数理解析研究所講究録1052 巻 1998 年 10-18

10;
FOURIER-JACOBI TYPE SPHERICAL FUNCTIONS ON
THE CASE OF-PRINCIPAL SERIES AND DISCRETE SERIES$Sp(2, \mathrm{R})$$P_{J}$

東大数理平野(MIKI HIRANO)

幹

Contents
1. Introduction
2. Preliminaries
3. Fourier-Jacobi type spherical functions
4. Differential
5. Result$\mathrm{e}\mathrm{Q}_{\wedge}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$l.Introduction
In this note, we study a kind of generalized Whittaker models, or equally, ofgeneralized spherical functions associated with automorphic forms on the real symplectic group of degree two. We call these spherical functions ’Fourier-Jacobi type’,
since these are closely connected with the coefficients of the ’Fourier-Jacobi expansions’ of (holomorphic or non-holomorphic) automorphic forms. Also these can beconsidered as a non-holomorphic analogue of the local Whittaker-Shintani functions
onof Fourier-Jacobi type in the paper of Murase and Sugano [6].
$Sp(2, \mathrm{R})$2. Preliminaries
the set of integers
such that
2.1. Groups and algebras. We denote by$n\geq m$ . Moreover, we use the convention that unwritten components of a matrix
are zero.
Let be the real symplectic groupof degree two given by$n$$\mathrm{z}_{\geq m}$$G$$Sp(2, \mathrm{R})$$Sp(2, \mathrm{R})=\{g\in M_{4}(\mathrm{R})|{}^{t}gJ_{2}g=J_{2}=,$ $\det g=1\}$.

Let
be a Cartan involution of and $K$ be the set of fixed pointsof . Then $K$ becomes a maximal compact subgroup of which is isomorphic to
the unitary group $U(2)$ .
Let $\mathrm{g}=\{X\in M_{4}(\mathrm{R})|J_{2}X+{}^{t}XJ_{2}=0\}$ be the Lie algebra of . If we denotethe differential of again by , then we have
. Let and be
and-l eigenspaces of in , respectively, and hence$G$$\theta(g)={}^{t}\overline{g}^{-1}(g\in G)$$G$$\theta$$G$$\theta$$\mathrm{t}\mathrm{h}\mathrm{e}+1$$\theta(X)=-^{t}\overline{X}(X\in \mathrm{g})$$\theta$$\mathrm{f}$$\mathfrak{p}$$\theta$$\mathrm{g}$$\mathrm{f}=\{X\in|\mathrm{A},$ $B\in M_{2}(\mathrm{R}),{}^{t}A=-A,{}^{t}B=B\}$ ,$\mathfrak{p}=\{X\in|A,$ $B\in M_{2}(\mathrm{R}),{}^{t}A=A,{}^{t}B=B\}$ .
Typeset by$A_{\mathcal{M}}\theta \mathrm{I}\mathrm{r}\mathrm{x}$11Then we have a Cartan decomposition$K$ which is isomorphic to the unitary
algebraFor a L.ie algebra 1, we denote by
a compact Cartan subalgebra of given by$\mathfrak{g}=\mathrm{f}\oplus \mathfrak{p}$. Of course, is the Lie algebra of.
the complexification of . Let be$\mathrm{g}$$\mathfrak{u}.(2)$$\mathfrak{l}_{\mathbb{C}}=\{\otimes_{\mathrm{R}}\mathrm{C}$$\mathfrak{l}$$\mathfrak{h}$$\mathrm{g}$$\mathfrak{h}=\{H(\theta_{1}, \theta_{2})=$

Now we identify a linear form :
with
where
. Then the set of roots
given by$\beta$$\theta_{i}\in \mathrm{R}\}$

via$(\beta_{1}, \beta_{2})\in \mathrm{C}^{2}$$\mathfrak{h}_{\mathbb{C}}arrow \mathrm{C}$$e_{i}(H(\theta_{1,2}\theta))=\sqrt{-1}\theta_{i}$.

$\beta=\beta_{1}e_{1}+\beta_{2}e_{2}$$\triangle=\triangle(\mathfrak{h}_{\mathbb{C},\mathrm{g}_{\mathbb{C}}})$

of

,

is

$(\mathfrak{h}_{\mathbb{C},9\mathbb{C}})$.

$\cdot$$\triangle=\{\pm(2,0), \pm(0,2), \pm(1,1), \pm(1, -1)\}$.

Fix a positive root system $\triangle^{+}=\{(2,0))(0,2), (1,1), (1, -1)\}$ , and put
the set of compact and non-compact positive roots, respectively. Then$\triangle_{c}^{+}$

and$\triangle_{n}^{+}$$\triangle_{c}^{+}=\{(1, -1)\}$,$\triangle_{n}^{+}=\{(2,0), (0,2), (1,1)\}$.

