C^{\infty}(G, \sigma)$



.

$S(\mathfrak{g}_{c})$

$ T\in$

generated by and 1. Let is an is a K-isomorphism, to . We define $p$

$v_{K}$

be the restriction of to onto isomorphism from . into from be the map For simplicity we denote the differential representation of also by the same symbol . $(S(\mathfrak{p}_{c})\otimes L(V_{\tau}, V_{\sigma}))^{K}$

$v$

$v_{K}$

$(\Lambda(S(p_{c}))\otimes L(V_{\tau}, V_{\sigma}))^{K}$

$(S(p_{c})\otimes L(V_{\tau}, V_{\sigma}))^{K}$

$\zeta_{K}$


$(S(\mathfrak{p}_{c})\otimes L(V_{\tau}, V_{\sigma}))^{K}$

$\mu_{0}\cdot v_{K}$

$v$

$\tau$

$\tau$

(resp. LEMMA 1.1. Let (resp. ) be a finite-dimensional representation of $K$ on map $G/K$ ) the . Then (resp. over ) be the associated bundle to , and let E. (resp. is an onto isomorphism. to from $(S(p_{c})\otimes L(V_{\tau}, V_{\sigma}))^{K}$

. Let


be an arbitrary element in $D$ in whose restriction to ([15, Proposition 5.4.11]). Let be a basis of (resp. f). By the decomposition $D$ is written as 2.4.15]), with multi-indices and $PR\infty F$

$\sigma$

$\tau$

$E_{\sigma}$

$V_{\sigma})$

$\zeta_{K}$

$V_{\tau}$

$\sigma$

$\tau$


$\Delta$



induces

$X_{1},$ $X_{2},$

$C^{\infty}(E_{\tau})\cong C^{\infty}(G, \tau)$

. Then there exists an element $\cdots,$

by the identification

$\Delta$

$X_{m}$

(resp.

$U(\mathfrak{g}_{c})=\Lambda(S(p_{c}))U(\mathfrak{k}_{c})([3$

$p$

$(A_{\alpha,\beta}\in L(V_{\tau}, V_{\sigma}))$ $D=\sum_{\alpha,\beta}\Lambda(X^{\alpha})Y^{\beta}\otimes A_{\alpha.\beta}$

to be the element Set and for any

$\sum_{\alpha,\beta}\Lambda(X^{\alpha})\otimes A_{\alpha,\beta}\tau((Y^{\beta})^{T})$

$D_{\mathfrak{p}}$

$f\in C^{\infty}(G, \tau)$

$g\in G$

and rewrite it as

,

$=\sum_{\alpha,\beta}A_{\alpha,\beta}\tau((Y^{\beta})^{T})(\Lambda(X^{\alpha})f)(g)$

$=(D_{p}f)(g)$

belongs to


.

,

$(Df)(gk)=\sigma(k^{-1})(Df)(g)$

Therefore,

$\cdots,$

$Y_{n}$

)

, Proposition

.

$(Df)(g)=\sum_{\alpha,\beta}A_{\alpha,\beta}(\Lambda(X^{\alpha})Y^{\beta}f)(g)$

$\Delta$

$Y_{2},$

$\beta,$

$\alpha$

Since

$Y_{1},$

$(g\in G, k\in K)$

.

$\sum_{\alpha}\Lambda(X^{\alpha})\otimes B_{\alpha}$

. Then

234

KATSUHIRO MINEMURA

$(Df)(gk)=\sigma(k^{-1})(Df)(g)$ $=\sigma(k^{-1})(D_{\mathfrak{p}}f)(g)$

On the other hand, since

$f$

belongs to


.

,

$(Df)(gk)=(D_{\mathfrak{p}}f)(gk)$

$=\sum_{a}B_{\alpha}\tau(k^{-1})(Ad(k)\Lambda(X^{\alpha})f)(g)$

.

Therefore, we have $(D_{p}f)(g)=((\sum_{\alpha}\sigma(k)B_{\alpha}\tau(k^{-1})(Ad(k)\Lambda(X^{\alpha})))f)(g)$

$=((kD_{\mathfrak{p}})f)(g)$

.

Since $G/K$ is a reductive coset space, has sufficiently many elements in orde $k\in K$ to conclude that . Therefore for all belongs to $\mu_{0}(D_{p})=A$ and . Hence is surjective. Injectivity of is proved by the same arguments This finishes the proof. $C^{\infty}(G, \tau)$

$D_{p}=kD_{\mathfrak{p}}$

$(\Lambda(S(\mathfrak{p}_{c}))\otimes L(V_{\tau}, V_{\sigma}))^{\prime}$

$D_{\mathfrak{p}}$

$\zeta_{K}$

$\zeta_{K}$

From now on, we assume moreover that equals and the differential of , whicl is denoted also by , is irreducible. We ragard as a subset of by sending and let be the restriction of to . Extending to tht representation of canonically, let $J$ be the kernel of in $U(f,)$ . Since $Ad(k)(k\in K$ commutes with and $J$ is K-invariant, is also K-invariant and $S(p_{c})\otimes U(f_{c})/J^{T}i!$ regarded as a K-module by tensor product. Then the map given by $\sigma$

$\tau$

$\tau$

$U(\mathfrak{g}_{c})^{K}$

$\tau$

$q\mapsto q\otimes Id$

$\mu$

$(U(\mathfrak{g}_{c})\otimes EndV_{\tau})^{l}$


$\mu_{0}$


$\tau$

$\tau$

$J^{T}$

$T$

$\xi$

$\xi(q\otimes z)=q\otimes(z+J^{T})$

$(q\in S(p_{c}), z\in U(f_{c}))$

is clearly a K-homomorphism from onto gives ahomomorphism restriction of to

$S(\mathfrak{p}_{c})\otimes U(f_{c})/J^{T}$

$S(p_{c})\otimes U(\mathfrak{k}_{c})$

$(S(p_{c})\otimes U(\mathfrak{k}_{c}))^{K}$

$\xi$

$(S(p_{c})\otimes U(f_{c})/J^{T})^{K}$

Now since

$\tau$

$\xi_{K}$

from

into

.

is irreducible, the map

$\eta$

given by

$\eta(q\otimes(z+J^{T}))=q\otimes\tau(z^{T})$

$(q\in S(p_{c}), z\in U(\mathfrak{k}_{c}))$

is a well-defined K-isomorphism from onto restriction of to gives an isomorphism from onto . Let be the map given by $S(\mathfrak{p}_{c})\otimes U(\mathfrak{k}_{c})/J^{T}$

