数理解析研究所講究録
第 712 巻 1990 年 73-92

73

Invariant Hyperfunctions on a Rank One Semisimple Symmetric SpaceHIROYUKI OCHIAI
Department of Mathematics

Rikkyo University

1. Introduction

Let

$G$

be a connected semisimple Lie group,phism of$G$$\sigma$an involutive automor-

and $H$ an open subgroup of the group of fixed points ofThen $G/H$ is called a semisimple symmetric space. Let

gebra of$G$andwith respect to$g=\mathfrak{h}\oplus q$$\sigma$$g$$\sigma$.

be the Lie al-

the decomposition $into+1$ and-l eigenspaces. Choose a maximal abelian subspaceof semisimple elements. Then all such

$a\subset q$

consisting

have the same dimension, which$a$

we call the rank of $G/H$ . In what follows, we assume that the rank of$G/H$ is one.

It is clear that$q$

can be identified with the tangent spaceat the origin of $G/H$ . Moreover, $H$ acts on$q$by means of the restrictionof the adjoint representation to .
$q$

If $E$ is a real vector space, we put$S(E_{C})$the symmetric algebra ofusual, the differential operator

form on$g$$E_{C}$$\partial(p)$$E_{C}$

the complexification of $E$ and. For any
we define, as
on . Let $B( , )$ be the Killing$p\in S(q_{C}))$$q$, and write $\omega(X)=B(X,X)$ for all $X\in q$ , the Casimirpolynomial of. , then the polynomial$q$$\omega$

is an H-invariant nondegenerate/

74
quadratic form on

form,$\square =\partial(\omega)$$q.$.Identifying$q$with its dual by means of the Killing

can be viewed as an H-invariant constant coefficients

second order differential operator on
of

$q$

of

. We callthe pseudo-Laplacian$\square$. We set$(\tilde{H})_{1}=$

Let

$q$

$(\tilde{H})_{0}$$(\tilde{H})_{0}${ $T\in GL(q);\omega(Tx)=\omega(x)$ for all $x\in q$ }.

be the identity component of, then$ad_{q}(\mathfrak{h})\subset\sim \mathfrak{h}$$(\tilde{H})_{1}$, and$\sim \mathfrak{h}$be the Lie algebra.

The dimensions of H-invariant eigendistributions ofon$\square$$q$

is given

by careful considerations to the radial part of pseudo-Laplacian$\square$by

van Dijk [2]. On the other hand by a result of Cerezo [1] we can also
give the dimensions of H-invariant eigenhyperfunction of$\cdot$$\square$

on

$q$

. By

comparison of these results, Kowata introduce the conjecture that every
H-invariant eigenhyperfunction of$\square$

on

$q$

is

$(\tilde{H})0$-invariant, and he has

shown that this conjecture holds if the eigenvalue of 口 is not equal tozero.
In this paper we give the answer to this conjecture except for the caseof one of the exceptionat type $F_{4(4)}/Spin(4,5)$ . In \S 2 we rewrite theproblem in terms of the systems of differential equations (D-modules)

and in \S 3 we compare these two D-modules related to $H$ and$(\tilde{H})_{0}$inthe case of the classical type. In \S 4 we make more precise investigationby using the H-orbits decomposition ofconjecture of$I\langle owata$$q$. We must remark that thedoes not hold in one case (Proposition 18).

I express my sincere gratitude to‘ Prof.T.Oshima, whose constantencouragements and guidances made me write this paper. I would also
like to thank Prof.T.Kobayashi and Prof.N.Tose for many advices.

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2. Preliminaries

Let
$H$

$\overline{H}$be the subgroup of $GL(q)$ generated byis contained in$\tilde{H}$and$(\tilde{H})_{0}$$(\tilde{H})_{0}$and $Ad_{q}(H)$ . Thenis the identity component ofbe the sheaf of hyperfunctions on .

$\tilde{H}$. Let$\mathcal{B}$$q$

We define$\mathcal{B}^{H}(q),$$\mathcal{B}_{\lambda}^{H}(q),$$\mathcal{B}^{\tilde{H}}(q)$$\sigma\vee$and$\mathcal{B}^{H}(q)=${$f\in \mathcal{B}(q);f$$\mathcal{B}_{\lambda}^{H}(q)=${$f\in \mathcal{B}^{H}(q)${$f\in \mathcal{B}(q);f$$B^{\tilde{H}}(q)=$function$u\in \mathcal{B}_{\lambda}^{H}(q)$general, we haveLet$\mathcal{D}$$u\in \mathcal{B}^{H}(q)$;(ロー $\lambda)f=0$ }is$\overline{H}$-invariant}.

is said to be H-invariant and a hyper-

is called an invariant spherical hyperfunction. In$\mathcal{B}_{\lambda}^{\tilde{H}}(q)\subset \mathcal{B}_{\lambda}^{H}(q)$and$q_{C}$$\mathcal{B}^{\overline{H}}(q)\subset \mathcal{B}^{H}(q)$.