by , then we have a decomposition
If we denote the root space for
.
and
-with
the Jacobi maximal parabolic subgroup of with the Langlands decomPut$P_{J}=M_{J}A_{J}N_{J}$, whereposition$\beta\in\triangle$$\mathfrak{p}_{\mathbb{C}}=$$\mathfrak{g}_{\beta}$$\mathfrak{p}_{-}=\sum_{\beta\in\triangle}+n\mathcal{B}-\beta$$\mathfrak{p}_{+}=\sum_{\beta\in\triangle_{n}^{+}}\mathrm{B}\beta$$\mathfrak{p}_{+}\oplus \mathfrak{p}$$G$$P_{J}$$M_{J}=\{\epsilon\in\{\pm 1\}$ ,$\in SL(2, \mathrm{R})\}\simeq\{\pm I\}\cross SL(2, \mathrm{R})$,$N_{J}=\{n(x, y;z)=$. Remark that the unipotent radicalofand. The Levi partofis isomorphic to the 3-dimensional Heisenberg group
via the conjugate action, andgives the centralizer of the centeracts onin. Now we define the Jacobi groupofhereby the semidirect product$S’..L(2, \mathrm{R})$,.
is the identity component of$A_{J}=\{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(a, 1, a^{-},11)|a>0\}$$N_{J}$$M_{J}A_{J}$$\mathcal{H}_{1}$$P_{J}$$P_{J}$$M_{J}$$N_{J}$$Z(N_{J})=\{n(\mathrm{O}, 0;\mathcal{Z})|z\in \mathrm{R}\}\simeq \mathrm{R}$$N_{J}$$M_{J}A_{J}$$R_{J}$$M_{J}^{\mathrm{o}}\ltimes N_{J}\simeq SL(2,.\mathrm{R}.)\ltimes..\mathcal{H}_{1}.’.\mathrm{w}$$\mathit{2}.\mathit{2}.Rep\Gamma esentations$$M_{J}^{\mathrm{O}}..\cdot,\simeq$‘$M_{J}$. First we investigate the irreducible unitary representations. Since $Z(R_{J})$. $=Z(N_{J})\simeq \mathrm{R}$, the central characters ofof the Jacobi groupare parametrized by the real numbers. Then we call anandelements inandirreducible unitary representation ofof type $m$ if its central character
. Let
with
of type $m$ . According to theis of the form
$R_{J}$$\hat{R}_{J}$$\hat{N}_{J}$$R_{J}$$z\mapsto e^{2\pi\sqrt{-1}mz}$$m\in \mathrm{R}$$N_{J}$$I\text{ノ}\in\hat{N}_{J}$12
theorem of Stone-von Neumann (cf. Corwin-Greenleaf [1; pp.46-47, 51-52]), is acharacter if $m=0$ and is infinite dimensional if $m\neq 0$ . Moreover l ノ of type $m\neq 0$
is uniquely determined by $m$ up to unitary equivalence. Now we fix an irreducible
unitary representationofof type $m\neq 0$ . Rom the theory of the Weilrepresentation,
can be extended to a continuous true projective unitary
representation
of
by
for
with the Weil representationon. Herehas a factor set which is aproper 2-cocycle.
$\nu$

$\nu$

$N_{J}$$(\nu_{m}, \mathcal{U}_{m})$$(I\text{ノ_{}m’ m}\mathcal{U})$$R_{J}$$(\tilde{\nu}_{m},\mathcal{U}_{m})$$\tilde{l}\text{ノ_{}m}(\tilde{n})=W_{m}(g)_{l\text{ノ_{}m}}(n)$$W_{m}$$\tilde{n}=g\cdot n\in M_{J}^{\mathrm{o}}\ltimes N_{J}$$\tilde{l}\text{ノ_{}m}$$M_{J}^{\mathrm{o}}$$\alpha$Lemma 2.1. (Satake [7; Appendix I, Proposition 2]) Let
as above.
For every irreducible projective unitary representation ofwith factor set,
put. Then is an irreducible unitaryforrepresentation of.
, all irreducible unitaw representations ofof$m\neq0$
type
are obtained in this manner. Moreover is square-integrable iff is so.
$\tilde{\nu}_{m}(m\neq 0)$$\pi$$\rho(\tilde{n})=\pi(g)\otimes\tilde{\nu}_{m}(\tilde{n})$$\tilde{n}=g\cdot n\in M_{J}^{\mathrm{o}}\ltimes N_{J}$$\alpha^{-1}$$M_{J}^{\mathrm{o}}$$\rho$$c_{onve}r\mathit{8}ely$$R_{J}$$R_{J}$$\pi$$\rho$

Let
be an irreducible unitary representation ofof type $m\neq 0$ . Hkom
the above lemma, we can regard (as a tensor product representation
Here, if wefor the double cover of,is a unitary representation ofwhich is extended fromas above andis a unitary representation ofwhich does not factor. On the other hand, the unitary dual ofthrough
is given as follows.
$R_{J}$$(\rho, \mathcal{F}_{\rho})$$\rho,$$(\pi_{1}\otimes\tilde{l}\text{ノ_{}m}, \mathcal{W}_{\pi}1\otimes \mathcal{U}_{m}).$,$\mathcal{F}_{\rho}\underline{)}\in\hat{R}_{J}$$M_{J}^{\mathrm{O}}\simeq SL(2, \mathrm{R})$$\mathrm{w}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{e}\underline{M}_{J}^{\mathrm{O}}$$(\iota \text{ノ_{}m},\mathcal{U}_{m}\sim)$$(\nu_{m},\mathcal{U}_{m})\in$$M_{J^{\ltimes}}^{\mathrm{O}}NJ$$\hat{N}_{J}$$(\pi_{1}, \mathcal{W}_{\pi_{1}})$$\overline{M}_{J}^{\mathrm{o}}$$\overline{M}_{J}^{\mathrm{o}}$$M_{J}^{\mathrm{o}}$Proposition 2.2. (cf.
; Lemma 4.1, 4.2]) The following representations
exhaust a set of $re\underline{pr}esentatives$ for the equivalenceof irreducible unitaw
representations of.
(1) ( $unitar_{J}l$ principal se $r’ ies$)$\tau=0,1,$
except for the case$\mathrm{G}\mathrm{e}\mathrm{l}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{t}[2$$clas\mathit{8}es$$SL(2, \mathrm{R})$$\mathcal{P}_{s}^{\mathcal{T}},$$(s, \mathcal{T})=(0,1)$)$\mathit{8}e\dot{n}es$$C_{s}^{\mathcal{T}},$.