$\eta$

. Therefore, tht

$(S(p_{c})\otimes U(\mathfrak{k}_{c}))^{K}$

$(S(p_{c})\otimes U(f_{c})/J^{T})^{K}$

$S(p_{c})\otimes EndV_{\tau}$

$\eta_{K}$

and tht

$(S(p_{c})\otimes U(f_{c})/J^{T})^{J}$

$(S(p_{c})\otimes EndV_{\tau})^{K}$

$\psi$

$(q\in S(p_{c}), z\in U(\mathfrak{k}_{c}))$

$\psi(q\otimes z)=\Lambda(q)z$

is a K-isomorphism from Therefore, restricted to

Then

$\psi$

$S(p_{c})\otimes U(\mathfrak{k}_{c})$

$(S(p_{c})\otimes U(\mathfrak{k}_{c}))^{K},$

$\psi$

onto


.

by [3, Proposition 2.4.15],

gives an isomorphism

$\psi_{K}$

of

$(S(\mathfrak{p}_{c})\otimes U(\mathfrak{k}_{c}))^{X}$

235


onto


.

whose PROPOSITION 1.2. Let be a finite-dimensional representation of $K$ on $d\iota fferential$ is irreducible. Let be as above. Then and and the following diagram is into (i) is an algebra homomorphism of $V_{\tau}$

$\tau$

$\mu,$

$\zeta_{K},$

$\psi_{K}$

$\xi_{K}$

$\eta_{K},$

$D(E_{\tau})$


$\mu$

commutative: $(S(p_{c})\otimes U(\mathfrak{k}_{c}))^{K}\rightarrow^{\xi_{K}}(S(\mathfrak{p}_{c})\otimes U(\mathfrak{k}_{c})/J^{T})^{K}\rightarrow^{\eta_{K’},\cong}(S(p_{c})\otimes EndV_{\tau})^{K}$

$\zeta_{K}\downarrow\cong$

$\psi_{K}\downarrow\cong$

$\underline{\mu}$


(ii) The kernel of is (iii) Let be the algebra isomorphism

$U(\mathfrak{g}_{c})^{K}\cap U(\mathfrak{g}_{c})J^{T}$

$\mu$

$\mu_{K}$

by

$\mu$

and (ii). Then $PR\infty F$

$U(\mathfrak{k}_{c}))$

.

and set

(i)

$\mu_{K}$

is bijective

Take any

$D=\psi_{K}(p)$

if

.

.

of

$U(\mathfrak{g}_{c})^{K}/U(\mathfrak{g}_{c})^{K}\cap U(\mathfrak{g}_{c})J^{T}$

and only if,

$p\in(S(\mathfrak{p}_{c})\otimes U(\mathfrak{k}_{c}))^{K}$

. Then for any

$D(E_{\tau})$

,

$\xi_{K}$

into

$D(E_{\tau})$

given

is surjective.

write it as and

$f\in C^{\infty}(G, \tau)$

$D=\sum\Lambda(q_{i})z_{i}$

$p=\sum q_{i}\otimes z_{i}(q_{i}\in S(p_{c}),$

$g\in G$

,

$ z_{i}\in$

we have

,

$\eta_{K}(\xi_{K}(p))=\sum q_{i}\otimes\tau(z_{i}^{T})$

and $(\mu(D)f)(g)=(\sum\Lambda(q_{i})z_{i}f)(g)$

$=\sum\tau(z_{i}^{T})(\Lambda(q_{i})f)(g)$

.

On the other hand, we have $(\zeta_{K}(\eta_{K}(\xi_{K}(p)))f)(g)=(\zeta_{K}(\sum q_{i}\otimes\tau(z_{i}^{T}))f)(g)$

$=((\sum\Lambda(q_{i})\otimes\tau(z_{i}^{T}))f)(g)$

$=\sum\tau(z_{i}^{T})(\Lambda(q_{i})f)(g)$

.

Therefore we get $\mu(D)f=\zeta_{K}(\eta_{K}(\xi_{K}(p)))f$

.

and , we have ker $\mu=U(\mathfrak{g}_{c})^{K}\cap\psi(ker\xi)$ maps (ii) By the bijectivity of , by (i). Since ker is equal to $S(p_{c})\otimes J^{T}$ , ker is equal to Proposition 2.4.15]). (iii) This follows immediately from (i). This completes the proof. $th\underline{e}$

$\psi_{K},$

$\zeta_{K}$

$\eta_{K}$

$U(\mathfrak{g}_{c})^{K}\cap U(\mathfrak{g}_{c})J^{T}([3$

$\xi$

$\mu$

236

KATSUHIRO MINEMURA

Now we give some sufficient conditions for the surjectivity of

$\mu_{K}$

.

THEOREM 1.3.

Let $G$ be a connected Lie group and let $K$ be a closed subgroup $G$ such that $G/K$ is reductive. Let be a finite-dimensional representation of $K$ such th its differential representation is irreducible and let $J$ be the kernel of in . Let be the vector bundle over $G/K$ associated to and let be the algebra of invaria $d\iota fferential$ operators on the vector bundle . Then the algebra isomorphism into is surjective if one of the following conditions $($

$\tau$


$\tau$

$\lrcorner$

$D(E_{\tau})$

$\tau$

$E_{\tau}$

$\mu_{K}$

$U(\mathfrak{g}_{c})^{K}/U(\mathfrak{g}_{c})^{K}\cap U(\mathfrak{g}_{c})J^{T}$

$D(E_{\tau})$

satisfied:

(i) There exists a K-invariant subspace in $U(f,)$ such that sum), or equivalently (direct sum). (ii) The adjoint representation of $K$ on is semisimple. (iii) $Ad(K)$ is compact. $\mathscr{J}$

$U(\mathfrak{k}_{c})=J\oplus \mathscr{J}$

(dire

$U(f_{c})=J^{T}\oplus \mathscr{J}$

$\mathfrak{k}_{c}$

$PR\infty F$

.