. For any $X\in g1(q)$ , define a vector field$(D_{X}f)(y)= \frac{d}{dt}f(e^{-tX}y)|_{t=0}$$D_{X}$

cients onbybe the sheaf of the ring of holomorphic coefficient differential

operators on

Then$\lambda\in C$

is H-invariant}$\mathcal{B}_{\lambda}^{\overline{H}}(q)=B^{\tilde{H}}(q)\cap \mathcal{B}_{\lambda}^{H}(q)$A hyperfunctionfor$\mathcal{B}_{\lambda}^{\overline{H}}(q)$on$D_{X}$$(y\in q, f\in C^{\infty}(q))$

$q$

by

.

is a first order differential operator with polynomial coeffi$q_{C}$. We define the systems of differential equations defining

H-invariants and H-invariants$\mathcal{M}=\mathcal{D}/\mathcal{I}=\mathcal{D}/\sum_{x\in \mathfrak{y}}\mathcal{D}D_{X}$,$\mathcal{N}=\mathcal{D}/\mathcal{J}=\mathcal{D}/$$\sum_{\sim,X\in \mathfrak{h}}\mathcal{D}D_{X}$and also define3

76$\mathcal{M}$$\mathcal{N}$\mbox{\boldmath $\lambda$}=D/I\mbox{\boldmath $\lambda$}=D/(I\mbox{\boldmath $\lambda$}+D(ロー)\mbox{\boldmath $\lambda$}=D/J\mbox{\boldmath $\lambda$}=D/(J\mbox{\boldmath $\lambda$}+D(ロー )).
$\lambda$

In general, we have surjective homomorphisms$\mathcal{N}_{\lambda}$$\lambda)$$\mathcal{M}arrow \mathcal{N}$and$\Lambda 4_{\lambda}arrow$.

In these notations, our problems are written as follows.

PROBLEM I. When is?

$\mathcal{B}_{\lambda}^{H}(q)=\mathcal{B}_{\lambda}^{\tilde{H}}(q)$PROBLEM II. When is$\mathcal{B}^{H}(q)=B^{\tilde{H}}(q)$PROBLEM III. When is$\mathcal{M}_{\lambda}arrow^{\sim}\mathcal{N}_{\lambda}$

PROBLEM IV. When is$\mathcal{M}arrow^{\sim}\mathcal{N}$

?

?

?

REMARK 1.
(1) ff (IV) holds, then (II) and (III) hold. If$(\Pi)$or (III) hold, then(I) hold.
(2) Problemof$(g, \mathfrak{h})$$\Pi I$an$d$

rv depend only on the complexiflcation$(g_{C}, \mathfrak{h}_{C})$. I

Our approach to these problems is case by case calculation. The
following table shows the non-Riemannian semisimple symmetric spaces$G/H$ of rank one ([4, Chapter X], [2,4\S 6.3]).

77
TABLE 2.

Here $p\geq 1,$ $q\geq 1,$ $m\geq 2$ .

We sometimes write $p+q=m$ in ases$c$

In cases 1o and$2^{o}$, then we have$1^{o},$ $2^{o},$ $3^{o},$ $4^{o}$$ad_{q}(\mathfrak{h})=\sim \mathfrak{h}$.

, hence Problem IV andtherefore all problems are true. On the other hands, we have$ad_{q}(\mathfrak{h})\neq\sim \mathfrak{h}$in the rest cases, hance the whole problems have meanings.

3. Problem III and IV

3.1. The Spaces$6^{o}$: $G/H=SL(m+1, R)/S(GL_{+}(m, R)\cross GL_{+}(1, R))$ . I

Let $(x, y)=(x_{1}, \ldots, x_{m}, y_{1}, \ldots, y_{m})$ be a coordinate system in$R^{2m}$. We denote the action of $T\in GL(q)$ on$(\begin{array}{l}xy\end{array})\in q$$q\cong$by left multipli-

cation. The Casimir polynomial is $\omega=x_{1}y_{1}+\cdots+x_{m}y_{m}$ . We realize$\mathfrak{h}$and6 as subalgebras of gl(q).
$\{\begin{array}{l}\mathfrak{h}=g1(m,R)\cong\{(0A-0_{tA\prime})A\in M(m,R)\}\overline{\mathfrak{h}}=\epsilon o(m,m)=\{(_{C-t}^{AB_{A}})A,B,C\in M(m,R),{}^{t}B=-B,{}^{t}C=-C\}\end{array}$First we deal with Problem IV.

5

78

PROPOSITION 3. If $m\geq 3$ , then

PROOF: We write$\mathcal{I}$and$\mathcal{J}$$\mathcal{M}arrow^{\sim}N$

explicitly on$\mathcal{I}=\sum_{i,j=1}^{m}D(x;\partial_{x_{j}}-y_{j}\partial_{y:})$.

$q_{C}$

with the above coordinates.

$\mathcal{J}=\mathcal{I}+\sum^{m}D(x;\partial_{y_{j}}-x_{j}\partial_{y:})+\sum^{m}\mathcal{D}(y_{i}\partial_{x_{j}}-y_{j}\partial_{x_{i}})$

We will show.

$i,j=1$$i,j=1$$x_{i}\partial_{y_{j}}-x_{j}\partial_{y;}\in \mathcal{I}$for $1\leq i<j\leq m$ . Since $m\geq 3$ , there

is an integer $k(1\leq k\leq m)$ such that $i,j,$$k$

are different to one another.

Since the right hand side of the formula$x_{i}\partial_{y_{j}}-x_{j}\partial_{y;}=(x_{j}\partial_{y_{k}}-x_{k}\partial_{y_{j}})(x;\partial_{x_{k}}-y_{k}\partial_{y;})$$+(x_{k}\partial_{yi}-x;\partial_{y_{k}})(x_{j}\partial_{x_{k}}-y_{k}\partial_{y_{j}})$$+(x_{i}\partial_{y_{j}}-x_{j}\partial_{y_{i}})(x_{k}\partial_{x_{k}}-y_{k}\partial_{y_{k}})$is contained in , then we have$\mathcal{I}$. Hence Proposition 3$x_{i}\partial_{y_{j}}-x_{j}\partial_{yj}\in \mathcal{I}$is proved. 1
Next we deal with the case $m=2$ . We define two left$\mathcal{D}$-Modules$\mathcal{B}_{\{x=0\}|q_{\mathbb{C}}}=\mathcal{D}/\mathcal{D}(x_{1}, x_{2}, \partial_{y_{1}}, \partial_{y_{2}})$$\mathcal{B}_{\{y=0\}[q_{C}}=\mathcal{D}/\mathcal{D}(y_{1}, y_{2}, \partial_{x_{1}}, \partial_{x_{2}})$and two morphisms as left$\mathcal{D}$,-Modules(3.1)$\mathcal{B}_{\{x=0\}|q_{C}}arrow\Lambda t$(3.2)$\mathcal{B}_{\{y=0\}|q_{C}}arrow \mathcal{M}$,$P P(y_{1}\partial_{x_{2}}-y_{2}\partial_{x_{1}})$,$P P(x_{1}\partial_{y_{2}}-x_{2}\partial_{y\iota})$

6

.