(3) ((limit of) discrete series)(4) (quotient representation)(5) The trivial representation of$0<s<1$ for $\tau=0,1$ and$D_{k}^{\pm},$$D_{\frac{1}{2}}^{-},$In the above, thefactor through$SL(2, \mathrm{R})$$\tau=\pm\frac{1}{2}$$D_{\frac{+1}{2}}$

with

.

$C_{s}^{\prime r}$.

for $\tau=0,1,$$D_{k}^{\pm}$for$k\in \mathrm{z}_{\geq 1}$

one of the irreducible unitary representations.

$k \in\frac{1}{2}\mathrm{Z}\backslash \mathrm{Z},$for$0<s< \frac{1}{2}$.

, and the otherwise not.

$(\pi_{1}, \mathcal{W}_{\pi_{1}})$$D_{k}^{\pm}$$k \in\frac{1}{2}\mathrm{z}_{\geq 2}$$SL(2, \mathrm{R})$$repre\mathit{8}entati_{on}s\mathcal{P}_{s}^{\mathcal{T}},$

Hence we take as
with
and

$\pm.\frac{1}{2}$.

(2) (complementaw$\tau=\pm\frac{1}{2}$$s\in\sqrt{-1}\mathrm{R},$and (5)$P_{s}^{\mathcal{T}},$$C_{s}^{\mathcal{T}}$$k \geq\frac{1}{2}$Remark 2.3. The Weil representationconsidered as the representation ofhas the following irreducible decomposition;$W_{m}$$D_{\frac{+1}{2}}\oplus D_{\frac{+3}{2}}$$W_{m}=\{$$D_{\frac{1}{2}}^{-} \oplus D^{-}\frac{3}{2}$,if $m>0$ ,,if $m<0$ .

$\overline{M}_{J}^{\mathrm{o}}$Next, we treat the irreducible unitary representations of $K$ . Sinceis also a
positive system of, then the set of the-dominant weights, and thus$\triangle_{c}^{+}$$\triangle(\epsilon_{\mathbb{C}}, \mathfrak{h}_{\mathbb{C}})$$\triangle_{c}^{+}$

13$\hat{K}$, is parametrized by the set$\Lambda=\{\lambda=(\lambda_{1}, \lambda_{2})|\lambda_{i}\in \mathrm{z}, \lambda_{1}\geq\lambda_{2}\}$; Theorem 4.28]). We denote by
the element of corresponding to. Herewith.
$K$
Both ofbecome -modules via the adjoint representation of $K$ , and we haveisomorphisms
and. For a given irreducible $K$-modulewith the parameter, the tensor product $K$-modules
and-have the irreducible decompositions(cf.

$\mathrm{K}\mathrm{n}\mathrm{a}\mathrm{p}\mathrm{P}[4$$\hat{K}$$(\tau_{\lambda}, V_{\lambda})$$\lambda=(\lambda_{1}, \lambda_{2})\in\Lambda$$\dim V_{\lambda}=d_{\lambda}+1$$d_{\lambda}=\lambda_{1}-\lambda_{2}$$\mathfrak{p}_{\pm}$$\mathfrak{p}_{+}\simeq V_{(2,0)}$$\mathfrak{p}_{-}\simeq V_{(0,-2)}$$V_{\lambda}$$\lambda=(\lambda_{1}, \lambda_{2})\in\Lambda$$V_{\lambda}\otimes \mathfrak{p}_{+}$$V_{\lambda}\otimes \mathfrak{p}$,$V_{\lambda}\otimes \mathfrak{p}_{+\bigoplus_{\beta\triangle^{+}}}\simeq\in nV\lambda+\beta$$V_{\lambda}\otimes \mathfrak{p}_{-}\simeq\beta\in\oplus V_{\lambda-\beta}\triangle n+\cdot$