Clearly (iii) implies (ii) and (ii) implies (i) (cf. [1]). Now

assume that

is a K-invariant subspace such that . Then since $Ad(k)(k\in with , we have and a K-homomorphism such that is identity. Therefore is surjective and consequently by Proposition 1.2. This finishes the proof. $U(\mathfrak{k}_{c})=J\oplus \mathscr{J}$

$T$

$U(\mathfrak{k}_{c})=J^{T}\oplus \mathscr{J}^{T}$

$\ell$

commutt

$\alpha:S(p_{c})\otimes U(f_{c})/J^{T}\rightarrow S(p_{c})($

$\xi\cdot\alpha$


K)$

$\xi_{K}$

$\mu_{K}$

is surjectit

REMARK. Let be the trivial representation of $K$ and let $D(G/K)$ denote the algebr of invariant differential operators on $G/K$. Then, since $J=U(f_{c})f$ , we have a K-invariar , and by the above theorem, we have th direct sum decomposition well-known isomorphism $\tau$

$U(\mathfrak{k}_{c})=U(f_{c})\mathfrak{k}\oplus C$

$U(\mathfrak{g},)^{K}/U(\mathfrak{g}_{c})^{K}\cap U(\mathfrak{g}_{c})f\cong D(G/K)$

for the reductive coset space

.

$G/K$

2. Eigensections and spherical sections. In this section we assume that $G$ is a connected Lie group and $K$ is a compat subgroup of $G$ . Then the coset space $G/K$ is automatically reductive ([7]). We keep the notation in \S 1. Let denote the set of the equivalence classes of finite-dimensional tions of the algebra $D(E_{f})$ . We denote a representation of on a vector space $(\chi, H)$ , and denote its equivalence class by $[\chi, H]$ (or simply by ). by $G/K$ Let over . Let $G$ act be the space of hyperfunctional sections of , and $x\in G/K$ (see [15]). Then by $\pi(g)u(x)=gu(g^{-1}x)$ for $g\in G,$ , let in regarded as a G-module by . For any denote the -value $G$ $u^{\sim}(g)=g^{-1}u(gK)(g\in G)$ corresponds by to in the sam hyperfunction on which way as \S 1. With the consideration of Lemma 3.4 we give the following

$t$

$R(D_{\tau})$

$represent^{\prime}$

$\chi$

$D(E_{\tau})$

$\rfloor$

$[\chi]$

$9(E_{\tau})$

$0$

$E_{\tau}$

$g(E_{\tau})$

$u\in ae(E_{\tau})$ $u$

$\pi$

$g(E_{\tau})$

$\mathscr{B}(E_{\tau})$

$u^{\sim}$

$V_{\tau}$

$u$

DEFINITION 2.1.

Let

$[\chi, H]$

be in

$R(D_{\tau})$

. A (hyperfunctional) section in $u$


237


-invariant subspaces called an eigensection of type ifthere exists a finite number of -module, each is isomorphic to , and 2) as a such that 1) belongs to -module of $H$. a quotient $D(E_{\tau})$

$\chi$

$H_{i}$

$D(E_{\tau})$

$\sum {}_{t}H_{i}$

$u$

$H_{i}$

$D(E_{\tau})$

Let

of

$\mathscr{B}(E_{\tau}, \chi)$

$\Delta\in D(E_{\tau}),$

be the space of eigensections of type . Because of the G-invariance . isaG-invariant subspace of $\chi$



LEMMA 2.2.

All sections in


are analytic.

. Then there PROOF. Let be an $Ad(K)$ -invariant subspace of such that $K$ exists an $Ad(K)$ -invariant inner product on since is compact. Let be an orthonormal basis of with respect to this inner product. Then the element is an elliptic . Then . Put belongs to which is isomorphic to a quotient differential operator. Let be a subspace of $H$ of and take an arbitrary element in . Then, since $\dim(S)$ is finite, there exists a $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$

$p$

$\mathfrak{g}$

$X_{1},$

$\mathfrak{p}$

$X_{2},$

$\cdots,$

$X_{m}$

$\mathfrak{p}$

$q=(\sum_{i}X_{i}^{2})\otimes 1$

$S$

$\Delta$

$\Delta=\zeta_{K}(q)$

$(S(\mathfrak{p}_{c})\otimes EndV_{\tau})^{K}$


$u$

$S$

monomial $P(t)$ in such that $P(\Delta)u=0$ . Since analytic. This finishes the proof. $t$

$P(\Delta)$

is also elliptic, we see that

$u$

is

. instead of Hereafter we write $dk$ be the Haar measure on $K$ so normalized let and of be the dimension Let that the total volume of $K$ is equal to 1. Let $f$ be a -valued analytic function on $G$ and consider the following three conditions: $\mathscr{B}(E_{\tau}, \chi)$

$\mathscr{A}(E_{\tau}, \chi)$

$d(\tau)$

$V_{\tau}$

$V_{\tau}$

(C1) (C2)

$f(gk)=\tau(k^{-1})f(g)$

(C3)

$d(\tau)\int_{K}\overline{tr(\tau(k))}f(k^{-1}g)dk=f(g)$

$f(kgk^{-1})=f(g)$

$(g\in G,k\in K)$

,

$(g\in G, k\in K)$ , $(g\in G)$

.

) be the spaoe of such analytic functions on $G$ that satisfies (C1) (resp. and (C3) (resp. (C2) and (C3)). Let denote the identity element in $G$ .

Let

$S$

$\mathcal{T}$

$e$

LEMMA 2.3.

Let

$S$

and

$T$

be the linear maps

$(Sf)(g)=\int_{K}f(kgk^{-1})dk$

defined by

$(f\in S, g\in G)$

$(Tf)(g)=d(\tau)^{2}\int_{K}\tau(k)f(gk)dk$

,

$(f\in \mathcal{T}, g\in G)$

.

Then (i) (ii)

$SisanisomorphismofSonto\mathcal{T}andTisltsinverse$ .

Let $f$ be an element of $S$ and let $D$ be an element of is equal to $f(e;D)$ . In particular, satisfies $f(e;D)=0$ for all equal to zero.


$\iota ff$

PROOF.

The first assertion (i) is clear if we notice that

. Then

$D\in U(\mathfrak{g}_{c})^{K}$

$(Sf)(e;D)$

, then

$tr(\tau(k^{-1}))=\overline{tr(\tau(k))}$

$f$

is

and that

238

KATSUHIRO MINEMURA

$d(\tau)\int_{K}\overline{tr(\tau(k))}\tau(k)dk=Id$

,

$d(\tau)\int_{K}\tau(kk_{1}k^{-1})\&=(tr(\tau(k_{1})))Id$

(here Id

denotes the identity operator on

$V_{\tau}$

),

since is irreducible. Let $\tau$

$D$

be in

$U(\mathfrak{g}_{c})^{R}$

Then $(Sf)(e;D)=\int_{K}f(e;Ad(k)D)dk=f(e;D)$

Now suppose $f(e;D)=0$ for all


. Given

$D\in U(\mathfrak{g})$

$D^{K}=\int_{K}Ad(k)Ddk$

Since $(Sf)(e;Ad(k)D)=(Sf)(e;D)$ , we have have $(Sf)(e;D)=f(e;D^{K})=0$ for all by (i). This completes the proof.