79
By (3.1) and (3.2), we get the complex of left
(3.3)

-modules.

$\mathcal{D}$$0arrow \mathcal{B}_{t^{x=0\}|qc}}\oplus \mathcal{B}_{\{y=0\}|qc}arrow \mathcal{M}arrow \mathcal{N}arrow 0$.

PROPOSITION 4. The complex (3.3) is exact.
PROOF:

It is sufficient to prove the injectivity of the first morphism,

which follows from Proposition 6 below. I

Next we consider Problem III.
PROPOSITION 5. $Hm=2$ and
PROOF:$\lambda\neq 0$

Since the pseudo-Laplacianby the explicit coordinates, then ロー
that$(y_{1}\partial_{x_{2}}-y_{2}\partial_{x_{1}})\in \mathcal{I}_{\lambda}$, then$\square$$\mathcal{M}_{\lambda}arrow^{\sim}\mathcal{N}_{\lambda}$

is written as.

$\square =\sum_{i=1}^{m}\partial_{x;}\partial_{y;}$$\lambda=\partial_{x_{1}}\partial_{y_{1}}+\partial_{x_{2}}\partial_{yz}-\lambda$. We findby the formula$\lambda(y_{1}\partial_{x_{2}}-y_{2}\partial_{x_{1}})=-(y_{1}\partial_{x_{2}}-y_{2}\partial_{x_{1}})(\square -\lambda)$$+\partial_{x_{1}}^{2}(x_{1}\partial_{x_{2}}-y_{2}\partial_{y_{1}})-\partial_{x_{2}}^{2}(x_{2}\partial_{x_{1}}-y_{1}\partial_{y_{2}})$$-\partial_{x_{1}}\partial_{x_{2}}(x_{1}\partial_{x\text{、}}-y_{1}\partial_{y_{1}})+\partial_{x_{1}}\partial_{x_{2}}(x_{2}\partial_{x_{2}}-y_{2}\partial_{y_{2}})$.

Hence Proposition 5 is proved. 1
The rest is in case$\lambda=0$.

PROPOSITION 6. The following complex (3.4) induced by the complex(3.3) is exact.

(3.4)
PROOF:.

$0arrow \mathcal{B}_{\{x=0\}|q_{C}}\oplus \mathcal{B}_{\{y=0\}|q_{\mathbb{C}}}arrow \mathcal{M}_{0}arrow \mathcal{N}_{0}arrow 0$

It is sufficient to prove the injectivity of the first morphism.

First we consider the complex (3.4) outside the origin. We may assume

7

8

$U$

$y_{2}\neq 0$

without loss of generality. Then remark that$\mathcal{B}_{\{y=0\}|q_{C}}=0$

on theopen set $\{y_{2}\neq 0\}$ . Moreover the following coordinate transformation
makes sense on the open set(3.5)$\{y_{2}\neq 0\}$$\{\begin{array}{l}x_{1}\sim=x_{1}\sim_{2}x=x_{l}y_{1}+x_{2}y_{2}\sim_{l}y=y_{1}y=y_{2}\sim_{2}\end{array}\}$:$x_{2}^{1}= \frac{\overline x\sim^{1_{2}}\sim_{1}x-x\overline{y}_{1}}{\sim_{1},yy\sim_{2}}x_{2}=y^{1}=yy=\sim_{2}\}$.

We rewrite everything in the new coordinates. Then$\mathcal{B}_{t^{x=0\}|q_{\mathbb{C}}}}=D/\mathcal{D}(x\sim_{1},x\sim_{2}, \partial_{\overline{y}_{1}}, y\sim_{2}\partial_{\overline{y}_{2}}-1)$,$\mathcal{M}_{0}=D/\mathcal{D}(\partial_{\overline{y}_{1}},\partial_{\overline{y}_{2}},x_{1}\sim\partial_{\overline{x}_{1}}, x_{2}\sim\partial_{\overline{x}_{1}}, \partial_{\tilde{x}_{2}}^{2}\overline{x}_{2})$and the morphism (3.1) is the right multiplication by,$y\sim_{2}\partial_{\overline{x}_{1}}$, which isthe composition of the following two injective morphisms$\mathcal{B}_{\{x=0\}|q_{\mathbb{C}}}arrow y\mathcal{D}/D(x_{1}\sim, x_{2}\sim, \partial_{\overline{y}_{1}}, \partial_{\overline{y}_{2}})\sim_{2}\sim\cdot\underline{\partial_{\tilde{x}_{1}}}\mathcal{M}_{0}$.

This implies the complex (3.4) is exact outside the origin, i.e. the kernelof$\mathcal{B}_{\{x=0\}|q_{\mathbb{C}}}\oplus \mathcal{B}_{\{y=0\}|q_{C}}arrow\Lambda t_{0}$is supported in the origin. Moreover

it follows from the involutivity of the characteristic varieties that thekernel must be zero. Thus we finish the proof of Proposition 6 andProposition 4. 1

8

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3.2. The Spaces$3^{o}$: $SU(p, q+1)/S(U(p, q)\cross U(1))$

By Remark 1, we can get the following Proposition 7 for spaces$6^{o}$

We take coordinates $z=(z_{1}, \ldots, z_{m})$ of

and those of$C^{2m}$as $(z, w)\in$$q_{C}$

with $z,$ $w\in C^{m},$ $q=\{(z, w)\in q_{C};z=\overline{w}\}$ .