For each
, let
:
and
:
be theprojectors into the irreducible factors ofIn this note, we consider the following two series of representations of ; oneis the principal series induced from, and the other is the ‘crete series... We
explain these representations in the remaining of this section.
Letbe a representation ofwith a character
and a discrete series representationof,and take a quasi-characterofsuch that. Then we can construct a induced representationoffrom the Jacobi maximal parabolic subgroup $P_{J}=M_{J}A_{J}N_{J}$ by the usual manner(cf.
; Chapter VII]), and callthe-principal series
representation of . The following lemma is derived from thereciprocity
for induced representations.
$P^{\beta}$$\beta\in\triangle_{n}^{+}$$P^{-\beta}$$V_{\lambda}\otimes \mathfrak{p}_{+}arrow V_{\lambda+\beta}$$V_{\lambda}\otimes \mathfrak{p}_{-}arrow V_{\lambda-\beta}$$V_{\lambda}\otimes \mathfrak{p}\pm\cdot$$G$$P_{J}$$\sigma=(\epsilon, D)$$\mathrm{d}\mathrm{i}\mathrm{s}$$M_{J}\simeq\{\pm I\}\cross SL(2, \mathrm{R})$$\epsilon:\{\pm I\}arrow \mathrm{C}^{\mathrm{x}}$$D=D_{n}^{\pm}(n\in \mathrm{z}_{\geq 2})$$\nu_{z}(z\in \mathrm{C})$$A_{J}$$SL(2, \mathrm{R})$$\nu_{z}(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(a, 1, a^{-1},1))=$$a^{z}$$\mathrm{I}\mathrm{n}\mathrm{d}_{P_{j}}^{G}(\sigma\otimes\nu_{z}\otimes 1_{N_{J}})$$\mathrm{K}\mathrm{n}\mathrm{a}\mathrm{p}\mathrm{P}[4$$\mathrm{I}\mathrm{n}\mathrm{d}_{Pj}^{G}(\sigma\otimes\nu_{z}\otimes 1_{N_{J}})$$G$$P_{J}$$G$$\mathrm{H}\mathrm{Y}_{0}.\mathrm{b}\mathrm{e}\mathrm{n}\mathrm{i}\mathrm{u}\mathrm{S}$Lemma 2.4.

with the parameter
does not occur in the $K$ -type of. The ’comer’ $K$ -types
ofgiven below occur with multiplicity one.
(1) $\lambda=(n, n)$ for
and $D=D_{n}^{+}$ ,(2) $\lambda=(n, n-1)$ for $\epsilon(\gamma)=-(-1)n$ and $D=D_{n}^{+}$ ,(3) $\lambda=(-n, -n)$ for
and $D=D_{n}^{-}$ ,(4) $\lambda=(-n+1, -n)$ for $\epsilon(\gamma)=-(-1)n$ and $D=D_{n}^{-}$ .
Here.
$\tau_{\lambda}\in\hat{K}$$\mathrm{I}\mathrm{n}\mathrm{d}_{PJ}^{G}(\sigma\otimes\nu_{z}\otimes 1_{N_{J}})$$\tau_{\lambda}\in\hat{K}$$D_{n}^{-})$$\mathrm{I}\mathrm{n}\mathrm{d}_{P_{j}}^{G}(\sigma\otimes\nu_{z}\otimes 1_{N_{J}})$

that(resp.
(resp.
forwith the parameter$\lambda=(\lambda_{1}, \lambda_{2})\in\Lambda \mathit{8}uch$$\lambda_{2}>-n)$$\lambda_{1}<n$$D=D_{n}^{+}$$\lambda\in\Lambda$$\epsilon(\gamma)=(-1)^{n}$$\epsilon(\gamma)=(-1)^{n}$$\gamma=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(-1,1, -1,1)$

In order to parametrize the discrete series representations of
all the positive root systems compatible to$\triangle_{c}^{+}:$

(I)
(II)
(III)
(IV)

$\triangle_{\mathrm{I}}^{+}=\{(1, -1), (2,0), (1,1), (0,2)\}$,$\triangle_{\mathrm{I}\mathrm{I}}^{+}=\{(1, -1), (2,0), (1,1), (0, -2)\}$,$\triangle_{\mathrm{I}\mathrm{I}\mathrm{I}}^{+}=\{(1, -1), (2,0), (0, -2), (-1, -1)\}$,$\triangle_{\mathrm{I}\mathrm{v}^{=}}^{+}\{(1, -1), (0, -2), (-1, -1), (-2,0)\}$.

$G$, we enumerate

14
Let be a variable running over the set of indices I, II, III, IV, and let us denotethe set of non-compact positive roots for the index by. Defineaof-dominant weights by
$J$

$J$

$\Delta_{J,n}^{+}=\triangle_{J}^{+}-\triangle_{c}^{+}$$\triangle_{c}^{+}$$\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{e}\mathrm{t}^{-}--J$

$—J=$

{

$\mathrm{A}=(\Lambda_{1},$$\triangle_{C}+$$\Lambda_{2}),$–dominant weight$|\langle\Lambda,\beta\rangle>0,\forall\beta\in\Delta_{J,n}^{+}$}.

The setgives the Harish-Chandra parametrization of the discrete series
representation of . Let us write by
the discrete series representation of with
the Harish-Chandra parameter. Thenis called the holomorphic
discrete series representation if A
and the anti-holomorphic one if A.
, a discrete series representation
Moreover if
is called large (in the.
sense of
The Blattner formula gives the description of the $K$-types of . In particular,, wherethe minimal $K$-typeofis given by the formula(resp.
) is the half sum of compact (resp. non-compact) positive roots in.
$\dot{B}iattner$
parameter of.
We cail such the$\bigcup_{J=\mathrm{I}^{-J}}^{\mathrm{I}\mathrm{V}-}-$

$G$

$G$

$\pi_{\Lambda}$$\Lambda\in\bigcup_{J=\mathrm{I}}^{\mathrm{I}\mathrm{V}}-_{J}--$$\pi_{\Lambda}$$\in--1-$$\in---\mathrm{I}\mathrm{V}$$\Lambda\in--1-\mathrm{I}\bigcup_{-\mathrm{I}\mathrm{I}\mathrm{I}}^{-}-$$\pi_{\Lambda}$$\mathrm{V}\mathrm{o}\mathrm{g}\mathrm{a}\mathrm{n}[8])$$\pi_{\Lambda}$$(\tau_{\lambda}, V_{\lambda})$$\lambda=\Lambda-\rho_{c}+\rho_{n}$$\pi_{\Lambda}$$\triangle_{J}^{+}$$\rho_{n}$$\rho_{c}$$\lambda$$\pi_{\Lambda}$

3.