.

.

$G$

Then using (i), being connected, $Sf=0$ . Hence $S=1$

Let $[\chi, H]\in R(D_{\tau})$ . Then from the definition of ker and . Therefore gives a map we can define a linear map of into $u\in \mathscr{A}(E_{\tau}, \chi)$

$s_{X}$

, put

$(Sf)(e;D^{K})=(Sf)(e;D)$ ,

$D\in U(\mathfrak{g}_{c})$

$\chi$

.


,

$w($

we have by

$u$

$D(E.)/ker\chi\rightarrow V_{\tau}$

$\mathscr{A}(E_{f}, \chi)$

$V_{\tau}\otimes(D(E_{\tau})/ker\chi)^{*}$

for A

$\Delta u=0$

$\Delta\mapsto(\Delta u)(eK)$

$($

. Thu

such that

$\langle s_{\chi}(u), \Delta+ker\chi\rangle=(\Delta u)(eK)$

for

$\forall u\in \mathscr{A}(E_{f}, \chi)$

and

$\forall\Delta\in D(E_{\tau})$

DEFINITION 2.4.

.

A section in $u$


is called spherical ifu satisfies the conditio]

$d(\tau)\int_{K}\overline{tr(\tau(k))}\pi(k)udk=u$

It is clear that

condition (C3). Let

.

satisfies the above condition if and only if denote the space of spherical sections in

$u\in \mathscr{A}(E_{f}, \chi)$

$d(E_{\tau}, \chi)^{e}$

$u^{\sim}$

satisfie

$d(E_{\tau}, \chi)$

.

THEOREM 2.5. Let $G$ be a connected Lie group and $K$ be a compact subgroup of Let be a finite-dimensional representation of $K$ and assume that the differential of irreducible. Let be the vector bundle over $G/K$ associated to and let be a finite dimensional representation of . Then the restriction of to is injective. particular, $G$

$\tau i$

$\tau$

$E_{\tau}$

$\tau$

$D(E_{\tau})$

dim $PR\infty F$

. Suppose

$\forall D\in U(\mathfrak{g}_{c})^{K}$

,

$d(E_{\tau}, \chi)^{\tau}\leq d(\tau)\cdot\dim(D(E_{\tau})/ker\chi)$

$u\in d(E_{\tau}, X)^{\tau}$

$\chi$

$Ii$

$\mathscr{A}(E_{\tau}, \chi)^{\tau}$

$s_{\chi}$

satisfies

$s_{X}(u)=0$

.

.

Then by the definition of

$s_{\chi}$

, fo

239


$(\mu(D)u)^{\sim}(e)=0$

Since $(\mu(D)u)^{\sim}(e)=u^{\sim}(e;D)$ , we get completes the proof.

$u^{\sim}=0$

.

by using Lemma 2.2 and Lemma 2.3, which

Spherical sections and the Poisson transform on Riemannian symmetric spaces.

3.

In this section we assume that $G$ is a connected semisimple Lie group with finite center. Let $K$ be amaximal compact subgroup of $G$ and let and be the Lie algebras of $G$ and $K$, respectively as before. Let be the orthogonal complement of in with respect to the Killing form of . Since is $Ad(K)$ -invariant, $G/K$ is reductive. We keep of , then to the notation in previous sections. Fix a maximal abelian subspace and respectively, and introduce a linear order in the dual and to complexify be the set of restricted roots and positive restricted roots, and space . Let the root space in corresponding respectively. For any root in , we denote by . Let $A$ and $N$ denote the analytic and to . We put and , respectively. Then by the Iwasawa desubgroups of $G$ corresponding to $\mathfrak{k}$

$\mathfrak{g}$

$\mathfrak{k}$

$\mathfrak{g}$

$\mathfrak{p}$

$\mathfrak{p}$

$\mathfrak{g}$

$\mathfrak{a}$

$\mathfrak{p}$

$\mathfrak{k}$

$\mathfrak{a}$

$g_{c},$

$\mathfrak{g},$

$\Sigma$

$\mathfrak{a}^{*}$

$f_{c}$

$\mathfrak{a}_{c}$

$\Sigma_{+}$

$\Sigma$

$\mathfrak{g}$

$\alpha$

$\rho=\neq\sum_{\alpha e\Sigma_{+}}\dim(\mathfrak{g}_{a})\alpha$

$\mathfrak{n}=\sum_{\alpha\in\Sigma_{+}}\mathfrak{g}_{\alpha}$

$\alpha$

$\mathfrak{g}_{\alpha}$

$\mathfrak{n}$

$\mathfrak{a}$

and with composition $G=KAN$, write any $g\in G$ as $n\in N$ . LetM be the centralizer ofA inKand put P $=MAN$. be the Let be an irreducible representation of $K$ on a vector space $V$ and let differential too, the denote we vector bundle over $G/K$ associated to . In this section $M$ . For a finite-dimensional group of to the denote the restriction of by . Let denote the vector bundle over $G/P$ , let and representation of $M$ on $P\ni man\mapsto e^{t-\lambda+\rho)H(a)}\sigma(m)(m\in M, a\in A, n\in N)$ . In associated to the representation and . Let instead of , we write simply particular, if over $G/K$ and $G/P$ , respecand be the spaces of hyperfunctional sections of are considered as G-modules and we denote and tively. As in \S 2, $G$ on and , respectively. Let and by and the action of denote the spaces of V-valued hyperfunctions $f$ on $G$ such that $f(gk)$ $=\tau(k^{-1})f(g)(g\in G, k\in K)$ and $f(gman)=e^{(\lambda-\rho)H\langle a)}\sigma(m^{-1})f(g)(g\in G,$ $m\in M,$ $a\in A$ , , we have the canonical $n\in N)$ , respectively. Then in the same way as . Throughout this section, we and isomorphisms , respectively, by these and identify and with isomorphisms. Let be the set of equivalence classes of irreducible representations of $M$ and , is positive. For , the multiplicity of in such that be the set of we denote by the fixed M-module in , and for simplicity we denote the representation also by . of $M$ on be the universal enveloping algebras of and Let determined by let be the element in and respectively. For with the algebra . Then identifying the condition $\kappa(g)\in K,$