PROPOSITION 7.
(1) II $m\geq 3$ , then$\Lambda 4arrow^{\sim}\mathcal{N}$.

If $m=2$ , then we have th follo wing exact seq uence,$e$.

$0arrow \mathcal{B}_{\{z=0\}|q_{C}}\oplus \mathcal{B}_{\{w=0\}|q_{C}}arrow \mathcal{M}arrow \mathcal{N}arrow 0$(2) If $m\geq 3$ or$\lambda\neq 0$

If $m=2$ and, then$\lambda=0$$\mathcal{M}_{\lambda}arrow^{\sim}\mathcal{N}_{\lambda}$.

, then we have the exact sequence,$0_{\wedge}arrow \mathcal{B}_{\{z=0\}|q_{\mathbb{C}}}\oplus \mathcal{B}_{\{w=0\}|q_{C}}arrow \mathcal{M}_{0}arrow \mathcal{N}_{0}arrow 0$(3) We have an isomorpism
In particular,.

$\underline{Hom}_{D}(\mathcal{N},\mathcal{B})arrow^{\sim}\underline{Hom}_{D}(\mathcal{M}, \mathcal{B})$$\mathcal{B}^{\overline{H}}(q)=\mathcal{B}^{H}(q)$and$\mathcal{B}_{\lambda}^{\tilde{H}}(q)=\mathcal{B}_{\lambda}^{H}(q)$.

.

PROOF: (1) and (2) follow from Proposition 3, 4, 5 and 6.
(3) is led by$\underline{Hom}_{\mathcal{D}}(\mathcal{B}_{\{z=0\}|q_{C}}, B)=\underline{Hom}_{D}(\mathcal{B}_{\{w=0\}|q_{\mathbb{C}}}, \mathcal{B})=0$

9

as a

. Let $m=p+q,$ $q=C^{m}$ .

corollary of the previous results for spaces$q$$3^{o}$. I

82

3.3. The Spaces

We set$8^{o}$: $Sp(m+1, R)/Sp(m, R)\cross Sp(1, R)$ .

$Sp(m, R)=\{g\in SL(2m, R);g(\begin{array}{ll}0 l_{m}-1_{m} 0\end{array}){}^{t}g=(\begin{array}{ll}0 1_{m}-1_{m} 0\end{array})\}$. We

take coordinates$(\begin{array}{ll}x_{l} x_{2}y_{1} y_{2}\end{array})\in M(2m, 2;R)\cong q$,$x_{1}=(\begin{array}{l}x_{11}|x_{m1}\end{array})x_{2},$ $y_{1},$In this coodinates, the Casimir polynomial$y_{2}\in R^{m}$.

is written as$\omega$$m$$\omega=t(\begin{array}{l}x_{1}y_{1}\end{array})(\begin{array}{ll}0 l_{m}-1_{m} 0\end{array}) (\begin{array}{l}x_{2}y_{2}\end{array})=\ell_{X_{1}}\cdot y_{2}-{}^{t}y_{2}\cdot x_{2}=\sum(x_{k1}y_{k2}-y_{k1}x_{k2})$

.

$k=1$

The group $H=Sp(m, R)\cross Sp(1, R)$ acts on(3.6)$(A, B)\cdot C=ACB^{-1}$

$q$

by

$(A\in Sp(m, R),$ $B\in Sp(1, R),$ $C\in q$ ).

THEOREM 8. For $m\geq 1$ , we have$\mathcal{M}arrow^{\sim}N$.

PROOF: First we consider the problem outside the origin. This step

contains rather tedious calculation. We may assume $x_{11}\neq 0$ and the:following coordinate transformation makes sense on the locus$\{x_{11}\neq 0\}$$\{\begin{array}{l}\sim_{j1}x=x_{j1},y=y_{j1}\sim_{jl}(1\leq j\leq m)\sim_{12}x=x_{12},y\sim_{12}=\omega(x,y)\sim_{k2}x=x_{k2}-\frac{x_{12^{X}kl}}{x_{l1}},y\sim_{k2}=y_{k2}-\frac{x_{12}y_{kl}}{x_{ll}}(2\leq k\leq m)\end{array}$

10

83

We recall the generators of$\mathcal{I}$by old coordinates (cf. (3.6)).

In new coordinates, we exchange generators of :$\mathcal{I}$(3.7)$\{\begin{array}{l}(e)=x_{11}\partial_{\overline{x}_{12}}((c),i=j=1)=2x_{11}\partial_{\tilde{y}_{11}}((a),i=j=1)=x_{l1}\partial_{\tilde{x}_{11}}+x_{12}\partial_{\overline{x}_{12}}-y_{11}\partial_{\overline{y}_{11}}((c),i\geq 2,j=1)=x_{ll}\partial_{\overline{y}i1}+x_{i1}\partial_{\overline{y}_{11}}((a),i\geq 2,j=1)=x_{11}\partial_{\overline{x}_{j1}}-y_{i1}\partial_{\overline{y}_{11}}\end{array}$$\partial\partial_{\overline{x_{\overline{y}_{i1}}}^{12}}\partial^{\overline{x}_{11}}\partial\partial_{\overline{x}^{11}}^{\overline{y}_{j1}}\in \mathcal{I}\in \mathcal{I}\in \mathcal{I}\in \mathcal{I}\in \mathcal{I}.\cdot$

.

(3.8)

$(d)=- \sum(x_{k2}\sim\partial_{\overline{x}_{k2}}+y\sim_{k2}\partial_{\overline{y}_{k2}})m$$k=2$$+ \sum(x_{k1}\partial_{\overline{x}_{k1}}+y_{k1}\partial_{\overline{y}_{k1}})+x_{11}\partial_{\overline{x}_{11}}+y_{11}\partial_{\overline{y}_{11}}m$$k=2$$\sum_{k=2}(\overline{x}_{k2}\partial_{\overline{x}_{k2}}+y\sim_{k2}\partial_{\overline{y}_{k2}})\in \mathcal{I}m$

11

.