-Jacobi type spherical functions$\mathrm{F}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{r}$3.1. Radial parts. Let
be an irreducible unitary representation ofand letbe a finite dimensional $K$-module. We denote by
the space of smooth functions
with the property$(\rho, \mathcal{F}_{\rho})$$(\tau, V_{\mathcal{T}})$$R_{J}$$C_{\rho,\tau}^{\infty}(R_{J}\backslash c/K)$$F:Garrow \mathcal{F}_{\rho}\otimes V_{\tau}$$F(rgk)=(\rho(r)\otimes\tau(k)^{-1})F(g)$On the other hand, letsatisfying$C^{\infty}(A_{J};\rho, \tau)$,.

$(r, g, k)\in R_{J}\cross G\cross K$

be the space of smooth functions$\varphi$:$A_{J}arrow$$\mathcal{F}_{\rho}\otimes V_{\tau}$$(\rho(m)\otimes\tau(m))\varphi(a)=\varphi(a))$$m\in R_{J}\cap K=M_{J}^{\mathrm{o}}\cap K,$$a\in A_{J}$.

Because of an Iwasawa decomposition of , we have $G=R_{J}A_{J}K$ . Also we remarkthat all elements inare commutative with $a\in A_{J}$ . Then the restriction to
gives a linear map fromto, which is injective. For, we calleach
the radialof , wheremeans the restriction to.
Let
be also a finite dimensional $K$-module. For each -linear map:, we have a unique -linear map:
with the property $(uf)|_{A_{J}}=\mathcal{R}(u)(f|_{A_{j}})$ for $f\in$. We callthe radial part of .
$G$

$M_{J}^{\mathrm{o}}\cap K$$A_{J}$$C_{\rho,\tau}^{\infty}(R_{j}\backslash G/K)$$C^{\infty}(A_{J};\rho, \mathcal{T})$$f|_{A_{J}}\in C^{\infty}(A_{J;\rho,\tau})$$f\in C_{\rho,\tau}^{\infty}(R_{J}\backslash c/K)$$pc_{\iota}rt$$(\tau’, V_{\tau}’)$

$u$

$f$

$A_{J}$$|_{A_{J}}$$\mathrm{C}$$C_{\rho,\tau}^{\infty}(R_{J}\backslash c/K)arrow C_{\rho,\tau}^{\infty},$$(R_{J}\backslash G/K)$$\mathrm{C}$$\mathcal{R}(u)$$C^{\infty}(A_{J;\rho,\tau})arrow C^{\infty}(A_{J;}\rho, \mathcal{T})/$$\mathcal{R}(u)$$C_{\rho,\tau}^{\infty}(R_{J\backslash G}/K)$$u$3.2. Fourier-Jacobi type spherical functions. Let
be as above and considera-induced representation
with the representation space$(\rho, \mathcal{F}_{\rho})$$C^{\infty}$$C^{\infty}\mathrm{I}\mathrm{n}\mathrm{d}_{R_{J}}^{G}(\rho)$$C_{\rho}^{\infty}(R_{J}\backslash G)=\{F:Garrow \mathcal{F}_{\rho}, C^{\infty}|F(rg)=\rho(r)F(g), (r,g)\in R_{J}\cross G\}$

on which
acts by the right translation. Then-module and a-module naturally. Moreover let
$G$

$G$

$C_{\rho}^{\infty}(R_{j}\backslash G)$$(\mathrm{g}_{\mathbb{C}}, K)$

becomes a smoothand take an$(\tau, V_{\tau})\in\hat{K}$15
irreducible Harish-Chandra module ofwith the $K$-type , whereis thecontragredient representation of . Now we consider the intertwining space
$G$

$\pi$

$\tau^{*}$$\tau^{*}$$\tau$$\mathcal{I}_{\rho,\pi}:=\mathrm{H}\mathrm{o}\mathrm{m}_{(_{9}K)}\mathrm{c},(\pi, C^{\infty}\mathrm{I}\mathrm{n}\mathrm{d}G(R_{J}\rho))$-modules and its restriction to the $K$-typeof .
$i:\tau^{*}arrow\pi|_{K}$ be a $K$ -equivariant map and let
be the ullback via . ThenbetweenLet
the map$\tau^{*}$$(\mathrm{g}_{\mathbb{C}}, K)$$\pi$$i^{*}$$i$$\dot{\mathrm{p}}$$\mathcal{I}_{\rho_{)}K}\pi^{arrow \mathrm{H}((\backslash }i^{*}\mathrm{o}\mathrm{m}\mathcal{T}^{*},$$c_{\rho}^{\infty}R_{j}G))\simeq C_{\rho,\tau}^{\infty}(R_{J}\backslash G/K)$

to the $K$-type
and we denote the image of ingives the restriction ofof the algebraic Fourier-Jacobi typeby . Now the spaceon is defined byspherical functions of type
$T_{i}$$C_{\rho,\tau}^{\infty}(R_{J\backslash G}/K)$$T$$\tau^{*}$$T\in \mathcal{I}_{\rho,\pi}$$I_{\rho_{)}\pi}(\tau)$$G$$(\rho, \pi;\tau)$$J_{\rho,\pi}( \tau):=i\in \mathrm{H}\mathrm{o}\mathrm{m}_{K}(\bigcup_{*,\mathcal{T}\pi|_{K})},\{T_{i}|T\in \mathcal{I}_{\rho,\pi}\}$.