$g=\kappa(g)e^{H\langle g)}n$

$H(g)\in \mathfrak{a}$

$E_{\tau}$

$\tau$

$\tau$

$\tau$

$\tau$

$\tau$

$\tau_{M}$

$\lambda\in \mathfrak{a}_{c}^{*}$

$V_{\sigma}$

$\sigma$

$F_{\sigma,\lambda}$

$E_{\tau}$




$F_{\tau_{M},\lambda}$

$F_{\tau,\lambda}$

$\sigma=\tau_{M}$

$\mathscr{B}(F_{\sigma,\lambda})$

$F_{\sigma,\lambda}$


$\mathscr{B}(F_{\sigma.\lambda})$

$\pi$

$\mathscr{B}(G, \tau)$

$\pi_{\sigma,\lambda}$

$\mathscr{B}(G, (\sigma, \lambda))$

$C^{\infty}(E_{\tau})\cong C^{\infty}(G, \tau)$

$\mathscr{B}(F_{\sigma,\lambda})\cong \mathscr{B}(G, (\sigma, \lambda))$

$\mathscr{B}(E_{\tau})\cong \mathscr{B}(G, \tau)$


$\mathscr{B}(G, \tau)$


$\mathscr{B}(G, (\sigma, \lambda))$

$\hat{M}$

$R_{\tau}$

$\sigma\in\hat{M}$

$\sigma in\hat{M}$

$[\tau:\sigma]$

$V_{\sigma}$

$V_{\sigma}$

$U(\mathfrak{g}_{c}),$

$\mathfrak{k}_{c}$

$\sigma$

$\tau_{M}$

$\sigma$

$\sigma$

$U(\mathfrak{n}_{c}),$

$U(\mathfrak{a}_{c})$



$D-\omega(D)\in \mathfrak{n}U(\mathfrak{g}_{c})$

$\mathfrak{g},,$

$\omega(D)$

$U(\mathfrak{a}_{c})U(\mathfrak{k}_{c})$

$U(\mathfrak{a}_{c})U(\mathfrak{k}_{c})$

$n_{c},$

$a_{c}$

240

KATSUHIRO MINEMURA

, we have ([3], [9], [12]). Let

an algebra anti-isomorphism

of into $J$ be the kernel of in and $End_{M}V$ be the set $T\in EndV$ such that $\tau(m)\cdot T=T\cdot\tau(m)$ for all $m\in M$ . Let denote the automorphisn given by $\#(H)=H+p(H)(H\in a)$ and put of where is regarded as a homomorphism of onto End $V$ and is the canonica ) is an algebra homomorphism (resp (resp. anti-automorphism of . Then , and the kemel of anti-homomorphism) of is equal into $K$ ( , Corollary 4.5]). Sinoe is compact and connected and is irreducible, : is a surjective algebra isomorphism Theorem 1.3. Therefore there exists a unique algebra isomorphism of into such that . $1ete_{\lambda}ktheevaluationmapofU(\mathfrak{a}_{c})intoCdefinedbye_{\lambda}(H)=\langle\lambda,H_{\text{ノ}}^{\backslash }$ For , let . For denote $Hom_{M}(V_{\sigma}, V)$ . Since $V$ is isomorphic to as an M-module, we have . Define to be the projection $End_{M}V$ to according to this decomposition. Put $U(\mathfrak{a}_{c})\otimes U(f_{c})$


$\omega$

$U(\mathfrak{a}_{c})\otimes U(\mathfrak{k}_{c})^{A}$

$U(f_{c})$

$\tau$

$0$

$\#$

$\omega_{f}=(\#\otimes(\tau\cdot T))\cdot\omega,$

$U(\mathfrak{a}_{c})$

$T$

$U(f_{c})$

$\tau$


$\omega_{\tau}^{\prime}$

$\omega_{\tau}$


$U(\mathfrak{g}_{c})^{K}\cap U(\mathfrak{g}_{c})J^{T}$

$\omega_{t}^{\prime}=(\$\otimes\tau)\cdot\omega$

$U(\mathfrak{a}_{c})\otimes End_{M}V$

$te$

$\omega_{\tau}$

$[9$

$\tau$

$U(\mathfrak{g}_{c})^{K}/U(\mathfrak{g}_{c})^{K}\cap U(\mathfrak{g}_{c})J^{T}\rightarrow D(E_{\tau})$

$b!$

$\mu_{K}$

$D(E_{\tau})$

$\chi_{\tau}$

$U(\mathfrak{a}_{c})\otimes End_{M}V$

$\chi_{l}\cdot\mu=\omega_{\tau}$

$\lambda\in \mathfrak{a}_{c},$

$(H\in \mathfrak{a}_{c})$

$\sigma\in R_{\tau}$

$H_{\sigma}$

$\sum_{\sigma\epsilon R_{*}}V_{\sigma}\otimes H$

$End_{M}V\cong\sum_{\sigma\in R_{\tau}}EndH_{\sigma}$

$0$

$\omega_{\sigma}$

$EndH_{\sigma}$

,

$\chi_{\tau.\lambda}=(e_{\lambda}\otimes Id)\cdot\chi_{\tau}$

$\chi_{\tau,\sigma.\lambda}=(e_{\lambda}\otimes\omega_{\sigma})\cdot\chi_{\tau}$

$\omega_{\tau.\lambda}=\chi_{\tau,\lambda}\cdot\mu$

,

,

$\omega_{\tau,\sigma.\lambda}=\chi_{\tau.\sigma.\lambda}\cdot\mu$

.

unde] , let and denote the space of K-finite sections of , which we shall denote also by . Then by the differential representation of is regarded as an admissible -module and hence $Hom_{K}(V, V(\sigma, \lambda))i^{t}|$ regarded as a -module canonically. Let and denote the contragredien , , respectively, and put representation of and on and Since is irreducible, the bilinear form

For

$\sigma\in\hat{M}$

$\lambda\in \mathfrak{a}^{*}$

$V(\sigma, \lambda)$

$\mathcal{B}(F_{\sigma.\lambda})$

$\pi_{\sigma,\lambda}$

$\pi_{\sigma.\lambda}$

$V(\sigma, \lambda)$

$\pi_{\sigma.\lambda}$

$(\mathfrak{g}_{c}, K)$

$\tau^{*}$


$V^{*}$

$\sigma$

$\tau$

$\sigma^{*}(\sigma\in R_{\tau})$

$V_{\sigma}^{*}$

$\omega_{.t,\lambda}^{\prime},=(e_{\lambda}\otimes\omega_{\sigma}.)\cdot\omega^{\prime}$

$\sigma^{*}$

$\langle b, a\rangle=tr(a^{t}\cdot b)$

is non-singular, where

as

LEMMA 3.1. Let -module by

$d$

$\sigma$

denotes the transpose of

$\omega_{\tau.\sigma,\lambda}$

For

$D\in U(\mathfrak{g}_{c})^{K},$

$a$

.

and be in , and regard , respectively. and and we have

be in


(i)

$(b\in H_{\sigma}., a\in H_{\sigma})$

$\mathfrak{a}^{*}$

$\lambda$

$R_{\tau}$

.