84

Since

$\omega$is H-invariant, then$\mathcal{J}\subset D$($\partial_{\overline{x}:;}$,$\partial_{\overline{y}:;}$; $1\leq i\leq m,$ $1\leq j\leq 2$ , except for$\partial_{\overline{y}_{12}}$).

If we can prove(3.10)$\mathcal{I}\supset \mathcal{D}$then$\mathcal{I}=\mathcal{J}$($\partial_{\overline{x}_{ij}},$$\partial_{\overline{y}_{ij}}$; $1\leq i\leq m,$ $1\leq j\leq 2$ , except for$\partial_{\overline{y}_{12}}$),. We prepare two Lemmas.

LEMMA 9. Two $grou$ps $Sp(n, R)\subset SL(2n, R)$ act onplica tion $(n\geq 1)$ . Then the system$\mathcal{M}_{1}$$Sp(n, R)$ -invarian $tseq$ uals the system$R^{2n}$by left $mul$ ti-of differential $eq$ uations defining$\mathcal{M}_{2}$of differential equations defin-

ing $SL(2n, R)- inva.\dot{n}ants$ .
PROOF: We set$\mathcal{M}_{1}=\mathcal{D}/\mathcal{I}_{1},$$\mathcal{M}_{2}=D/\mathcal{I}_{2}$12. We can express$\mathcal{I}_{1}$and$\mathcal{I}_{2}$85
in coordinates.

$\mathcal{I}_{1}=\mathcal{D}(\begin{array}{llll}x_{i}\partial_{x_{j}} -y_{j}\partial_{y_{i}} \leq i,j\leq m1x.\cdot\partial_{yj} +x_{j}\partial_{y}.

\leq i,j\leq m1y_{i}\partial_{x_{j}} +y_{j}\partial_{x}.

l\leq i,j\leq m\end{array})$$\mathcal{I}_{2}=\mathcal{D}(\begin{array}{llllll}x_{i}\partial_{x} -y_{j}\partial_{y_{j}} l\leq i,j\leq mx.\cdot\partial_{y_{j}} y_{j}\partial_{x}.

l\leq i,j\leq mx.\cdot\partial_{x_{j}} ’ y_{|}\cdot\partial_{yj} i\neq j\end{array})$.

One after another,$x^{2}:\partial_{x;}=x_{i}(x_{i}\partial_{x_{i}}-y;\partial_{yj})+y_{i}x_{i}\partial_{y:}$$\in \mathcal{I}_{1}$.

$x_{i}^{2}\partial_{y_{j}}=x_{i}(x_{i}\partial_{yj}+x_{j}\partial_{yi})-x_{j}x_{i}\partial_{y:}\in \mathcal{I}_{1}$$x_{i} \partial_{y_{j}}=\frac{1}{2}(\partial_{x:}x_{i}^{2}\partial_{yj}-\partial_{y:}J^{X^{2}}\partial_{x;})\in \mathcal{I}_{1}$Lie brackets of these elements in$\mathcal{I}_{1}$,.

$y;\partial_{x_{j}}\in \mathcal{I}_{1}$span all generators of$\mathcal{I}_{2}$.
. I

LEMMA 10. Let us consider th at the group $GL_{+}(n, R)$ acts on$mult$

iplication. Then the system of differential $eq$ uations$GL_{+}(n, R)$-invarian

PROOF: We set$ts$$\mathcal{M}_{1}=\mathcal{D}/\mathcal{I}_{1},$$\mathcal{I}_{1}=\sum_{j}^{m_{=1}}\mathcal{D}\partial_{j}$$\mathcal{M}_{1}$by leftdefining

is de Rham system.
$\mathcal{I}_{1}=\sum_{ij=1}^{n_{)}}D(x_{i}\partial_{j})$$\partial_{j}=\partial;x_{i}\partial_{j}-\partial_{j}x_{i}\partial;\in \mathcal{I}_{1}$Then$R^{n}$.

(for $i\neq j$ ).

.I

Now we return to the proof of (3.10). Applying Lemma 9 to (3.9) andthen applying Lemma 10 to (3.8), we conclude(3.11)$\partial_{\overline{x}:2}$,$\partial_{\overline{y}_{i2}}\in \mathcal{I}$13$(2\leq i\leq m)$.

(3.7) and (3.11) imply (3.10). Thus$\mathcal{M}|_{q-\{0\}}arrow^{\sim}\mathcal{N}|_{q-\{0\}}$phism and following Proposition 11 imply$\mathcal{M}arrow^{\sim}\mathcal{N}$. This isomor-. Hence Theorem 8

is proved. I
PROPOSITION 11. Let$H_{1}\subset H_{2}\subset SL(V)$$V$

be an n-dimensional vector space overbe two closed subgroups. Suppose$H_{1}$

$R$

and

and
$H_{2}$

are reductive in $SL(V)$ , by defini tion ${}^{t}H_{1}=H_{1},{}^{t}H_{2}=H_{2}$ , here ${}^{t}H;=$$\{{}^{t}h^{-1}\in SL(V);h\in H_{i}\}$defining$H_{i}$$\mathcal{M}_{1}|_{V_{\mathbb{C}}-\{0\}}arrow^{\sim}\mathcal{M}_{2}|_{V_{C}-\{0\}}$$\mathcal{M}_{1}arrow^{\sim}\mathcal{M}_{2}$PROOF:

be the system of differenti $aleq$ uations$\mathcal{M}_{i}$-invarian $ts(i=1,2)$ . We assume(3.12)

then

. Let

.

