数理解析研究所講究録1052 巻 1998 年 112-127

112

MATRIX COEFFICIENTS OF THE MIDDLE DISCRETE SERIES OF $SU(2,2)$
東京大学数理科学早田孝博 (TAKAHIRO HAYATA),
三重大学教育古関春隆 (HARUTAKA KOSEKI),
東京大学数理科学織田孝幸 (TAKAYUKI ODA)

1. INTRODUCTION
Among the discrete series representation of the non-compact real unitary group $SU(2,2)$of signature $(2+, 2-)$ , there are representations whose Gel’fand-Kirillov dimensions are 5.
Because other discrete series representations have Gel’fand-Kirillov dimension 6 (the large
discrete series), or 4 (holomorphic or anti-holomorphic discrete series), and also becausethe-cohomology of these representations have “Hodge type” $(2, 2)$ (others $(3,1)$ , $(1,3)$ ,$(4,0)$ , $(0,4))$ , we call them the middle discrete
series.
We detemine the -radial part of the matrix coefficients with minimal $K$ -type of a representation belonging to the middle discrete series in this paper. It is written in terms of Gaussian hypergeometric series (Main Theorem 5.5). Our method of proof is a direct computationof the -radial part of the Schmid operator, a gradient-type operator which characterize theminimal $K$ -type vectors in the representation space of a discrete series representation (\S 3).
The obtained operators constitute a holonomic system of 2 variables with rank 2 (\S 4). It israther complicated difference-differential equations. We honestly solve this system step bystep.
$(\mathfrak{g}, \mathrm{t})$

$A$

$A$

2. THE GROUP $SU(2,2)$ AND ITS DISCRETE SERIES

2.1. Structure of $SU(2,2)$ and its Lie algebra. Let
realized as$G=\{g\in sL4(\mathbb{C})|g^{*}I_{2,2}g=I_{2,2}\}$,$G$be the special unitary group $SU(2,2)$$I_{2,2}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(1,1, -1, -1)$

,

where
denotes the adjoint of a matrix . Let $U(n)$ be the unitary group of degree .
Take a maximal compact subgroup $K=G\cap U(4)=S(U(2)\cross U(2))$ . We denote by
the Lie algebra of $G,$ $K$ , respectively. Let
be a Cartan involution and
the Cartan decomposition of .
We setwith $H_{1}=X_{23}+X_{32},$ $H_{2}=X_{14}+X_{41}$ , where theareelementary matrices given by$g^{*}={}^{t}\overline{g}$

$g$

$n$

$\mathrm{g}$$\mathfrak{g},$$\theta(X)=-^{t}\overline{X}$$\mathfrak{g}=\+\mathfrak{p}$$\mathfrak{g}$$a=\mathbb{R}H_{1}+\mathbb{R}H_{2}$$X_{ij}’ \mathrm{s}$with Kronecker’s delta.
Then is a maximally -split abelian subalgebra of contained in . Then the restrictedroot system
is expressed as$X_{ij}=(\delta_{ip}\delta_{j})_{1\leq}qp,q\leq 4$$a$$\delta_{ip}$$\mathbb{R}$$\mathfrak{g}$$\mathfrak{p}$$\triangle=\triangle(\mathfrak{g}, a)$$\Delta=\triangle(\mathfrak{g}, a)=\{\pm\lambda_{1}\pm\lambda 2, \pm 2\lambda_{1}, \pm 2\lambda_{2}\}$

where$\triangle_{\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{d}}$$\lambda_{j}$ofis the dual of:$H_{j}$. We choose a positive system$\Delta^{+}$.

and a fundamental system$\triangle$$\triangle^{+}=\{\lambda_{1}\pm\lambda_{2},2\lambda 1,2\lambda 2\}$,$\triangle_{\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{d}}=\{\lambda 1-\lambda_{2},2\lambda_{2}\}$.

113
We also denote the corresponding nilpotent subalgebra byroot subspace of corresponding to. Then oneof and :$0$$G$$\mathfrak{g}$$\mathfrak{g}=\mathfrak{n}+a+\mathrm{e}$with$A=\exp \mathfrak{a},$ $N=\exp \mathfrak{n}$,. Hereis thean Iwasawa decomposition$\mathfrak{n}=\sum_{\beta\in\Delta}+9\beta$.btains$\beta\in\triangle^{+}$$\mathfrak{g}$$\mathfrak{g}_{\beta}$$G=NAK$ ,. Now let$E_{1}=H_{13}-\sqrt{-1}X13+\sqrt{-1}X31$,$E_{2}=H_{24}-\sqrt{-1}X_{2}4+\sqrt{-1}x42$,$E_{3}=1/2(X_{12^{-}}X_{21}-x_{1}4+X_{23}+X_{32}-x_{4}1-x34+X_{43})$ ,$E_{4}=\sqrt{-1}/2(X_{12}+X_{21^{-}}X14-x23+X_{32}+X_{41}-x34-X_{4}3)$ ,$E_{5}=1/2(X_{12^{-}}x_{2}1+X_{14}+X_{23}+X_{32}+X_{41}+X_{34^{-}}X_{4}3)$ ,$E_{6}=\sqrt{-1}/2(X_{12}+X_{21}+x_{14}-X_{2}3+x_{32}-X_{4}1+X_{34}+X_{43})$ ,where $H_{ij}=\sqrt{-1}(X_{ii}-x_{jj})$ for $1\leq i<j\leq 4$ . Then it is easy to see that$\mathfrak{g}_{2\lambda_{j}}=\mathbb{R}E_{j}$$(j=1,2)$ ,$\mathfrak{g}_{\lambda_{1}+\lambda_{2}}=\mathbb{R}E_{3}+\mathbb{R}E_{4}$,$\mathfrak{g}_{\lambda_{1}-\lambda_{2}}=\mathbb{R}E_{5}+\mathbb{R}E_{6}$.

2.2. Parametrization of the discrete series. Let us
parametrize the discrete series of$SU(2,2)$ . Take a compact Cartan subalgebra defined by$\mathrm{n}\mathrm{o}\dot{\mathrm{w}}$$\mathrm{t}$with $h^{1}=x_{11^{-}}x_{22},$ $h2=x_{33}-X_{4}4$$\mathrm{t}=\mathbb{R}\sqrt{-1}h^{1}+\mathbb{R}\sqrt{-1}h^{2}+\mathbb{R}\sqrt{-1}I_{2,2}$

and letbe its complexification. Then the absolute root system, of type$\mathrm{t}_{\mathbb{C}}$$\triangle=\triangle(9\mathbb{C}\sim\sim, \mathrm{t}_{\mathbb{C}})=\{[\pm 2, \mathrm{o};0], [0, \pm 2;0], [\pm 1, \pm 1;\pm 2]\}$$A_{3}$, is given by.