Moreover put$J_{\rho,\pi}^{\mathrm{O}}(\tau)=${$f\in J_{\rho,\pi}(\mathcal{T})|f|_{A_{J}}(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(a,$$1,$$a-1,1))$ is of moderate growth as $aarrow\infty$}.

.
a Fourier-Jacobi type spherical
We callof type$(p,\pi;\tau)$:
for the following triplet
In this note, we investigate the space, we take either the -principal series representation and theAsandcorner $K$-type or the discrete series representation and the minimal $K$-type, andthe one with the non-trivial central character, i.e. of type $m\neq 0$ .
also as$(\rho, \pi;\tau)$$function\mathit{8}$$f\in J_{\rho,\pi}^{\mathrm{O}}(\tau)$$J_{\rho,\pi}^{\mathrm{O}}(\tau)$$\pi\in\hat{G}$$\tau^{*}\in\hat{K}$$P_{J}$$\rho\in\hat{R}_{J}$4.

$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}$equationsoperators. In this subsection, we introduce some differential op.
erators acting onform of . Now
of with respect to theTake an orthonormal basis
we define a first order gradient type differential operator$\mathit{4}\cdot \mathit{1}.Differential$$C_{\rho,\tau}^{\infty}(R_{J\backslash G}/K)$$\dot{\mathrm{K}}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}$$\{X_{i}\}$$\nabla_{\rho,\tau}$:$\mathfrak{p}$$\mathrm{g}$$c_{\rho,\tau}^{\infty}(RJ\backslash G/K)arrow C_{\rho_{)}\tau\otimes}^{\infty}\mathrm{A}\mathrm{d}_{\mathfrak{p}_{\mathrm{C}}}(R_{J}\backslash G/K)$

by$\nabla_{\rho,\tau}..f=\sum_{i}R_{X_{i}}f\otimes X_{i}$,$f\in C_{\rho}^{\infty},(T\backslash R_{J}G/K)$

,

where
$R_{X}f(g)= \frac{d}{dt}f(g\cdot\exp(tx))|_{t=0}$,$X\in \mathfrak{g}_{\mathbb{C}},$$g\in G$.

can beis called the Schmid operator. ThenThis differential operator:
withdecomposed as, the shift oper. For eachcorresponding to the decomposition
is defined as the composition of:ator$\nabla_{\rho_{)}\tau}$$\nabla_{\rho,\tau}$$\nabla_{\rho,\tau}^{+}\oplus\nabla_{\rho,\tau}^{-}$$\nabla_{\rho,\tau}^{\pm}$$C_{\rho,\tau}^{\infty}(R_{J}\backslash G/K)arrow C_{\rho,\tau\otimes \mathrm{A}\mathrm{d}\mathfrak{p}\pm}^{\infty}(R_{J}\backslash G/K)$$\mathfrak{p}_{\mathbb{C}}=\mathfrak{p}_{+}\oplus \mathfrak{p}_{-}$$\nabla_{\rho,\tau_{\lambda}}^{\pm\beta}$$C_{\rho,\tau_{\lambda}}^{\infty}(R_{J}\backslash c/K)arrow C_{\rho,\pm\beta}^{\infty_{\tau_{\lambda}}}(R_{J}\backslash G/K)$$\beta\in\triangle_{n}^{+}$

16$\nabla_{\rho,\tau_{\lambda}}^{\pm}$with the projector$P^{\pm\beta}$$\nabla_{\rho,\tau_{\lambda}}^{\pm\beta}=(1_{F_{\rho^{\otimes}}}P\pm\beta)\nabla_{\rho,\tau_{\lambda}}^{\pm}$fromthe irreducible component$V_{\tau_{\lambda}}\otimes \mathfrak{p}_{\pm^{\mathrm{i}\mathrm{n}}}\mathrm{t}\mathrm{o}$.

$V_{\tau_{\lambda\pm\beta}}$;

On the other hand, the Casimir element is defined by, whereis an orthonormal basis of with respect to the Killing form of . It is wellknown that is in theof the universal enveloping algebra of .
$\Omega$$\{Y_{j}\}$$\Omega=\sum x_{i^{-}}\sum Y_{j}$$\mathrm{f}$$\mathfrak{g}$center.