$\mathcal{M}_{i}=\mathcal{D}/\mathcal{I}_{i},$$H_{i}(i=1,2)$,$\mathcal{I}_{i}=\sum_{X\in b:}\mathcal{D}D_{X}$, here$\mathfrak{h}$; is the Lie algebra of.

Let A be a Weyl algebra overtomorphism defined by$V_{C}$$x_{i}rightarrow-\partial;,$and $F:Aarrow A$ be the algebra au-$\partial;\mapsto x_{i}$(cf. [5]). By the assumption(3.12) and Hilbert zero point theorem, there exists a positive integer $N$

such that$\alpha$$(x_{1}, \ldots , x_{n})^{N}D_{X}\subset \mathcal{I}_{1}$satisfying $(|\alpha|\geq N)$ , there arefor all$X\in \mathfrak{h}_{2}$$P_{j}(x, \partial)\in A$, i.e. for any multi-index,$x^{\alpha}D_{X}= \sum_{x_{j}\in b\iota}P_{j}(x, \partial)D_{X_{j}}$Let$P_{j^{o}}(x, \partial)\in$

A be the homogeneous part of.

$P_{j}(x, \partial)$

here we as sign the degree 1 and-l to the element; and$\partial$

Then(3.13)$x^{\alpha}D_{X}= \sum_{X_{j}\in \mathfrak{h}_{1}}P_{j}^{o}(x, \partial)D_{X_{j}}$14

of $degree-|\alpha|$ ,.

$x_{i}$

respectively.

87

The transform of (3.13) by $F$ is$(- \partial)^{\alpha}D_{-l}x=\sum_{X_{j}\in b\iota}P_{j}^{o}(-\partial, x)D_{-\iota_{X_{j}}}$

Then the left ideal { $P\in R;PD_{X}\in \mathcal{I}_{1}$ for any

both$(x_{1}, \ldots, x_{n})^{N}$

This means that

and$(\partial_{1}, \ldots, \partial_{n})^{N}$$D_{X}\in \mathcal{I}_{1}$for any.

$X\in \mathfrak{h}_{2}$} of A contains, but such an ideal must be A itself.
$X\in \mathfrak{h}_{2}$, hence Proposition 11 isproved. 1

3.4. The Spaces$4^{o}$: $Sp(p, q+1)/Sp(p, q)\cross Sp(1)$As a corollary of Theorem 8 and Remark 1,
THEOREM 12. For$p,$ $q\geq 1$, we have$\mathcal{M}arrow^{\sim}\mathcal{N}$. I

4. Problem I and II

In this section, we treat Problem I and II with the H-orbit decompositionof$q$. Define $q_{reg}=\{x\in q;co\dim_{q}H\cdot x=1\}$ and $S=q-q_{reg}$ , thenis H-invariant open subset of . Define$q$15$\omega’=\omega|_{q_{r\epsilon g}}$and$q_{reg}$$\omega_{1}=\omega|_{q-\{0\}}$

.

88

4.1.

We quote the following Proposition from [2, \S 6.3].
PROPOSITION 13. in cases(1):$\omega’$

grad
(2)
(3)

$q_{reg}arrow R$$\omega’\neq 0$$S\subset\omega^{-1}(0)$$3^{o}\sim 9^{o}$$6^{o}$

an $dm=2$ .

gives H-orbit parametrization, by definition,

everywhere andand

except for the case$\omega^{-1}(t)$decomposes fini$S$

is H-orbit for any $t\in R$ .

$te$H-orbits.

: $q-\{O\}arrow R$ gives H-orbit pararn etrization.

$\omega_{1}$(4) As a consequence,(4.1)$\{\begin{array}{l}\mathcal{B}^{H}(q_{reg})=\omega^{/*}(\mathcal{B}_{R}(R))-\mathcal{B}^{H}(q-\{0\})=\omega_{1}^{*}(B_{R}(R))\end{array}$I

We quote the following theorem due to Cerezo [1].
THEOREM 14. A connected Lie group $SO_{0}(p, q)$ acts on$R^{p+q}$

naturally.

ff $(p, q)\neq(1,1)$ , then bothand$\mathcal{B}^{SO_{0}(p,q)}(R^{p+q})arrow \mathcal{B}^{SO_{0}(p,q)}(R^{p+q}-\{0\})$$\mathcal{B}^{SO(p,q)}(R^{p+q})arrow \mathcal{B}^{SO(p,q)}(R^{p+q}-\{0\})$are surjective. 1

Now we prepare Lemma 15 for the proof of Proposition 16.

LEMMA 15. Set$9^{o}$$\mathcal{F}[0]=${$u\in \mathcal{F}(q)$; supp $u\subset\{0\}$ }. Then in case$1^{o}\sim$,$\mathcal{B}^{\tilde{H}}[0]=B^{H}[0]$.

PROOF: After Fourier transformation, Lemma 15 is easily shown by thefact that he ring of H-invariant polynomial on is generated by Casimir$t$polynomial$q$$\omega$

.I
16

89

PROPOSITION 16. In the same case in Proposition 13, if the assumption$(A)$: $\mathcal{B}^{H}(q-\{0\})[S-\{0\}]=0$ holds, then we have

PROOF: Takeis a$g\in \mathcal{B}(R)$there is an$f\in \mathcal{B}^{H}(q)$such that$f^{\sim}\in \mathcal{B}^{\tilde{H}}(q)$$f-\overline{f}\in \mathcal{B}^{H}(q)[S]$. For the restriction$f|_{q_{rcg}}=g\circ\omega’$such that. For$f|_{q_{reg}}\in \mathcal{B}^{H}(q_{reg})$by (4.1). For$f|_{q-\{0\}}\sim=g\circ\omega’$.

