数理解析研究所講究録909 巻 1995 年 51-53

51

TOWARD WAVE MODELS OF REPRESENTATIONSOF REAL SEMISIMPLE LIE GROUPS
by Takayuki Oda$\not\in \mathrm{R}*^{r}\backslash \mathrm{H}\hslash_{1}^{*}\mathfrak{c}\mathrm{k}_{\backslash }\mathcal{T}\backslash \cdot\#\backslash \mathrm{g}_{\mathrm{f}}\mathfrak{x}$)This is an introduction for a nulnber of talks given in this conference.

A typical situation to construct automorphic -funcions froIn automorphic forms,
is the following.
$\mathrm{L}$We are given a semisimple algebraic group over a field of Kronecker dimension
isthc case, when
1. Sometime
is assumed to be quasi-split.
one use Whittaker models. Also a closed algebraic subgroup $R$ of is given whichis large enough so that it is spherical.
$G$

$G$

$k$

$\mathrm{T}\}_{1}\mathrm{i}_{\mathrm{S}}$$\mathrm{P}^{\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{U}111}\mathrm{a}\cdot \mathrm{y}$$G$

Assume that is quasi-split for simplicity. Fix a minimal parabolic subgroupof over . Then theof is called spherical,
the orbit of the$R$extension of scalarsof on thc flag manifold
has openorbit. As shown by, this condition is equivalent to the follwing.
Brion
$P$

$G$

$G$

$k$

$G$

$\mathrm{s}\mathrm{u}\mathrm{b}_{\circ}\sigma \mathrm{r}\mathrm{o}\mathrm{u}\mathrm{p}R$$\mathrm{w}1_{1}\mathrm{e}\mathrm{n}$$\overline{R}=R\otimes_{k}\overline{h^{\wedge}}$$G\Theta\overline{k^{\wedge}}/P\otimes\overline{k}$$([\mathrm{B}1])$$\mathrm{M}$The restriction tois multiplicity-free.
$(^{*})$$\overline{R}$of any finite dimensional irreducible representation of$\overline{G}$Also as shown in [B1], this condition implies and is equivalent to the condition
that the double coset
is finite.
$\overline{R}\backslash \overline{G}/\overline{P}$Remark. The notion of”spherical” seems to be discussed only for the case of groups
over algebraically closed fields, in the literature. Of urse one can generalize thisconcept for any , and can use the terminology ” -spherical groups”, if one canprovc any non-trival statement about that.....
$\mathrm{c}o$

$k$

$k$

Now, let us replace the global field by its completion at aplace, which
we also denote by the same .
-valued points of algebraic groups $G,$ $P,$ aredenoted by the same symbol by an abuse of symbol.
$k$

$k$

$\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}_{\vee}$$R$$\mathrm{T}\mathrm{h}\mathrm{e}_{\vee}k$

An optimist might hope the following.

Folklore. Given an airre ducible repres entationof R. Let$Ind_{R()}^{G}\eta bc$ theooth in duction of to G. Then for any irreducible admissible
repres entation of , the dimension of in ertwin $ig$ spa ce$s\mathrm{m}$$\pi$$d\mathrm{m}$issi$\mathrm{b}le$$\eta$$C_{\eta}^{\infty}(R\backslash G)=$$\eta$

$G$

$t$

$Hom_{G(\pi},$ $C\infty(\eta\backslash RG))$

is at most 1.

andIn view of Frobenius reciprocity with respectkind of multiplicity free statement for the restriction of$G$

$R$
$\pi$

, this is no other than ato the subgroup $R$ .

52When is a -adic fielf, this problem is considered by Murase and Sugano in theirwork (see their articles in this proceedings) on theautomorphic Lfunctions, under the name ”Shintani function”. Among others(and S. Kato)
now have explicit formula of $K$ -invariant vector in thehe unique element$\Phi\in Hom_{G(}\Pi,$(.ill principal series, when is of class 1 (i.e. asrepresentation), in rather general situation.
$k$