$\Omega$$Z(\mathfrak{g}\mathbb{C})$$\emptyset \mathbb{C}$equations. In this subsection, we consider the system of differential equations satisfied by the Fourier-Jacobi type spherical functions.
First we discuss the case of the -principal series representationand the$K$
corner -type . It is well known that the Casimir element
acts on, hence on, as the scalar operator(cf.
; Corollary 8.14]). Letwith data, andbe the$K$$=\tau_{(n-2,n-2)}\in\hat{K}$ does notcorner -type of , i.e. $\lambda=(-n, -n)$ . Since)
occur in the $K$-types of from Lemma 2.4, an element in
is annihilated by
the action of the composition of the shift operators$\mathit{4}\cdot \mathit{2}.Differential$$\pi\in\hat{G}$$P_{J}$$\tau^{*}$$\pi$$\Omega\in Z(\emptyset \mathbb{C})$$J_{\rho,\pi}(\mathcal{T})$$\mathrm{K}\mathrm{n}\mathrm{a}\mathrm{p}\mathrm{P}[4$$\chi_{\Omega}$$\sigma=(\epsilon, D_{n}^{+}),$$\pi=\mathrm{I}\mathrm{n}\mathrm{d}_{PJ}^{c}(\sigma\otimes\nu_{z}\otimes 1_{N_{J}})$$\pi$$\mathcal{E}(\gamma)=(-1)^{n}$$\tau^{*}=\tau_{\lambda}^{*}$$\tau_{\lambda+(2}^{*},2$$\pi$$J_{\rho,\pi}(\mathcal{T})$:$\nabla^{(0,2})0\nabla^{(2,0)}\rho,\tau_{\lambda+(2,0})\rho,\mathcal{T}\lambda$$C_{\rho,\tau_{\lambda}}^{\infty}(R_{j}\backslash c/K)arrow C_{\rho,2,2)}^{\infty}(\tau_{\lambda+}(RJ\backslash c/K)$

Hence we have a system of differential equations satisfied by$\Omega f=\chi_{\Omega}f$

(4.1)

$f$

in

.

$J_{\rho,\pi}(\mathcal{T})$

;

,

$\{$
$\nabla_{\rho,\tau_{\lambda+()}}^{(}0,2)0\nabla_{\rho,\lambda}(2,0)2,0\mathcal{T}f=0$.

Letwith data, and$K$
be the corner -type of , i.e. $\lambda=(-n+1, -n)$ . Since
does not occur in the $K$-types of from Lemma 2.4, therefore an element invanishes by the action of the shift operator$\sigma=(\epsilon, D_{n}^{+}),$$\pi=\mathrm{I}\mathrm{n}\mathrm{d}_{P_{j}}^{G}(\sigma\otimes\nu_{z}\otimes 1_{N_{J}})$$\mathcal{E}(\gamma)=-(-1)^{n}$$\tau_{\lambda}^{*}+(1,1)=\tau_{(n-2},n-1)\in\hat{K}$

$\pi$

$\pi$

$\nabla_{\rho,\tau_{\lambda}}^{(1,1})+(1,1)$:$\tau^{*}=\tau_{\lambda}^{*}$$c_{\rho,\tau_{\lambda}}^{\infty}(RJ\backslash G/K)arrow o_{\rho,\tau_{\lambda+(}1,1)}^{\infty}(R_{j}\backslash c/K)$

Hence we have a system of differential equations satisfied by$\Omega f=\chi_{\Omega}f$

(4.2)

$f$

in

$J_{\rho,\pi}(\tau)$.

$I_{\rho},\pi(\tau)$

;

,

$\{$
$\nabla_{\rho_{\mathcal{T}}\lambda+(1,1\rangle}^{(1,1)},f=0$.

For the case with the data
, we have similar systems of equations fromthe Casimir operator and the shift operators.
Letbe a discrete series representaiton of with the Harish-Chandra
parameter A $\in---J$ andbe the minimal $K$-type of . Now we refer thefollowing proposition which enables us to
the intertwining spacewitha solution space of differential equations for any.
$\sigma=(\epsilon, D_{n}^{-})$$G$$\pi=\pi_{\Lambda}$$\tau^{*}=\tau_{\lambda}^{*}\in\hat{K}$$\pi$$\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\Psi$$\mathcal{I}_{\rho,\pi}$$p\in\hat{R}_{J}$Proposition 4.1. (Yamashita [9; Theorem 2.4]) Let
be as above. Then we have a linear$\pi=\pi_{\Lambda}\in\hat{G}$$i_{Somo}rphi_{\mathit{8}}m$$\mathcal{I}_{\rho,\pi}\simeq$$\cap$$\beta\in\triangle_{J^{*},n}^{+}$$\mathrm{k}\mathrm{e}\mathrm{r}(\nabla_{\rho,\tau}^{-\beta})\subset c_{\rho,\tau}\infty(R_{J}\backslash G/K)$

and$\tau^{*}=\tau_{\lambda}^{*}\in\hat{K}$17
for any$\rho\in\hat{R}_{J}$. In particular,$J_{\rho,\pi}(\mathcal{T})=\{F\in C_{\rho,\tau}^{\infty}(R_{J}\backslash G/K)|\nabla_{\rho,\tau}^{-\beta}F=0, \forall\beta\in\triangle_{Jn}^{+}*,\}$

Here the index

means IV, III, II and I for$J^{*}$$J=\mathrm{I},$$\mathrm{I}\mathrm{I}$.

, III and IV, respectively.

5. Result
Solving the systems of the differential equations given by (4.1), (4.2), and Proposition 4.1, we obtain the following theorem.