, there$g\circ\omega_{1}\in \mathcal{B}^{\tilde{H}}(q-\{0\})$,by Theorem 14. Then$(f^{\sim}-f)|_{q-\{0\}}\in \mathcal{B}^{H}(q-\{0\})[S-\{0\}]$be zero by the assumption (A). Then$f\in \mathcal{B}^{\overline{H}}(q)$$\mathcal{B}^{\tilde{H}}(q)=\mathcal{B}^{H}(q)$$f-f\in\sim \mathcal{B}^{H}[0]=\mathcal{B}^{\tilde{H}}[0]$, it must, therefore.I

We can check the assumption (A) in some cases.
PROPOSITION 17.
(1) The singular set $S=\{0\}$ in cases$1^{o},$$2^{o},$$3^{o},$$4^{o}$

and$5^{o}$. In

particular, the assumption $(A)$ is satisfied.
(2) The assumption $(A)$ is satisfi $ed$ in cases$6^{o},$$7^{o}$.

PROOF:(1) This is the definition of isotropic symmetric pairs. [2](2) If $m\geq 3$ , then Theorem 3 implies this. Thus we may assume$m=2$ . Since the problem is outside the origin, we adopt the coor-dinates (3.5). In this coordinates, $S=\{x_{1}\sim=x_{2}\sim=0,y\sim_{2}\neq 0\}$ and$\mathcal{M}=\mathcal{D}/\mathcal{I}=D/\mathcal{D}(\partial_{\overline{y}_{1}},\partial_{\overline{y}_{2}},\overline{x}_{1}\partial_{\overline{x}_{1}},x_{2}\sim\partial_{\overline{x}_{1}})$, then$Hom_{D}(\mathcal{M}, \Gamma_{S}\mathcal{B})\subset 1$$Hom_{D}(D/D_{X}^{\sim_{1}}\partial_{\overline{x}_{1}}, \Gamma_{\{x=0\}}\sim_{1}\mathcal{B})=0.1$

Summing up the results, we havefor the case$6^{o}$$\mathcal{B}^{\tilde{H}}(q)=\mathcal{B}^{H}(q)$and $m=2$ .

17

in cases$3^{o}\tilde{8}^{o}$except90

4.2. The Space : $SL(3, R)/S(GL_{+}(2, R)\cross GL_{+}(1, R))$

We consider case$m=2$ . We adopt the notations in \S 3.1. We$6^{o},$

introduce an element$f\in\Gamma(R^{4}-\{0\}, \mathcal{B})$$f=\{\begin{array}{l}\delta(x_{l}y_{1}+x_{2}y_{2})\{Y(x_{1})Y(y_{2})+Y(x_{2})Y(y_{1})\}\in\Gamma(\{x_{1}\neq 0ory_{2}\neq 0\},\mathcal{B})\delta(x_{1}y_{1}+x_{2}y_{2})\{Y(x_{2})Y(-y_{1})+Y(-x_{2})Y(y_{1})\}\in\Gamma(\{x_{2}\neq 0ory_{1}\neq 0\},\mathcal{B})\end{array}$

Then $f$ is H-invariant, $\square f=0$ , and$(x_{1}\partial_{y_{2}}-x_{2}\partial_{y_{1}})f=\delta(y)=\delta(y_{1}, y_{2})$$(y_{1}\partial_{x_{2}}-y_{2}\partial_{x_{1}})f=\delta(x)$

thus

$f$

is not H-invariant. Let$(\theta+2)f=0$.

We takethe mapping $(\theta+2)$ :$\mathcal{B}[0]$$\theta=x_{1}\partial_{x_{1}}+x_{2}\partial_{x_{2}}+y_{1}\partial_{y_{1}}+y_{2}\partial_{y_{2}}$$f\sim\in \mathcal{B}(R^{4})$$\mathcal{B}[0]arrow \mathcal{B}[0]$such that$\overline{f}|_{R^{4}-\{0\}}=f$, then. Sinceis bijective, there is a unique $g\in$such that $(\theta+2)g=(\theta+2)f\sim$. We define $f_{0}=f-g\sim\in \mathcal{B}(q)$ ,then $(\theta+2)f_{0}=0$ . For any$2)D_{X}f_{0}=D_{X}(\theta+2)f_{0}=0$

口ゐ,$\in \mathcal{B}[0]$$X\in \mathfrak{h},$$[D_{X}, \theta]=0$, then $D_{X}f_{0}=0$ , i.e.

is homogeneous of degree $-4$ , thenimpli es that$f_{0}\in \mathcal{B}^{H}(q)$$(\theta+$. Moreover$\square f_{0}=c\delta(x, y)$by someconstant $c\in$ C. The relation $(\square f_{0})(-x_{1}, x_{2}, y_{1}, -y_{2})=-\square f_{0}(x, y)$and $c\delta(-x_{1}, x_{2}, y_{1}, -y_{2})=c\delta(x, y)$ imply 口 $fo=0$ . Therefore $fo\in$$\mathcal{B}_{0}^{H}(q),$$0,$and supp $(f_{0})=\overline{N_{+}}=\{(x, y)\in R^{4};\omega(x, y)=$$f_{0}\not\in \mathcal{B}^{\overline{H}}(q)$$x_{1}y_{2}-x_{2}y_{1}\geq 0$

is contained in$\overline{N_{+}}-\{0\}$Unfortunately

tion because}. Conversely an element$\omega’$$\mathcal{B}^{H}(q-\{0\})$is a constant multiple ofwhose support$f_{0}|_{q-\{0\}}$.