where $\beta=[r, s;u]$ means $r=\beta(h^{1}),$ $s=\beta(h^{2})$ and $u=\beta(I_{2,2})$ . We write the set of
positive roots by
and we fix it hereafter. The Weyl groupis generated by $s_{2,3}S$ where$\triangle_{c}^{+}-=\{[2,0;\mathrm{o}], [0,2;0]\}$$\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\underline{\mathrm{a}\mathrm{c}\mathrm{t}}$$\overline{W}=W(\mathfrak{g}_{\mathbb{C}}, \mathrm{t}_{\mathbb{C}})$$s_{1},$$s_{1}[r, s;u]=[-r, S;u]$ ,$s_{2}[r, s;u]=[(r-s+u)/2, (-r+s+u)/2;r+s]$ ,$s_{3}[r, S;u]=[r, -S;u]$We
and thecompact Weyl group is given by$\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\iota \mathrm{i}\mathrm{f}\mathrm{y}\overline{W}$$\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{m}\mathrm{e}\mathrm{t}\underline{\mathrm{r}\mathrm{i}_{\mathrm{C}}}$$\mathfrak{S}_{2}\cross \mathfrak{S}_{2}$

group

.
$\mathfrak{S}_{4}$of degree 4 by the map: $s_{j}rightarrow(j,j+1)$ . The, also identified canonically with the subgroup$W_{c}=\langle S_{1}, s_{3}\rangle$.

There are exactly six positive systems, where$\triangle^{+}--\mathrm{I}\mathrm{I}’\ldots,$$\triangle_{\mathrm{I}}^{+},$$\triangle^{+}-\mathrm{v}\mathrm{I}$containing$\triangle_{c}^{+}-$, defined by$w_{J}\overline{\Delta}^{+}$$\triangle^{+}-=\{[2,0;\mathrm{o}], [0,2;0], [\pm 1, \pm 1;2]\}$and the elements$w_{\mathrm{I}}=1,$$w_{J}\in\overline{W}$$w_{\mathrm{I}\mathrm{I}}--_{S_{2}},$are given by$w\mathrm{I}\mathrm{I}\mathrm{I}=S2S3,$$w\mathrm{I}\mathrm{v}=s_{2}S1,$

We denote by
the noncompact positive roots in
By definition, the space of the Harish-Chandra$\triangle_{J,n}^{+}\wedge$$W_{}=s2s3S1,$ $w_{}\mathrm{I}=S_{2132}Sss$ .
$\triangle_{J}^{+}-$.

$\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{S}^{-}--_{C}$

$—c=$

{

$\Lambda\in \mathrm{t}_{\mathbb{C}}^{*}|$

A is$\triangle-$

is given by-regular, $K$ -analytically integral and$\triangle_{c}^{+}-$-dominant}.

$\triangle_{J}^{+}-=$

114$\mathrm{P}\mathrm{u}\mathrm{t}_{\cup J}^{-}\{\Lambda\in-_{C}-|\triangle_{J^{\frac{-}{}}}^{+_{\mathrm{d}}1}0\min_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{h}}\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{n}\mathrm{d}^{-=}\rho_{J},n=2-1^{-}\sum\beta\in\triangle+\beta\sim,\mathrm{a}1\mathrm{f}\mathrm{s}\mathrm{u}\mathrm{m}\mathrm{o}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{s},\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{h}\mathrm{a}\mathrm{s}^{J}\mathrm{u}\mathrm{m}\mathrm{o}\mathrm{f}-jn\mathrm{t}\}\mathrm{w}\mathrm{e}\mathrm{a}_{\mathrm{f}_{\mathrm{P}^{\mathrm{o}\mathrm{s}}}}1\mathrm{s}\mathrm{o}_{\mathrm{P}}\mathrm{u}\mathrm{t}_{\beta}c,J=2-\sum_{1\mathrm{r}}\beta\in\triangle+^{\beta,\rho_{c}}\sim \mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{c}\mathrm{t}\mathrm{p}\mathrm{o}=2^{-}1_{\sum_{\mathrm{S}}\beta\in\beta}-\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\triangle c+$roots and the the half sum of
positive roots,
divided into six parts:. For
discrete series by. We say thatis the middle. Particularly in this case, we have$–c-= \bigcup_{\mathrm{I}\leq J\leq \mathrm{V}^{\cup}}J$$\Lambda\in\bigcup_{\mathrm{I}\leq \mathrm{I}^{\cup J}’}j<\mathrm{V}$$\overline{\mathrm{d}\mathrm{i}}_{\mathrm{S}\mathrm{C}\mathrm{r}\mathrm{e}}\mathrm{t}\mathrm{e}$$\pi_{\Lambda}$

arewe denote the corresponding
series representation if A$\mathrm{r}\mathrm{e}\mathrm{s}_{\underline{\underline{\mathrm{P}}}}\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{y}$$\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{m}_{\underline{\underline{\mathrm{P}}}}\mathrm{a}\mathrm{c}\mathrm{t}$. The$\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}-_{C}\cup-\subset\{_{\mathbb{C}}^{*}$$\in$$\pi_{\Lambda}$$–_{\mathrm{I}\mathrm{I}\mathrm{I}^{\cup}\cup}---_{\mathrm{I}\mathrm{V}}$$\triangle_{\mathrm{I}\mathrm{I}\mathrm{I}}^{+}=\{[1, \pm 1;\pm 2]\},$$\triangle_{\mathrm{I}\mathrm{V}}^{+}=\{[\pm 1,1;\pm 2]\},$$\rho_{\mathrm{I}\mathrm{I}\mathrm{I},n}=[2,0;\mathrm{o}],$2.3. Representations of the maximal compact subgroup. LetFor$d=[d_{1}, d_{2};d3]\in \mathrm{t}_{\mathbb{C}}^{*}$(1), define$\tau_{d}\in\hat{K}$$\tau_{d}(h^{j})fk1k(d)2=(2k_{j}-d_{j})f_{kk}(d)12$$V_{d}=\{f_{k_{1}k}^{(d)}2|0\leq k_{j}\leq d_{j}\}_{\mathbb{C}}$$h^{1}$

are the generators of$k_{\mathbb{C}}${,$h^{2}$,$\tau_{d}(I_{2,2})f_{kk}(d)12=d_{3}f_{k_{1}k}^{(}d)2$’,$e_{+}^{2}=X_{34}$of $K$ on$\mathfrak{p}_{\mathbb{C}}=\mathfrak{p}_{+}+\mathfrak{p}_{-}$$|_{\mathfrak{p}\pm}$

.

’

.

$e_{-}^{j}={}^{t}e_{+}^{j}$$\hat{K}$$\tau_{[1,1;\pm 2}$] (writeis exhausted by
is even}.