$H_{\sigma}$

(ii)


If .

$PR\infty F$

$b\in H_{\sigma}$

$a\in H_{\sigma}$

-module, is isomorphic to is an irreducible $H_{\sigma}$

$V(\sigma^{*}, -\lambda)$

(i)

$V(\sigma^{*}, -\lambda)$

$\pi_{\sigma.-\lambda}$

$\langle\omega^{\prime},.’’\lambda(D)b, a\rangle=\langle b, \omega_{\tau.\sigma.\lambda}(D)a\rangle$

and as a

and $Hom_{K}(V^{*},$

$D=\sum_{i}h_{i}z_{i}+w$

$Hom_{K}(V‘‘, V(\sigma^{*}, -\lambda))$

$(\mathfrak{g}_{c}, K)$

$TakeanyD\in U(\mathfrak{g}_{c})^{K}$

,

-module, then

$H_{\sigma}$

.

is irreducible.

and writeD as

$(h_{i}\in U(\mathfrak{a}_{c}), z_{i}\in U(\mathfrak{k}_{c}),$

$w\in \mathfrak{n}U(\mathfrak{g}_{c}))$

.

241


Then for

$b\in H_{\sigma^{*}}$

and

$a\in H_{\sigma}$

,

we have

$\langle\omega_{\tau.\sigma^{*},\lambda}^{\prime}(D)b, a\rangle=tr(a^{t}\omega_{\tau,\sigma^{l},\lambda}^{\prime}(D)b)$

$=tr(\sum a^{t}e_{\lambda+\rho}(h_{i})\tau^{*}(z_{i})b)$

$=tr(\sum e_{\lambda+\rho}(h_{i})a^{t}\tau^{t}(z_{i}^{T})b)$

$=tr((\sum e_{\lambda+\rho}(h_{i})\tau(z_{i}^{T})a)^{t}b)$

$=tr((\omega_{\tau,\sigma,\lambda}(D)a)^{t}b)$

$=\langle b, \omega_{\tau,\sigma.\lambda}(D)a\rangle$

.

On the other hand, is adjoint to $Hom_{K}(V^{*}, V(\sigma^{*}, -\lambda))$ by [9, Theorem 5.5 and Theorem 7.2]. Therefore is isomorphic to $Hom_{K}(V^{*}, V(\sigma^{*}, -\lambda))$ as a module. (ii) This follows immediately from [10, Theorem 5.5]. This finishes the proof. $H_{\sigma^{*}}$

$U(\mathfrak{g}_{c})^{K_{-}}$

$H_{\sigma}$

Now we are in the position to give the definition of the Poisson transform for the following Okamoto [11]. Let $dk$ be the normalized Haar measure on vector bundle $K$. For , consider the function on $G$ given by $E_{\tau}$

$\mathscr{P}_{\tau,\lambda}\phi$

$\phi\in \mathscr{B}(F_{\tau,\lambda})$

$(g\in G)$

$(\mathscr{P}_{\tau,\lambda}\phi)(g)=\int_{K}\tau(k)\phi(gk)dk$

.

Then one can show that $(\mathcal{P}_{\tau,\lambda}\phi)(g)=\int_{K}e^{-\{\lambda+\rho)\langle H(g^{-}{}^{t}k))}\tau(\kappa(g^{-1}k))\phi(k)dk$

and that belongs to analytic and $K$ is compact, space of analytic sections of $\mathscr{P}_{\tau,\lambda}\phi$

DEFINITION 3.2. transform for .


. Put

$\mathscr{P}_{c,\lambda}\phi$

$E_{f}$

on

The map

$\Psi_{\tau,\lambda}(g)=e^{-(\lambda+\rho)(H\{g^{-1}))}\tau(\kappa(g^{-1}))$

is an analytic section of

$E_{f}$

. Let

. Since

$\mathscr{A}(E_{\tau})$

$\Psi_{\tau,\lambda}$

is

denote the

.

$G/K$

$\mathscr{P}_{\tau,\lambda}$

from

$\mathscr{B}(F_{\tau,\lambda})$

into

$\mathscr{A}(E_{\tau})$

is called the Poisson

$E_{\tau}$

LEMMA 3.3. $PR\infty F$

For any

. Write

$D\in U(\mathfrak{g}_{c})^{K},$


$\Psi_{\tau,\lambda}(g;D)=\Psi_{\tau,\lambda}(g)\cdot\omega_{\tau,\lambda}(D)(g\in G)$

as $D=\sum_{i}h_{i}z_{i}+\sum_{j}w_{j}h_{j}^{\prime}z_{j}$

where

$w_{j}\in \mathfrak{n}U(\mathfrak{n}_{c}),$

$h_{i}$

and

.

$h_{j}^{\prime}\in U(\mathfrak{a}_{c})$

,

, and $z_{j}\in U(f_{c})$ . Then for $n\in N,$ $a\in A$ and $k\in K$, we have

$\Psi_{\tau,\lambda}(nak;D)=\tau(k^{-1})\Psi_{\tau,\lambda}(na;D)$

$=\tau(k^{-1})\{\sum_{i}\Psi_{\tau,\lambda}(na;h_{i}z_{i})+\sum_{i}\Psi_{\tau,\lambda}(na;w_{j}h_{j}z_{j})\}$

242

KATSUHIRO MINEMURA

$=\tau(k^{-1})e^{(\lambda+\rho)(H\langle a))}\sum_{i}e_{\lambda+\rho}(h_{i})\tau(z_{i}^{T})$

since

$\Psi_{\tau.\lambda}(na;X)=0$

for

$X\in \mathfrak{n}$

,

. Therefore we get

$\Psi_{c,\lambda}(nak;D)=\Psi_{\tau,\lambda}(nak)\sum_{i}e_{\lambda}(\# h_{i})\tau(z_{i}^{T})$

$=\Psi_{\tau.\lambda}(nak)\cdot\omega_{\tau,\lambda}(D)$

,

which completes the proof. Now for

$\sigma\in R_{\tau}$

, regard

$H_{\sigma}$

as a trivial bundle over $G/K$. Then, since , we have

$F_{\tau,\lambda}$

is isomorphi

$tothedirectsumofF_{\sigma.\lambda}\otimes H_{\sigma}(\sigma\in R_{\tau})$

(direct sum)