$p$

$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{C}\mathrm{t}\mathrm{i}_{\mathrm{t}}$)$)\mathrm{f}$$1\mathrm{l}\langle$$\dagger 1_{1\mathrm{t}^{\backslash }},\mathrm{y}$$\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{t}^{\backslash }()\{\uparrow$$c\infty\eta(R\backslash G))$$\pi$$\mathrm{s}_{1^{\mathrm{J}1\iota 11}}’ \mathrm{i}$Naive analogue at archemean places does not seem to hold in general. The typical counter-example is the case when $R$ is a maxilnal unipotent subgroup and is a
non-degenerate character. Then the above space is the space of Whittaker vectors(or functionals if you prefer this terminology). As shown by Kostant, the dimension of the above intertwing space for irreducible principal series representationof is the order of the (little) Weyl group. In this case, to have multiplicity one
statement, we have impose an increasing condition for the functions in target space, to replace it by much smaller space. As shown by Wallach, wehave multiplicity-free statement after this modification.
$\eta$$([\mathrm{K}])$$G$$C_{\eta}^{\infty}(R\backslash c)$$([\mathrm{t}\mathrm{V}])$At the real place, Yamashita [Y] proved various sufficient conditions for thefiniteness of the above intertwining space. Because we need further terminology
to describe his results, we do not give the details of his results here. They covervarious important cases in application. I recommend the readers to consult withthe original paper. Here we quote a theorem of [B-O] wllic.h is easy to state.
Theorem. (Bien-Oshim a [B-O]) Set $k=$ R. Assume thathas open orbits on the Hag manifold. Then for any finite-dimensional represent ation of and any irreducible a dmissible
odule , the in terwining $sp$ ace
is of finite dimension.
$R_{\mathrm{C}}$$G_{\mathrm{C}}/P_{\mathrm{C}}$$\eta$$R$$(\mathrm{g}, K)- m$$\pi$$Hom\mathfrak{g},K(\pi, c_{\eta}\infty(R\backslash G))$Now let me explain the titles of some talks given here. But before that recall
the following. When $G=S_{p}($ 2 :Miyazaki-Oda $[]$ computed explicitely theholonomic system for the radial part of the principal series Whuittaker functions
with smallest $K$ -types.
and Miyazaki-Oda $[]$ give integral expressions ofWhittaker function with the oomer $K$ -type for the discrete series representations
and generalized principal series for the cuspidal parabolic subgroup corresponding
to long root.
$\mathbb{R}),$$\mathrm{O}\mathrm{d}\mathrm{a}[]$The talk of(Masatoshi) will discuss thesystem of the radial partof the matrix coefficients with smallest -types of the principal series representationof, and this shift operates. Oda’s talk is on the holonomic system of thematrix coefficients with minimal -type of the large discrete series representationsof.
$\mathrm{I}\mathrm{i}\mathrm{d}\mathrm{a}_{\mathit{1}}$$\mathrm{h}\mathrm{o}1_{01\mathrm{n}\mathrm{o}\mathrm{r}}\mathrm{P}\mathrm{h}\mathrm{i}\mathrm{c}$$k$$Sp(2;\mathbb{R})$$I1’$$Sp(2;\mathbb{R})$Takuya Miyazaki will talk about the holonomic system for the radial part ofgeneralized Whittaker functions onwith respect to the Siegel parabolicsubgroup.
$Sp(2;\mathbb{R})$Hayata (Takahiro) will treat principal series Whittaker function on $SU(2,2)$ withsmallest K-type.

53

Taniguchi (Kenji) works out completely the Whittaker functions with minimal-types of the discrete series on real unitary group $SU(n, 1)$ of split-rank 1 (andrecently he settled also the case of Spin $(2n, 1))$ .
$I’\backslash$

Tuzuki considers the case when $G=SU(2,1)$ and $H=S(U(1,1)\cross U(1)\cong U(1,1)$
when the representations of $H$ are of infinitc-dimension.
$?l$$\eta$REFERENCES
On spherical subqroups

The following references are noticed by Prof. T. Matsuki, Manuscrip[Brl] Brion, M.: Quelques proprie’te’s des espaces homog\‘enes
Math. 55 (1986) 191-198.
[Br2] Brion, M.: Classification des espaces homog\’enes sph\’eriques, Comp. Math.
63 (1987) 189-208.
[K] Kr\"amer, M.:
Untergruppen in Kompakten zusanlmenh\"angenden
Liegruppen Comp. Math. 38 (1979) 129-153.
$\mathrm{s}\mathrm{p}\mathrm{h}\mathrm{e}^{\text{ノ}}\mathrm{r}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{S}$$\mathrm{S}\mathrm{p}\mathrm{h}\ddot{\mathrm{a}}\mathrm{r}\mathrm{i}_{\mathrm{S}}\mathrm{c}\mathrm{h}\mathrm{e}$The section 4 of[M]
, T.: Orbits on Flag Manifolds, Proc. of Intern. Congress of Math.,
Kyoto (1990), 807-813.
is illuiminating. Here is a much simpler proof of the main theorem of [Brl]. The,
conjecture 1,2 on real spherical subgroups given there are solved by. B. Kimelfeld
and F. Bien.
$\mathrm{I}\mathrm{v}\mathrm{I}\mathrm{a}\mathrm{t}\mathrm{S}\mathrm{u}\mathrm{k}\mathrm{i}$On $multiplicit\cdot U$ finiteness[W] Wallach, : Asymptotic expansions of generalized matrix entries ofsentations of realgroups. Lie Group Representations I. Lecture
Notes in Math. 1024, Springer (1984), 284-369.
[B-O] Bien, F. and Oshima, T.: Multiplicities of induced representations of semisim
ple his groups, preprint.
[Y] Yamashita, H.: Criteria for the finiteness of restinction on modules tosubalgebras and application to Harish-Chandra modules. J. Funct. Annual.
121 (1994), 296-329.
$\mathrm{n}.$$\mathrm{r}\mathrm{e}_{1^{)\mathrm{r}\mathrm{e}-}}$$\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\uparrow\partial \mathrm{i}\mathrm{l}\Gamma \mathrm{e}$-