, where,],,$\mathfrak{p}_{-}=^{t}\mathfrak{p}_{+}$.

respectively. For later use, we fix the$:1,1;\pm 2$$f_{kl}=f_{kl}^{[]}$),: $(X_{41}, x_{31}, X_{42}, x32)\vdasharrow(f_{00}, f\mathrm{o}1, -f10, -f11)$ ,$(X23, X13, X_{2}4, X14)\vdasharrow(f\mathrm{o}0, f10, -f01, -f_{1}1)$$\iota_{+}:$$\iota_{-}$

is isomorphic to$\mathfrak{p}_{\pm}arrow V_{[1,1;\pm 2}$$\iota_{\pm}$$d_{3}\in \mathbb{Z}$is decomposed into a direct sum of$\mathfrak{p}_{\mathbb{C}}$$\mathfrak{p}_{+}=\mathbb{C}x_{13}+\mathbb{C}X_{1}4+\mathbb{C}X_{2}3+\mathbb{C}x_{24}$$\mathrm{A}\mathrm{d}_{\pm}=$,$V_{d})|d=[d_{1},$ $d_{2;d],d_{1}}3+d_{2}+d_{3}$$=\mathrm{A}\mathrm{d}_{\mathfrak{p}_{\mathrm{C}}}$In fact,
Ad$K$ -isomorphisms:.

is the standard basis (see [2, \S 3]) and$e_{+}^{1}=X_{12}$The adjoint representation Ad
two irreducible subrepresentations:an.d$\tau_{d}(e^{j}-)fk_{1}k_{2}=(d)kjf_{k_{1}-}(d)\delta 1j,k2-\delta_{2j}$’. Then according to [2, Prop. 3.1],$.(.\mathcal{T}_{d},$$d_{2}\in \mathbb{Z}_{\geq 0}$

by the following rule $0=1,2$ ):$\tau_{d}(e_{+}^{j})fk_{1}(d)k_{2}=(d_{j}-k_{j})fk(d)1+\delta_{1}j,k_{2}+\delta 2j$Here,$d_{1},$$\rho_{\mathrm{I}\mathrm{V},n}=[0,2;0]$The irreducible decomposition of$V_{d^{\otimes=}}\mathfrak{p}_{\mathbb{C}}$.

$\mathrm{t}_{\mathbb{C}}$-module$V_{d}\otimes \mathfrak{p}-\oplus V_{d}\otimes \mathfrak{p}_{-}$,([2, Prop. 3.10].)is given as$V_{d}\otimes \mathfrak{p}_{\mathbb{C}}$$V_{d^{\otimes \mathfrak{p}_{\pm}\simeq}} \bigoplus_{\epsilon_{1},\epsilon 2\in\{\pm 1\}}V[r+\epsilon 1,s+\epsilon_{2;}u\pm 2]$.

The projectors$P_{rs}^{(\epsilon_{1},\epsilon_{2})}$:$\epsilon_{2}$$\overline{P}_{rS}^{(\epsilon_{1}}’$$V_{d}\otimes \mathfrak{p}_{+}arrow V_{[2]}r+\epsilon_{1},s+\epsilon 2;u+$

’

)

:

$V_{d}\otimes \mathfrak{p}_{-}arrow V_{[]}r+\epsilon_{1},s+\epsilon_{2};u-2$,

are explicitly given by [2, Lemma 3.12].
2.4. $K$ -types of the middle discrete series representations. Let
be the discrete series
representation of with Harish-Chandra parameter. Then the Blatter parameter of
becomes $\lambda=\Lambda+\rho_{G,J}-2\rho C$ . In the following we put $d=[r, s;u]$ .
), then the $K$ -types
If(resp.
ofare parametrized by$\pi_{\Lambda}$$G$$\Lambda\in--J-$$\pi_{\Lambda}$$\Lambda\in--\mathrm{I}\mathrm{I}\mathrm{I}-$$–\mathrm{I}\mathrm{V}-$$\tau_{\lambda}$$\pi_{\Lambda}$$[r, s;u]$ with $r>s+2+|u|,$ $r\in \mathbb{Z}_{>0},$(resp.

$[r,$ $s;u]$with $s>r+2+|u|,$$r\in \mathbb{Z}_{\geq 0},$,$s\in \mathbb{Z}_{\geq 0},$$r+s+u\in 2\mathbb{Z}$$s\in \mathbb{Z}_{>0},$$r+s+u\in 2\mathbb{Z}.$

)

115

3.

$\mathrm{s}\mathrm{C}\mathrm{H}\mathrm{M}\mathrm{I}\mathrm{D}’ \mathrm{S}$DIFFERENTIAL.OPERATOR

3.1.
-matrix coefficient of the middle discrete series. Let
be a middle
discrete series representation, andits minimal $K$ -type. Put $d=[r, s;u]$ . Then thecontragredient representationof is isomorphic to
We identify the representation spaces
with their unique images inrespectively. Then the matrix
coefficient of is defined by$(\tau, \tau^{*})$$(\pi_{\Lambda}, H_{\Lambda})$$(\tau_{d}, V_{d})$$\tau_{d}^{*}$$V_{d},$$\tau_{d}$$\tau_{[r,s;u]}-\cdot$$V_{d}^{*}$$H_{\Lambda},$$H_{\Lambda}^{*}$$\pi$$\langle\pi_{\Lambda}(g)v, w^{*}\rangle$

for $v\in V_{d},$$w^{*}\in V_{d}^{*}\subset H_{\pi}^{*}$. Here we consider a more convenient vector-valued function:$\Phi_{\pi,\mathcal{T}}(g)=\sum\langle\pi_{\Lambda}(g)i,j,k,lf_{k}*l’ f_{i}*\rangle jfij\otimes fkl$

where
we find that$\{f_{ij}=f_{ij}^{[r,s;u}]\}_{ij}$$\Phi_{\pi,\tau}$(resp. $\{f_{kl}=f_{kl}^{[;u]}r,s-\}kl$ ) is a standard basis of

belongs to the following function space:$V_{d}$(resp.

$V_{d}^{*}$). Then$C_{\mathcal{T},\mathcal{T}}^{\infty}*(K\backslash c/K)=\{\phi:Garrow V_{d}\otimes V_{d}^{*}|\phi(k_{1}gk2)=\tau_{d}(k_{1})\otimes\tau(d^{*}k^{-}1)2\phi(g), k_{j}\in K\}$.

For simplicity, we write the index $M=(i,j;k, l)$ and coefficients.
Due to the Cartan decomposition $G=KAK,$ $c_{M}(g)$ is detemined uniquely by its restriction to .
$c_{M}(g)=\langle\pi_{\Lambda}(g)f_{ij}^{*}, f_{kl}^{*}\rangle$$A$

Lemma 3.1. If$c_{M}$

is not zero, it satisfies the condition:$k_{1}+l_{1}+k2+l_{2}=r+s$ .

Proof. The centralizer of in
$A$

$K$

is

$\{m=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(u,\overline{u}\epsilon, u,\overline{u}\epsilon)||u|=1, \epsilon=\pm 1\}$

Therefore$\phi\in C_{\mathcal{T},\tau}^{\infty}*(K\backslash G/K)$.

satisfies$\phi(mam^{-1})=\phi(a)$$a\in A,$ $m\in Z_{K}(A)$ ,which implies the assertion.