$\mathscr{B}(F_{\tau,\lambda})\cong\sum_{\sigma\in R_{\tau}}\mathscr{B}(F_{\sigma.\lambda})\otimes H_{\sigma}$

and

$\mathscr{B}(F_{\sigma,\lambda})\otimes H_{\sigma}$

is regarded as a subspaoe of

$\mathscr{B}(F_{\tau,\lambda})$

by the G-isomorphism

$\theta(F_{\sigma,\lambda})\otimes H_{\sigma}\ni\sum_{t}\phi_{i}\otimes a_{i}\mapsto\sum_{i}a_{i}\phi_{i}\in g(F_{\tau.\lambda})$

Define the map Let

to be the restriction of to denote $Hom_{M}(V, V_{\sigma})$ . For , the function given by $g_{\tau,\sigma.\lambda}$

$\mathscr{P}_{\tau,\lambda}$

$H_{\mathfrak{r}.\sigma}(\sigma\in R_{\tau})$

$(v_{i}\in V, b_{i}\in H_{\tau,\sigma})$

.

$g(F_{\sigma.\lambda})\otimes H_{\sigma}$

$\lambda\in \mathfrak{a}_{c}^{*}$

and

.

$w=\sum_{i}v_{i}\otimes b_{i}\in V\otimes H_{\tau}$

$\psi_{w}$

$\psi_{\nu}(g)=\sum_{i}e^{(\lambda-\rho)\langle H\langle g))}b_{i}\cdot\tau(\kappa(g)^{-1})v_{i}$

clearly belongs to

$g(F_{\sigma,\lambda})$

.

Let

be the set of the section

$g(F_{\sigma.\lambda})^{\tau}$

$d(\tau)\int_{K}\overline{tr(\tau(k))}\pi_{\sigma,\lambda}(k)udk=u$

$u\in g(F_{\sigma,\lambda})$

satisfying

.

Then the map : is an isomorphism of onto by the Frobeniu reciprocity theorem. Hereafter we identify with by . Let $(\chi, H)$ be a subrepresentation of . Sinoe is irreducible, the biline, form is non-singular. By this bilinear form can identify , dual of with . Then we have canonical isomorphisms $\Gamma_{\sigma,\lambda}$

$w\mapsto\psi_{w}$

$V\otimes H_{\tau,\sigma}$

$V\otimes H_{\tau,\sigma}$

$\mathscr{B}(F_{\sigma,\lambda})^{\tau}$


$(\chi_{\tau,\sigma},{}_{\lambda}H_{\sigma})$

$\Gamma_{\sigma,\lambda}$

$\sigma$

$\langle b, a\rangle=d(\tau)^{-1}tr(b\cdot a)(b\in H_{\tau,\sigma}, a\in H_{\sigma})$

$H_{\sigma}$

$H_{\tau,\sigma}^{*}$

$H_{\tau,\sigma}$

$ D(E_{\tau})/ker\chi\cong\chi(D(E_{\tau}))\subset$

Let be the linear map from of the above inclusion $p_{\chi}$

$v$

End

$H\cong H\otimes H^{*}\subset H_{\sigma}\otimes H^{*}\cong(H_{\tau.\sigma}\otimes H)^{*}$

$H_{e.\sigma}\otimes H$

onto

$\mathscr{B}(F_{\sigma,\lambda})\otimes H_{\sigma}$

defined to be the transpof . We regard as a subspae

$(D(E_{\tau})/ker\chi)^{*}$

$D(E_{\tau})/ker\chi\subset(H_{\tau,\sigma}\otimes H)^{*}$

of

$\mathscr{B}(F_{\sigma.\lambda})\otimes H$

canonically.

LEMMA 3.4. Let $(\chi, H)$ be a subrepresentation (i) is a G-homomorphism $\mathscr{P}_{\tau,\sigma.\lambda}$

.

of

$(\chi_{\tau.\sigma}.{}_{\lambda}H_{\sigma})(\sigma\in R_{f}, \lambda\in \mathfrak{a}_{c}^{*})$

$ofB(F_{\sigma,\lambda})\otimes Hinto\mathscr{A}(E_{\tau})$

and the diagram

.

243


$\mathscr{B}(F_{\sigma,\lambda})\otimes H\mathscr{B}(F_{\sigma,\lambda})\otimes H\underline{Id\otimes\chi(\Delta)}$

$\mathscr{A}(E_{\tau})\downarrow \mathcal{P}_{\tau.\sigma.\lambda}$ $\mathscr{A}(E_{\tau})\downarrow \mathcal{P}_{\tau,\sigma.\lambda}$

$\underline{\Delta}$

is commutative for all

$\Delta\in D(E_{\tau})$

. Therefore we have :

$\mathscr{P}_{\tau,\sigma,\lambda}$


:

$\mathscr{B}(F_{\sigma,\lambda})\otimes H\rightarrow \mathscr{A}(E_{\tau}, \chi)$

$\mathscr{B}(F_{\sigma,\lambda})^{\tau}\otimes H\rightarrow \mathscr{A}(E_{\tau}, \chi)^{\tau}$

into be the linear map from Then we have the following commutative diagram: (ii)

Let

,


$V\otimes(D(E_{\tau})/ker\chi)^{*}$

$s_{\chi}$

defined in \S 2.

$\mathscr{B}(F_{\sigma,\lambda})^{\tau}\otimes HV\otimes H_{\tau,\sigma}\otimes H\underline{\Gamma_{\sigma,\lambda}\otimes Id}$

$1^{Id\otimes p_{\chi}}$

$\downarrow \mathcal{P}_{\tau,\sigma.\lambda}$

$\mathscr{A}(E_{\tau}, \chi)^{\tau}\underline{s_{\chi}}V\otimes(D(E_{\tau})/ker\chi)^{*}$

.