: $q_{reg}arrow R$ does not give H-orbit parametriza-$\omega^{\prime-1}(0)$has two connected components $N\pm=\{(x, y)\in$

18

91

$R^{4}$; $\omega(x, y)=0,$ $\pm(x_{1}y_{2}-x_{2}y_{1})>0$ }, Therefore Proposition 13 must be

modified in this case. Letby taking copies of$R$

be the non-Hausdorff manifold obtained

$R’$

, say

$R_{-}$$R_{+},$, and sticking together the positive

parts of $R+and$ R-and the negative parts of those [1]. Let $R’arrow^{\rho}R$

be the canonical map and define

tions on$R’$$\mathcal{B}_{R’}=\rho^{-1}\mathcal{B}_{R}$, the sheaf of hyperfunc-. We define a real analytic mapping$\omega^{\prime\prime-1}(t)=\omega^{\prime-1}(t)$for any $t\in R^{x}$ and$\omega’’$: $q_{reg}arrow R’$ by$\omega^{\prime\prime-1}(O\pm)=(N_{\pm})$$q_{reg}$$\subset$$q-\{0\}$$\subset$$\omega’’\downarrow$$\omega’\backslash$$\downarrow\omega_{1}$$J\omega$$R^{l}$$arrow$

.

$q$

$R$

$\rho$

Then$\omega^{ll}$: $q_{reg}arrow R^{l}$ gives H-orbit parametrization, and therefore(4.2)$\mathcal{B}^{H}(q_{reg})=\omega^{l/*}(\mathcal{B}_{R’}(R^{l}))$.

PROPOSITION 18. Retain the above notation.
(1)(2)$\mathcal{B}^{H}(q)=\mathcal{B}^{\tilde{H}}(q)\oplus Cf_{0}$.

$\mathcal{B}_{0}^{H}(q)=\mathcal{B}_{0}^{\tilde{H}}(q)\oplus Cf_{0}$.

PROOF: (1) Take$\mathcal{B}(R’)$

such that$f\in \mathcal{B}^{H}(q)$$f|_{q_{r\cdot g}}=g\circ\omega^{\prime l}$$(g|_{R_{-}})\circ\omega_{1}\in \mathcal{B}^{\tilde{H}}(q-\{0\})$$(g|_{R_{-}})\circ\omega_{1}$. Then$f|_{q_{r\epsilon g}}\in \mathcal{B}^{H}(q_{reg})$by (4.2). For, then there is anby Theorem 14. Then$\mathcal{B}^{H}(q-\{0\})[\overline{N_{+}}-\{0\}]$$cf_{0}$. For, there is a $g\in$$g|_{R_{-}}\in \mathcal{B}_{R’}(R_{-})=\mathcal{B}_{R}(R)$$f\in\sim \mathcal{B}^{\tilde{H}}(q)$$f-\overline{f}\in \mathcal{B}^{H}(q)[\overline{N_{+}}]$,

such that $f|_{q-\{0\}}\sim=$. For$(f-\overline{f})|_{q-\{0\}}\in$, there is a constant $c\in C$ such that $(f-f)|_{q-\{0\}}\sim=$$(f-f-cf_{0})\sim\in \mathcal{B}^{H}[0]=\mathcal{B}^{\tilde{H}}[0]$(2) It is obvious from (1) because. Therefore$f_{0}\in \mathcal{B}_{0}^{H}$

19

. I

$f\in \mathcal{B}^{\tilde{H}}(q)\oplus Cf_{0}$.

92Becauseis a distribution and tempered, the table in [2, \S 6.3] must$f_{0}$

be corrected: case 7, $m=3$ and$\dim D_{\lambda,H}’(q)=3$

ノ ‘ $r$$\oint$$>\varphi_{\backslash },$$[P]$$\tau*\cdot$, in his notation, then, and $\dim S_{\lambda,H}’(q)=3$ .
記した\‘o $l,$$-\sim t\emptyset$

正し \langle \sim 手,$T\iota\downarrow\tau T_{\iota}$$\lambda=0$)$\underline{\backslash }\simeq O$$\simeq$

二の64,) $\neq 7$$F\grave{0}1_{\vee}^{-}\Gamma\delta^{t}$。

$Z$$|=Rf\eta 1\mathcal{F}$L|\sim っかして$\wedge\nu\ddagger$

正

$b$

?

REFERENCES[1] Cerezo, A., Equations with constant coefficients invariant undera group of linear transformations, Trans. Amer. Math. Soc. 204(1975), 267-298.
[2] van Dijk, G., Invariant eigendistributions on the tangent space ofa rank one semisimple symmetric space, Math. Ann. 268 (1984),

405-A 16.
[3] van Dijk, G., On a class of generalized Gelfand pairs, Math. Z. 193(1986), 581-593.
[4] Helgason, S., “Differential geometry, Lie groups, and symmetricspaces,” Academic Press, London, 1978..
[5] Hotta, Rand M. Kashiwara, The invariant holonomic system on asemisimple Lie algebra, Invent. Math. 75 (1984), 327-358.
[6] Muro, M., Singular invariant tempered distributions on regular prehomogeneous vector spaces, J. Funct. Anal. 76 (1988), 317-345.
$\cdot$[7] $Seki\dot{g}uchi$ , J., Invariant spherical hyperfunctions on the tangent

space of a symmetric space, Advanced Studies in Pure Math. 4(1985), 83-126.
[$h_{n}k1$$\theta 30ct_{\mathfrak{U}^{\backslash }\mathcal{M}’,}H,$
$,$

コ

)

$=f^{\backslash }$$R^{\theta^{9}}\sim^{A_{\oplus_{\backslash }^{\backslash }}}E\backslash$のプー$\eta ff\zeta_{\backslash }I^{\backslash \ovalbox{\tt\small REJECT}_{\emptyset}}A7^{\backslash }\downarrow\ovalbox{\tt\small REJECT}\not\in\ *_{{}^{t}r}$.

$\neq r^{p_{\}}A$$\ovalbox{\tt\small REJECT} M$

20$( \int 7?P)+\uparrow-53$$\mathfrak{y}_{\delta}$