$\square$We can construct two intertwing operators$\Phi_{\pi}^{R},$$\Phi_{\pi^{*}}^{L}$

using the matrix coefficients:$\Phi_{\pi}^{R}\in \mathrm{H}\mathrm{o}\mathrm{m}(9^{K)},(\pi\Lambda, C_{\tau_{d}}\infty(K\backslash G))$,,$\Phi_{\pi^{*}}^{L}\in \mathrm{H}\mathrm{o}\mathrm{m}_{(_{9^{K)}}},(\pi\Lambda’ c\tau_{d}**\infty(G/K))$by$\Phi_{\pi}^{R}(v)(g)=\sum_{ij}\langle\pi(g)v, f_{i^{*}}j\rangle f_{ij}$,$\Phi_{\pi^{*}}^{L}(w)(g)=\sum_{kl}\langle f_{k^{*}}l’\pi^{*}(g-1)w\rangle fkl$

If we put$\Phi_{\pi,\mathcal{T}}^{R}(g)=\sum_{kl}\Phi^{R}(f_{kl})(g\pi)\otimes fkl$,$\Phi_{\pi,\tau}^{L}(g)=\sum_{ij}fij\otimes\Phi^{L}\pi*(f_{ij})(g)$

,

.

116

then

$\Phi_{\pi,\tau}^{R}(g)$and$\Phi_{\pi^{*},\mathcal{T}}^{L}(g)$are identical to$\Phi_{\pi,\tau}(g)$.

.

3.2. Some functions on $A$ We put $a_{i}=\exp(t_{i})$ for the element $a=\exp(t_{1}H_{1}+t_{2}H_{2})$ of
the -split torus , We use for notation the following symbols:$A$$\mathbb{R}$$\mathrm{s}\mathrm{h}(x)=(x-x^{-1})/2$,$\mathrm{c}\mathrm{h}(x)=(x+x^{-1})/2,$$\mathrm{c}\mathrm{t}\mathrm{h}(x)=\mathrm{c}\mathrm{h}(x)/\mathrm{S}\mathrm{h}(x)$,$,$$\dot{D}=D(a)=\mathrm{S}\mathrm{h}(a_{1}2)-\mathrm{S}\mathrm{h}(a_{2}^{2}),$$p=p(a)=\mathrm{C}\mathrm{h}(a_{1})\mathrm{c}\mathrm{h}(a_{2}),$$z_{\pm}(\mathrm{t})=(\mathrm{c}\mathrm{h}(a_{1})/\mathrm{c}\mathrm{h}(a_{2})\pm_{\mathrm{C}}\mathrm{h}(a_{2})/\mathrm{C}\mathrm{h}(a_{1})),$

3.3. The Schmid operator. Let
and orthonormal basisof ,$\mathcal{T}_{d_{1}},$$\{X_{k}\}$$\tau_{d_{2}}$$t=t(a)=(\mathrm{C}\mathrm{h}(a_{1})/\mathrm{C}\mathrm{h}(a_{2}))2$$\partial_{j}=a_{j^{\frac{\partial}{\partial a_{j}’}}}\partial_{t}=t\frac{\partial}{\partial t},$be representations of $K$ . For.

$F(g)\in C_{d_{1},d_{2}}^{\infty}(K\backslash G/K)$$\mathfrak{p}$$\nabla_{d_{1},d_{2}}^{R}F(g)=\sum_{k}R_{X}F(\mathit{9})\otimes Xkk$$\nabla_{d_{1},d_{2}}^{L}F(g)=\sum_{k}LX_{k}F(g)\otimes Xk$

are called the Schmid operator. Here$R_{g}$, (resp.

$D_{d_{1}}^{(j)},’=P_{ddd’ d,d_{2}}d_{2}R(J)(\nabla^{R},D)2\circ’=P_{dd_{1},d}121jL(J1)\circ\nabla^{L}2$$P_{d}^{(J)}$$\partial_{p}=p\frac{\partial}{\partial p}$,$L_{g}$,,) is a right (resp. left) translation. Put

with defining the projectors:$V_{d} \otimes \mathfrak{p}_{\mathbb{C}}arrow V_{d}^{-}=\in-\bigoplus_{\beta\triangle^{+}nJ},V_{d\beta}-\cdot$

Theorem 3.2 ([7]). Let

where

$d$

. Then,$\mathrm{H}\mathrm{o}\mathrm{m}_{(9},K)(\pi\Lambda, c\infty(\mathcal{T}dK\backslash G))\simeq \mathrm{k}\mathrm{e}\mathrm{r}(D_{d,d^{*}}^{(}),R)j$,$\mathrm{H}_{\mathrm{o}\mathrm{m}_{()}}K(\mathfrak{g},\Lambda’ C_{\mathcal{T}_{d}}\infty*\pi*(G/K))\simeq \mathrm{k}\mathrm{e}\mathrm{r}(D_{d,d^{*}}^{(J),L})$,is the Blattner parameter of

We see that$\mathrm{P}++\mathfrak{p}_{-}$$\Lambda\in--J-$$\pi_{\Lambda}$.

is also decomposed intoalong the decomposition..
(see [2, \S 6]). The following formula is found in [5]:$\nabla^{L/R}$$\nabla_{+}^{L/R}+\nabla_{-}^{L/R}$$\mathfrak{p}_{\mathbb{C}}=$

Theorem 3.3 (Koseki-Oda). Let
be the Schmid operators andtion to A. Put $Z_{13}=2^{-1}(I_{2,2}+h^{1}-h2),$ $z_{24}=2^{-1}(I_{2,2^{-h}}1+h^{2})$ andThen, we have$\nabla_{\pm}^{R/L}$$\rho_{A}(\nabla_{\pm}^{R/L})$their restric.