. Let is clear from the definition of PROOF. (i) The G-equivarianoe of $a\in H$ and such that , take a . For . Put $\mu(D)=\Delta$ . Then by Lemma 3.3, $\mathscr{P}_{\tau,\lambda}$


$\phi\in \mathscr{B}(F_{\sigma,\lambda})$

$u=\mathscr{P}_{\tau,\sigma,\lambda}(\phi\otimes a)$


$\Delta\in D(E_{\tau})$

$(\Delta u)(g)=u(g;D)$

$=\int_{K}\Psi_{\tau,\lambda}(k^{-1}g)\cdot\omega_{\tau,\sigma,\lambda}(D)\cdot a\phi(k)dk$

$=\int_{K}\Psi_{\tau,\lambda}(k^{-1}g)\cdot(\chi_{\tau,\sigma,\lambda}(\Delta)a)\phi(k)dk$

$=\int_{K}\Psi_{\tau,\lambda}(k^{-1}g)\cdot(Id\otimes\chi(\Delta))(\phi\otimes a)(k)dk$

,

which shows that the above diagram is commutative and that belongs to is contained in Hence, by the G-equivariance, the image of . Then for (ii) Let $v\in V,$ and $a\in H$. Put $u$

$\mathscr{B}(F_{\sigma,\lambda})^{\tau}\otimes H$

$b\in H_{\tau,\sigma}$

$u=\mathscr{P}_{\tau,\sigma,\lambda}(v\otimes b\otimes a)$


$\mathscr{A}(E_{\tau}, \chi)^{\tau}$

.

$\forall\Delta\in D(E_{\tau})$

,

.

244

KATSUHIRO MINEMURA

$(\Delta u)(e)=\int_{K}\Psi_{\tau.\lambda}(k^{-1})\chi_{\tau.\sigma.\lambda}(\Delta)a\Gamma_{\sigma,\lambda}(v\otimes b)(k)dk$

$=\int_{K}\tau(k)(\chi_{\tau,\sigma.\lambda}(\Delta)a)b\tau(k^{-1})v\ovalbox{\tt\small REJECT}$

$=d(\tau)^{-1}tr(\chi_{c.\sigma.\lambda}(\Delta)ab)v$

$=\langle b, \chi(\Delta)a\rangle v$

On the other hand by the definition of

$p_{\chi}$

. we have

,

$\langle p_{\chi}(b\otimes a), \Delta+ker\chi\rangle=\langle b, \chi(\Delta)a\rangle$

Therefore we obtain that

$s_{\chi}\cdot 9_{\tau.\sigma,\lambda}(v\otimes b\otimes a)=v\otimes p_{\chi}(b\otimes a)$

.

. This completes the proof.

THEOREM 3.5. Let $G$ be a connected semisimple Lie group with finite center $an/$ let $K$ be a maximal compact subgroup of G. Let be a finite-dimensional representatio’ of $K$ and assume that the differential of is irreducible, and let be the vector bundl over $G/K$ associated to . Let $(\chi, H)$ be a subrepresentation of for in an‘ in . Then we have the followings: (i) The map is an isomorphism of onto $V\otimes(D(E_{f})/ker\chi)^{*}$ . (ii) with we have $\tau$

$\tau$

$E_{\tau}$

$\tau$

$(\chi_{f\sigma},{}_{\lambda}H_{\sigma})$

$\sigma$

$R_{t}$

$\lambda$

$\mathfrak{a}_{c}^{*}$

$d(E_{\tau}, \chi)^{c}$

$s_{\chi}$

$Ident\iota fying\mathcal{B}(F_{\sigma,\lambda})^{c}\otimes H$

$V\otimes H_{\tau.\sigma}\otimes H$

$\Psi_{c,\sigma,\lambda}(V\otimes H_{\tau.\sigma}\otimes H)=.\mathscr{A}(E_{\tau}, \chi)^{l}$

ker

,

$\mathscr{P}_{\tau,\sigma,\lambda}\cap V\otimes H_{\tau.\sigma}\otimes H=V\otimes kerp_{\chi}$

(iii) The following three conditions 1) is irreducible. 2) dim . 3) is injective on

.

are mutually equivalent:

$\chi_{\tau,\sigma,\lambda}$

$\mathscr{A}(E_{\tau}, \chi_{\tau.\sigma.\lambda})^{\tau}=d(\tau)[\tau :

\sigma]^{2}$

$V\otimes H_{f\sigma}\otimes H_{\sigma}$

$9_{\tau.\sigma,\lambda}$

.

. This theorem follows immediately from Theorem 2.5, Lemma 3.4 and surjectivity of . $PR\infty F$

$th|$

$p_{\chi}$

References [1] [2] [3] [4] [5] [6]

N. BOURBAKI, \’El\’emems de Math\’ematique, Groupes et de Lie, Chap. 1, Hermann, Paris, 1960 W. CASSELMAN and D. MILICIcI, Asymptotic behavior of matrix coefficients of admissible representa tions, Duke Math. J., 49 (1982), 869-930. DIXMIER, Alg\‘ebres Enveloppantes, Gauthier-Villards, Paris, 1974. J. HARISH-CHANDRA, Representations of a semisimple Lie groups on a Banach space, I, Trans. AmeI Math. Soc., 75 (1953), 185-243. HARISH-CHANDRA, Spherical functions on a semisimple Lie group, I, II, Amer. J. Math., 80 (1958) 241-310, 553-613. HARISH-CHANDRA, Harmonic analysis on real reductive groups, I, J. Funct. Anal., 19 (1975), 104-204 $Al\mathscr{A}bres$


245

S. HELGASON, Differemial Geometry and Symmetric Spaces, Academic Press, 1962. S. HELGASON, Invariant differential operators on homogeneous manifolds, Bull. Amer. Math. Soc., 83 (1977), 751-774. [9] J. LEPOWSKY, Algebraic results on representations of semisimple Lie groups, Trans. Amer. Math. Soc., 176 (1973), 1-A4. [10] J. LEPOWSKY and G. W. McCoLLUM, On the deterImination of irreducible modules by restriction to a subalgebra, Trans. Amer. Math. Soc., 176 (1973), $4\succ 57$ . [11] K. OKAMOTO, Harmonic analysis on homogeneous vector bundles, Lecture Notes in Math., 266 (1972), Springer, 255-271. [12] C. RADER, Spherical Functions on aSemi-Simple Lie Group, Lecture Notes in Repr. Th., Univ. Chicago,

[7] [8]

1976. [13]

M. SATO, Theory of hyperfunctions, I, II, J. Fac. Sci. Univ. Tokyo Sect. I, 8 (1959), 139-193, 8 (1960),

[14]

M. SATO, T. KAWAI and M. KASHIWARA, Microfunctions and Pseudo-Differential Equations, Lecture Notes in Math., 287 (1973), Springer. N. R. WALLACH, Harmonic Analysis on Homogeneous Spaces, Marcel Dekker, 1973.

387-437.

[15]

Present Address: JAPAN WOMEN’S UNIVERSITY MEJIRODAI, BUNKYO-KU, TOKYO 112, JAPAN