Theorem 5.1. Let be a -principal series representation (resp. a discrete series
representation) of $G=Sp(2, \mathrm{R})$ andbe the ’comer’ $K$ -type (resp. the minimal$K$ -type)of . For each ir’reducible unitary representation of of type $m\neq 0$ ,we have$P_{J}$$\pi$$\tau^{*}$$\pi$$\dim J_{\rho,\pi}\mathrm{O}(\mathcal{T})\leq 1$

Moreover the radial parts of the functions in$G$-function$G_{2,3}^{3,0}(x|_{b_{1,2}}^{a_{1}}b’,ba_{2}3)$

Here the Meijer’s
parameters
the integral$a_{i},$$R_{J}$$\rho$$G$-function.

$J_{\rho,\pi}^{\mathrm{O}}(\tau)$

are expressed by the Meijer’s

or more degenerate similar functions.
$G_{2,3}^{3_{)}0_{(X}}$)$=G_{2,3}^{3,0}(x|_{b_{1},b_{2},b_{3}}^{a_{12}}’ a)$$b_{j}(1\leq i\leq 2,1\leq j\leq 3)$

is the many-valued

with the complex$\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}_{0}\mathrm{n}$defined by$c_{2,3}^{3,0_{(x)=G}3}2,3)0(x|^{a_{1}}b_{1},b_{2}’,2ab_{\mathrm{s}})= \frac{1}{2\pi\sqrt{-1}}\int_{L}\frac{\prod_{j=1}^{3}\mathrm{r}(b_{j}-t)}{\prod_{i=1}^{2}\mathrm{r}(a_{i}-t)}x^{t}dt$

of Mellin-Barnes type, where the contour is a loop starting and ending atand encircling all poles of $\Gamma(b_{j}-t)(1\leq j\leq 3)$ once in the negative direction. It
is known that, up to constant multiple,
is the unique solution of the linear
differential equation of 3-rd order$L$$+\infty$$G_{2,3}^{3,0}(X)$$\{x^{3}\frac{d^{3}}{dx^{3}}+\alpha_{2}(x)X^{2}\frac{d^{2}}{dx^{2}}+\alpha_{1}(x)x\frac{d}{dx}+\alpha_{0}(X)\}y=0$

with$\alpha_{2}(x)=3-b1^{-}b2-b3+x$ ,$\alpha_{1}(x)=(1-b_{1})(1-b_{2})(1-b_{3})+b_{1}b_{2}b_{3}+(3-a_{1}-a_{2})x$$\alpha_{0}(x)=-b_{1}b_{2}b_{\mathrm{s}}+(1-a_{1})(1-a_{2})_{X}$which decays exponentially as $|x|arrow\infty$ in
original paper [5] for details).

,,$- \frac{3}{2}\pi<\arg x<\frac{1}{2}\pi$(See the Meijer’s

Remark 5.2. Let be a holomorphic discrete series representation of andbeas in \S 2. For eachthe minimal $K$-type of . Moreover, put$m\neq 0$ , there is at most finitely manysuch that, and then thefactors of suchare the holomorphic discrete series representations of.
Moreover, the radial parts of the functions inare expressed by the function.
of the form
for some constant
$G$

$\pi$

$\rho=\pi_{1}\otimes\tilde{\nu}_{m}\in\hat{R}_{J}$$\pi$$\dim J_{\rho,\pi}\mathrm{O}(\tau)=1$$\rho$$\pi_{1^{-}}$$\overline{SL}(2, \mathrm{R})$$\rho’ \mathrm{s}$$J_{\rho,\pi}^{\mathrm{O}}(\tau)$$x^{p}e^{qx}$$\tau^{*}$

$p,$

$q$

18
REFERENCES
1. Corwin, L., Greenleaf, F. P., Representations of Nilpotent Lie Groups and their Applications
mathematics, vol.18,
Partl: Basic Theory and Examples, Cambridge studies in
Cambridge University Press, 1990.
2. Gelbert, S. S., Weil’s Representation and the Spectrum of the Metaplectic Group, Lecture
Note in Math., vo1.530, Springer Verlag, 1976.
, Thesis Univ. of Tokyo (1998).
3. Hirano, M., Fourier-Jacobi type spherical functions on
4. Knapp, A. W., Representation Theory of Semisimple Groups; An Overview Based on Examples, Princeton Univ. Press, 1986.
. (1946), 124-134, 213-225, 312-324,
5. Meijer, C. S., On the -function. I- VIII, Indag.
391-400, 468-475, 595-602, 661-670, 713-723.
6. Murase, A., Sugano, T., Whittaker-Shintani Functions on the Symplectic Group of FourierJacobi Type, Compositio Math. 79 (1991), 321-349.
7. Satake, I., Unitary Representations of a Semi-Direct Product of Lie Groups on -Cohomology
Spaces, Math. Ann. 190 (1971), 177-202.
8. Vogan, D. A., Gelfand-Kirillov Dimension for Harish-Chandra Modules, invent. Math. 48(1978), 75-98.
9. Yamashita, H., Embeddings of Discrete Series into Induced Representations of Semisimple LieGroups -General Theory and the Case of $SU(2,2)$ , Japan J. Math. 16 (1990), 31-95.

adva.nced$Sp(2, \mathrm{R})$$G$$\dot{\mathrm{M}}\mathrm{a}\mathrm{t}\mathrm{h}$$8$$\overline{\partial}$$I$

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