$\tau_{\pm}^{(*)}=\tau^{(*)}\otimes \mathrm{A}\mathrm{d}_{\pm}$$\rho_{A}(\nabla_{+}^{R})\phi=\frac{1}{2}(\partial_{1}-\mathrm{s}\mathrm{h}(a)12-1(\mathcal{T}Z_{1}3)-\mathrm{C}\mathrm{t}\mathrm{h}(a_{1}2)\tau(+)*Z_{13}+2\mathrm{C}\mathrm{t}\mathrm{h}(a21)+\frac{2}{D}\mathrm{S}\mathrm{h}(a_{1}2))(\phi\otimes X_{13})$$+ \frac{1}{2}(\partial_{2}-\mathrm{s}\mathrm{h}(a_{2})-1(\mathcal{T}Z24)-\mathrm{c}\mathrm{t}\mathrm{h}(2)a_{2}^{2}\mathcal{T}_{+}^{*}(z_{24})+2\mathrm{C}\mathrm{t}\mathrm{h}(a)2-\frac{2}{D}2\mathrm{S}\mathrm{h}(a^{2}2))(\phi\otimes X_{24})$$+ \frac{1}{D}(\mathrm{c}\mathrm{h}(a_{1})_{\mathrm{S}\mathrm{h}(}a2)\mathcal{T}(e_{-}1)+\mathrm{S}\mathrm{h}(a_{1})\mathrm{c}\mathrm{h}(a2)\tau(e^{2}-)+\mathrm{s}\mathrm{h}(a_{2})\mathrm{c}\mathrm{h}(a_{2})\tau_{+}^{*}(e_{-)}1$$+\mathrm{s}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{1})\tau_{+(}*e_{-}2))(\phi\otimes X14)$$- \frac{1}{D}(\mathrm{s}\mathrm{h}(a_{1})\mathrm{c}\mathrm{h}(a2)\tau(e_{+})1+\mathrm{c}\mathrm{h}(a_{1})\mathrm{S}\mathrm{h}(a2)\tau(e_{+}2)+\mathrm{S}\mathrm{h}(a_{1})\mathrm{c}\mathrm{h}(a_{1})\mathcal{T}(+*e^{1})+$$+\mathrm{s}\mathrm{h}(a_{2})_{\mathrm{C}}\mathrm{h}(a_{2})\mathcal{T}^{*}+(e_{+})2)(\phi\otimes X23)$

,

117

$\rho_{A}(\nabla_{-}^{R})\phi=\frac{1}{2}(\partial_{1}+\mathrm{S}\mathrm{h}(a_{1})^{-1}\mathcal{T}(z_{13})+\mathrm{C}\mathrm{t}\mathrm{h}(a_{1}^{2})\mathcal{T}_{2}^{-}(2z13)+2\mathrm{c}\mathrm{t}\mathrm{h}(a_{1}^{2})+\frac{2}{D}\mathrm{S}\mathrm{h}(a^{2})1)(\phi\otimes X31)$$+ \frac{1}{2}(\partial_{2}+\mathrm{s}\mathrm{h}(a_{2})2-1(\mathcal{T}z_{24})+\mathrm{c}\mathrm{t}\mathrm{h}(a)\mathcal{T}_{2}^{-}(Z_{24})+2\mathrm{C}\mathrm{t}\mathrm{h}(2a_{2}22)-\frac{2}{D}\mathrm{S}\mathrm{h}(a^{2}2))(\phi\otimes X_{42})$$- \frac{1}{D}(\mathrm{c}\mathrm{h}(a_{1})\mathrm{s}\mathrm{h}(a_{2})\mathcal{T}(e^{1})++\mathrm{s}\mathrm{h}(a1)\mathrm{c}\mathrm{h}(a_{2})_{\mathcal{T}}(e^{2})++\mathrm{s}\mathrm{h}(a_{2})\mathrm{c}\mathrm{h}(a_{2})\mathcal{T}2^{-}(e^{1})+$$+\mathrm{s}\mathrm{h}(a_{1})\mathrm{c}\mathrm{h}(a_{1})_{\mathcal{T}}2^{-}(e_{+})2)(\phi\otimes x_{4}1)$$+ \frac{1}{D}(\mathrm{s}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a2)_{\mathcal{T}}(e^{1}-)+\mathrm{C}\mathrm{h}(a1)\mathrm{s}\mathrm{h}(a_{2})\mathcal{T}(e^{2}-)+\mathrm{s}\mathrm{h}(a_{1})\mathrm{c}\mathrm{h}(a_{1})_{\mathcal{T}_{2^{-}}(e_{-}^{1}})$,$+\mathrm{s}\mathrm{h}(a_{2})_{\mathrm{C}}\mathrm{h}(a_{2})\mathcal{T}_{2}^{-}(e_{-})2)(\phi\otimes x32)$$\rho_{A}(\nabla_{+}^{L})\phi=-\frac{1}{2}(\partial_{1}-\mathrm{s}\mathrm{h}(a_{1}^{2})-1\tau(z13)-*\mathrm{C}\mathrm{t}\mathrm{h}(a_{1}^{2})\tau_{+}(z_{1}3)+2\mathrm{C}\mathrm{t}\mathrm{h}(a21)+\frac{2}{D}\mathrm{S}\mathrm{h}(a_{1}2))(\phi\otimes X_{13})$$- \frac{1}{2}(\partial_{2}-\mathrm{s}\mathrm{h}(a^{2})2\mathcal{T}-1*(z24)-\mathrm{C}\mathrm{t}\mathrm{h}(a^{2})_{\mathcal{T}}2+(Z_{2}4)+2\mathrm{C}\mathrm{t}\mathrm{h}(a^{2})2-\frac{2}{D}\mathrm{S}\mathrm{h}(a_{2})2)(\phi\otimes x_{2}4)$$- \frac{1}{D}(\mathrm{c}\mathrm{h}(a_{1})\mathrm{S}\mathrm{h}(a_{2})_{\mathcal{T}^{*}}(e_{-})1\mathrm{s}+\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})\mathcal{T}(*)e^{2}-+\mathrm{s}\mathrm{h}(a_{2})\mathrm{c}\mathrm{h}(a_{2})\tau+(e_{-)}1$$+\mathrm{s}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a1)\tau_{+}(e^{2}-))(\phi\otimes X_{14})$$+ \frac{1}{D}(\mathrm{s}\mathrm{h}(a_{1})\mathrm{c}\mathrm{h}(a_{2})\tau^{*}(e_{+}^{1})+\mathrm{c}\mathrm{h}(a_{1})\mathrm{s}\mathrm{h}(a_{2})\mathcal{T}^{*}(e_{+}^{2})+\mathrm{S}\mathrm{h}(a_{1})\mathrm{c}\mathrm{h}(a_{1})\tau+(e_{+}^{1})$$+\mathrm{s}\mathrm{h}(a_{2})_{\mathrm{C}}\mathrm{h}(a_{2})\tau+(e_{+}2))(\phi\otimes X_{23})$,$\rho_{A}(\mathrm{v}_{-)\phi=-}^{L}\frac{1}{2}(\partial_{1}+\mathrm{S}\mathrm{h}(a^{2})-1*1(\tau z13)+\mathrm{c}\mathrm{t}\mathrm{h}(a_{1}^{2})\tau-(z_{1}3)+2_{\mathrm{C}}\mathrm{t}\mathrm{h}(a_{1})2\frac{2}{D}\mathrm{S}+\mathrm{h}(a_{1}2))(\phi\otimes x_{3}1)$$- \frac{1}{2}(\partial_{2}+\mathrm{S}\mathrm{h}(a^{2})2\tau-1*(z24)+\mathrm{c}\mathrm{t}\mathrm{h}(a)\tau^{-}(Z_{2}4)+2\mathrm{c}\mathrm{t}\mathrm{h}(2a_{2}^{2})2-\frac{2}{D}\mathrm{S}\mathrm{h}(a_{2})2)(\phi\otimes X42)$$+ \frac{1}{D}(\mathrm{c}\mathrm{h}(a_{1})\mathrm{s}\mathrm{h}(a_{2})\mathcal{T}^{*}(e_{+}^{1})+\mathrm{S}\mathrm{h}(a_{1})\mathrm{c}\mathrm{h}(a_{2})\mathcal{T}^{*}(e_{+}^{2})+\mathrm{S}\mathrm{h}(a_{2})\mathrm{c}\mathrm{h}(a_{2})\mathcal{T}-(e_{+}^{1})$$+\mathrm{s}\mathrm{h}(a_{1})_{\mathrm{C}}\mathrm{h}(a_{1})\tau^{-}(e_{+})2)(\phi\otimes X_{4}1)$$- \frac{1}{D}(\mathrm{s}\mathrm{h}(a_{1})\mathrm{c}\mathrm{h}(a_{2})_{\mathcal{T}^{*}}(e_{-})1\mathrm{c}+\mathrm{h}(a_{1})\mathrm{s}\mathrm{h}(a_{2})\mathcal{T}*(e_{-)+\mathrm{s}}2\mathrm{h}(a_{1})\mathrm{c}\mathrm{h}(a_{1})\mathcal{T}-(e_{-)}1$$+\mathrm{s}\mathrm{h}(a_{2})\mathrm{C}\mathrm{h}(a_{2})_{\mathcal{T}^{-}}(e^{2}-))(\phi\otimes X_{32})$.

4. HOLONOMIC SYSTEM FOR THE SPHERICAL FUNCTIONS
We treat the case of A
) is $d=\Lambda+[1,$ $-1$ ;(resp.

$\in--\mathrm{I}\mathrm{I}\mathrm{I}-\cup--\mathrm{I}\mathrm{V}-$$\Lambda\in--\mathrm{I}\mathrm{V}-$

Lemma 4.1. The projector$P_{d}^{(\mathrm{I})}\mathrm{I}\mathrm{I}$. Then, the Blattner parameter of$\pi_{\Lambda}$(resp. $\Lambda+[-1,1;0]$ ).

decomposes into four projectors as follows:$P_{d}^{()(}\mathrm{I}\mathrm{I}\mathrm{I}=P\oplus(-,+)P(-,-)-,+)(-,-)\oplus\overline{P}\oplus\overline{P}$,$P_{d}^{(\mathrm{I}\mathrm{V})}=P^{(+,-)}\oplus P(-,-)\overline{P}^{(+,-}\oplus\oplus\overline{P}))(-,-$

in A$\in-_{\mathrm{I}\mathrm{I}\mathrm{I}}$118

Proof. We find that$\triangle_{n}^{+}-$, III$=\{[1,1_{1}\pm 2],$$[1,$ $-1;\pm 2]\}$,, IV $=\{[1,1;\pm 2],$$[-1,1_{1}\pm 2]\}$$\triangle_{n}^{+}-$

Thus the lemma follows.

$\square$According to Theorem 3.2, spherical functions are characterized by the differential equations derived by the composition of the Schmid operator and projectors which appears in thefor $M=(k_{1}, l_{1}; k2, l_{2})$ .
. Letdecomposition of),Thensatisfy the following system which is equivalent to$P_{d}^{(\mathrm{I})}\mathrm{I}\mathrm{I}$$\Phi_{\pi_{\Lambda},\tau_{d}}(a)=\sum_{M}c_{M}(a)f_{k,l_{1}}1\otimes f_{k_{2},l_{2}}$$R/L=0\pi_{\Lambda},\mathcal{T}_{d}:d_{2}$$D_{d_{1}}^{(\mathrm{I}\mathrm{I}\mathrm{I}},\Phi$$c_{M}’ \mathrm{s}$Lemma 4.2.
(2)$(r_{2}-k2) \{\partial_{1}-\frac{1}{2}(u1+2k_{1}-r_{1}-2l_{1}+s_{1}).\frac{1}{\mathrm{s}\mathrm{h}(a_{1}^{2})}$$- \frac{1}{2}(u_{2}+2k_{2^{-r_{2}-}}2l_{2}+s_{2})\mathrm{c}\mathrm{t}\mathrm{h}(a_{1}^{2})+(k_{2}.+1)\frac{\mathrm{s}\mathrm{h}(a_{1}^{2})}{D}\}c_{k,l_{1;k_{2}}}1,l_{2}$$+(k_{2}+1)(S_{2^{-}}l_{2}+1) \frac{\mathrm{s}\mathrm{h}(a_{2}^{2})}{D}c_{kl}1,1;k2+1,l2-1$

(3)

+2

$(k_{2}+1)(r1-k_{1}+1) \frac{\mathrm{s}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})}{D}ck1-1,l_{1};k2+1,l2$+2$(k_{2}+1)(_{S}1^{-l+}11) \frac{\mathrm{c}\mathrm{h}(a_{1})\mathrm{S}\mathrm{h}(a_{2})}{D}c_{k,l1;}-k2+1,l_{2}=110$,$(k_{2}+1) \{\partial_{2}-\frac{1}{2}(u_{1}-2k_{1}+r_{1}+2l_{1}-S_{1})\frac{1}{\mathrm{s}\mathrm{h}(a_{2}^{2})}$$- \frac{1}{2}(u_{2^{-}}2k2+r_{2}+2l_{2}-s_{2}-4)\mathrm{c}\mathrm{t}\mathrm{h}(a_{2}^{2})-(r_{2}-k_{2})\frac{\mathrm{s}\mathrm{h}(a_{2}^{2})}{D}\}c_{kl_{1}}1,;k2+1,l2-1$$-(r_{2}-k_{2})l2 \frac{\mathrm{s}\mathrm{h}(a_{1}^{2})}{D}c_{k_{1}},l1;k_{2},l_{2}$$-2(r_{2}-k_{2})(k_{1}+1) \frac{\mathrm{c}\mathrm{h}(a_{1})\mathrm{S}\mathrm{h}(a_{2})}{D}c_{k_{1}1}+,l_{1;}k2,l_{2}-1$,$-2(r2-k_{2})(l1+1) \frac{\mathrm{s}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})}{D}c_{kl}=01,1+1;k2,l_{2}-1$(4)$(k_{2}+1) \{\partial_{1}+\frac{1}{2}(u_{1}+2k_{1}-r_{1^{-}}2l_{1}+s_{1})\frac{1}{\mathrm{s}\mathrm{h}(a_{1}^{2})}$$+ \frac{1}{2}(u_{2}+2k_{2}-r_{2^{-}}2l_{2}+s_{2}+4)_{\mathrm{C}\mathrm{t}\mathrm{h}}(a_{1}^{2})+(r_{2}-k2)\frac{\mathrm{s}\mathrm{h}(a_{1}^{2})}{D}\}Ck1,l_{1};k2+1,l2-1$$+(r_{2}-k_{2})l2 \frac{\mathrm{s}\mathrm{h}(a_{2}^{2})}{D}c_{k_{1}},l1;k_{2},l_{2}$

(5)

+2

$(r_{2}-k_{2})(k_{1}+1) \frac{\mathrm{s}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})}{D}c_{k_{1}1}+,l_{1;}k2,l_{2}-1$+2$(r_{2}-k_{2})(l1+1) \frac{\mathrm{c}\mathrm{h}(a_{1})\mathrm{S}\mathrm{h}(a_{2})}{D}c_{kl_{1}1}+;k_{2},l_{2}-1=01,$$(r_{2^{-}}k_{2}) \{\partial 2+\frac{1}{2}(u_{1^{-}}2k1+r_{1}+2l_{1}-S_{1})\frac{1}{\mathrm{s}\mathrm{h}(a_{2}^{2})}$

,

119

$+ \frac{1}{2}(u_{2}-2k_{2}+r_{2}+2l_{2}-S_{2})\mathrm{C}\mathrm{t}\mathrm{h}(a^{2}2)-(k_{2}+1)\frac{\mathrm{s}\mathrm{h}(a_{2}^{2})}{D}\}c_{k,l_{1;k_{2}}}1,l_{2}$$-(k_{2}+1)(_{S}2^{-l+}21) \frac{\mathrm{s}\mathrm{h}(a_{1}^{2})}{D}C_{k,l_{1}}1;k2+1,l_{2}-1$$-2(k_{2}+1)(r1-k1+1) \frac{\mathrm{c}\mathrm{h}(a_{1})\mathrm{S}\mathrm{h}(a_{2})}{D}ck1-1,l_{1};k2+1,l_{2}$.

$-2(k_{2}+1)(S_{1}-l_{1}+1) \frac{\mathrm{s}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})}{D}C_{k}1,l1-1;k_{2}+1,l_{2}=0$

As for left equation systems, we have the following system:(6)$(r_{1}-k_{1}) \mathrm{f}\partial_{1}-\frac{1}{2}(u_{2}+2k_{2}-r_{2^{-}}2l_{2}+s_{2})\frac{1}{\mathrm{s}\mathrm{h}(a_{1}^{2})}$$- \frac{1}{2}(u_{1}+2k_{1}-r_{1^{-}}2l_{1}+s_{1})\mathrm{c}\mathrm{t}\mathrm{h}(a_{1}^{2})+(k_{1}+1)\frac{\mathrm{s}\mathrm{h}(a_{1}^{2})}{D}\}c_{kl_{1;k_{2}}}1,,l_{2}$$+(k_{1}+1)(_{S}1^{-l+}11) \frac{\mathrm{s}\mathrm{h}(a_{2}^{2})}{D}Ck_{1}+1,l_{1}-1;k_{2},l2$+2$(k_{1}+1)(r2^{-k_{2}+}1) \frac{\mathrm{s}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})}{D}c_{k_{1+}1,l_{1}};k2-1,l_{2}$,$+2(k_{1}+1)(s2^{-l_{2}+1}) \frac{\mathrm{c}\mathrm{h}(a_{1})\mathrm{S}\mathrm{h}(a_{2})}{D}ck1+1,l_{1;}k2,l2-1=0$(7)$(k_{1}+1) \{\partial_{2}-\frac{1}{2}(u_{2}-2k_{2}+r_{2}+2l_{2}-S_{2})\frac{1}{\mathrm{s}\mathrm{h}(a_{2}^{2})}$$- \frac{1}{2}(u_{1}-2k_{1}+r_{1}+2l_{1}-s_{1}-4)\mathrm{C}\mathrm{t}\mathrm{h}(a^{2}2)-(r_{1}-k_{1})\frac{\mathrm{s}\mathrm{h}(a_{2}^{2})}{D}\}C_{k_{1}1}+,l_{1}-1;k_{2},l2$$-(r_{1}-k_{1})l1 \frac{\mathrm{s}\mathrm{h}(a_{1}^{2})}{D}c_{k_{1}},l1;k_{2},l_{2}$$-2(r_{1}-k1)(k2+1) \frac{\mathrm{c}\mathrm{h}(a_{1})\mathrm{S}\mathrm{h}(a_{2})}{D}ck_{1},l1-1,\cdot k_{2}+1,l2$$-2(r_{1}-k_{1})(l_{2}+1) \frac{\mathrm{s}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})}{D}C_{k,l1k_{2)}}-;l2+1=110$(8),$(k_{1}+1) \{\partial_{1}+\frac{1}{2}(u_{2}+2k_{2}-r_{2^{-}}2l_{2}+s_{2})\frac{1}{\mathrm{s}\mathrm{h}(a_{1}^{2})}$$+ \frac{1}{2}(u_{1}+2k_{1}-r_{1^{-}}2l_{1}+s_{1}+4)_{\mathrm{C}\mathrm{t}\mathrm{h}}(a_{1}^{2})+(r_{1}-k_{1})\frac{\mathrm{s}\mathrm{h}(a_{1}^{2})}{D}\}Ck1+1,l_{1}-1;k_{2},l2$$+(r_{1}-k_{1})l1 \frac{\mathrm{s}\mathrm{h}(a_{2}^{2})}{D}c_{k_{1}},l1;k_{2},l_{2}$$+2(r_{1}-k_{1})(k_{2}+1) \frac{\mathrm{s}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})}{D}c_{k_{1}},l_{1}-1;k_{2}+1,l_{2}$+2(9)$(r_{1}-k_{1})(l2+1) \frac{\mathrm{c}\mathrm{h}(a_{1})\mathrm{S}\mathrm{h}(a_{2})}{D}C_{k,l1}-;k2,l2+1=110$$(r_{1}-k_{1}) \{\partial_{2}+\frac{1}{2}(u_{2^{-}}2k2+r_{2}+2l_{2}-S_{2})\frac{1}{\mathrm{s}\mathrm{h}(a_{2}^{2})}$

,

120

$+ \frac{1}{2}(u_{1}-2k_{1}+r_{1}+2l_{1}-s_{1})_{\mathrm{C}\mathrm{t}}\mathrm{h}(a^{2}2)-(k_{1}+1)\frac{\mathrm{s}\mathrm{h}(a_{2}^{2})}{D}\}c_{k,l_{1;k_{2}}}1,l_{2}$$-(k_{1}+1)(_{S}1^{-l+}11) \frac{\mathrm{s}\mathrm{h}(a_{1}^{2})}{D}ck_{1}+1,l_{1}-1;k_{2},l2$$-2(k_{1}+1)(r_{2}-k_{2}+1) \frac{\mathrm{c}\mathrm{h}(a_{1})\mathrm{S}\mathrm{h}(a_{2})}{D}c_{k_{1}+1},l1;k_{2}-1,l_{2}$$-2(k_{1}+1)(s2^{-l_{2}+1}) \frac{\mathrm{s}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})}{D}ck1+1,l_{1;}k2,l2-1=0$.

4.1. Going
equations. We can reduce the obtained equations to the following
going up system (10), (11), (12), (13) as follows:$\mathrm{u}\mathrm{p}/\mathrm{d}\mathrm{o}\mathrm{W}\mathrm{n}$Lemma 4.3.
(10)$(r-k_{2})\{\mathrm{c}\mathrm{t}\mathrm{h}(a1)\partial_{1}-(s-l_{1^{-}2}l)\mathrm{c}\mathrm{t}\mathrm{h}^{2}(a_{1})$$- \frac{1}{2}(-u-k_{1}+k2+l_{1^{-l)2}}2+(k_{2}+1)\frac{\mathrm{c}\mathrm{h}^{2}(a_{1})}{D}\}ck_{1},l1;k_{2},l_{2}$+2$(k_{2}+1)(r-k_{1}+1) \frac{\mathrm{c}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})}{D}C_{k_{1}}-1,l_{1;}k_{2+1,l_{2}}$$=-2(k_{2}+1)(S-l_{2}+1) \frac{\mathrm{c}\mathrm{t}\mathrm{h}(a_{1})\mathrm{S}\mathrm{h}(a2)\mathrm{c}\mathrm{h}(a_{2})}{D}Ck1,l_{1};k2+1,l2-1$,$-2(k_{2}+1)(S-l_{1}+1) \frac{\mathrm{c}\mathrm{t}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a1)\mathrm{s}\mathrm{h}(a_{2})}{D}ck_{1},l1-1;k_{2}+1,l_{2}$(11)$(r-k_{2})\{\mathrm{c}\mathrm{t}\mathrm{h}(a2)\partial_{2}-(s-l_{1}-l2)_{\mathrm{C}}\mathrm{t}\mathrm{h}^{2}(a_{2})$$- \frac{1}{2}(u-k1+k2+l_{1^{-l)2}}2-(k_{2}+1)\frac{\mathrm{c}\mathrm{h}^{2}(a_{2})}{D}\}ck_{1},l1;k_{2},l_{2}$$-2(k_{2}+1)(r-k_{1}+1) \frac{\mathrm{c}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})}{D}ck_{1}-1,l_{1};k2+1,l_{2}$$=2(k_{2}+1)(S-l_{2}+1) \frac{\mathrm{s}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{1})\mathrm{c}\mathrm{t}\mathrm{h}(a_{2})}{D}ck_{1},l1;k_{2+}1,l_{2}-1$

+2
(12)

$(k_{2}+1)(s-l_{1}+1) \frac{\mathrm{s}\mathrm{h}(a_{1})\mathrm{C}\dot{\mathrm{h}}(a_{2})\mathrm{c}\mathrm{t}\mathrm{h}(a_{2})}{D},ck_{1},l1-1;k_{2}+1,l_{2}$,$(r-k_{1})\{\mathrm{c}\mathrm{t}\mathrm{h}(a_{1})\partial_{1}-(s-l_{1^{-}}^{\backslash }l2)_{\mathrm{C}}\mathrm{t}\mathrm{h}^{2}(a_{1})$$- \frac{1}{2}(u+k1^{-}k_{2^{-}}l1+l2)+2(k_{1}+1)\frac{\mathrm{c}\mathrm{h}^{2}(a_{1})}{D}\}ck_{1},l1;k_{2},l_{2}$+2$(k_{1}+1)(r-k_{2}+1) \frac{\mathrm{c}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})}{D}c_{k_{1}1}+,l_{1;}k2-1,l_{2}$$=-2(k_{1}+1)(s-l_{1}+1) \frac{\mathrm{c}\mathrm{t}\mathrm{h}(a_{1})\mathrm{S}\mathrm{h}(a2)\mathrm{c}\mathrm{h}(a_{2})}{D}c_{k_{1}}+1,l_{1}-1;k_{2},l2$..
$-2(k_{1}+1)(_{S}-l_{2}+1) \frac{\mathrm{c}\mathrm{t}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a1)\mathrm{s}\mathrm{h}(a_{2})}{D}Ck_{1}+1,l1;k_{2},l_{2}-1$

(13)

,

$(r-k_{1})\{\mathrm{C}\mathrm{t}\mathrm{h}(a2)\partial_{2^{-}}(s-l_{1}-l2)_{\mathrm{C}}\mathrm{t}\mathrm{h}^{2}(a_{2})$$- \frac{1}{2}(-u+k_{1}-k_{2}-l_{1}+l_{2})-2(k_{1}+1)\frac{\mathrm{c}\mathrm{h}^{2}(a_{2})}{D}\}c_{k}1,l_{1;}k_{2},l2$121$-2(k_{1}+1)(r-k_{2}+1) \frac{\mathrm{c}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})}{D}c_{k_{1}}+1,l_{1};k_{2}-1,l2$$=2(k_{1}+1)(s-l_{1}+1) \frac{\mathrm{s}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a1)\mathrm{c}\mathrm{t}\mathrm{h}(a_{2})}{D}c_{k_{1}}+1,l_{1}-1;k_{2},l2$.

$+2(k_{1}+1)(s-l_{2}+1) \frac{\mathrm{s}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})\mathrm{c}\mathrm{t}\mathrm{h}(a_{2})}{D}ck_{1}+1,l_{1};k2,l_{2}-1$

Going down equations are as follows:(14)$k_{2}\{\mathrm{c}\mathrm{t}\mathrm{h}(a_{1})\partial_{1}+(s-l_{1^{-}}l2)_{\mathrm{C}}\mathrm{t}\mathrm{h}^{2}(a_{1})$$+ \frac{1}{2}(-u-k_{1}+k2+l_{1}-l2)+2(r-k_{2}+1)\frac{\mathrm{c}\mathrm{h}^{2}(a_{1})}{D}\}c_{k}1,l_{1;}k_{2},l2$+2$(r-k_{2}+1)(k_{1}+1) \frac{\mathrm{c}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})}{D}c_{k_{1}1}+,l_{1};k_{2}-1,l2$$=-2(r-k_{2}+1)(l_{2}+1) \frac{\mathrm{c}\mathrm{t}\mathrm{h}(a_{1})\mathrm{S}\mathrm{h}(a2)\mathrm{c}\mathrm{h}(a_{2})}{D}ck1,l_{1};k2-1,l_{2+}1$$-2(r-k2+1)(l1+1) \frac{\mathrm{c}\mathrm{t}\mathrm{h}(a_{1})_{\mathrm{C}}\mathrm{h}(a_{1})\mathrm{s}\mathrm{h}(a_{2})}{D}c_{k_{1},l_{1+}}1;k_{2}-1,l_{2}$

(15)

,

$k_{2}\{\mathrm{c}\mathrm{t}\mathrm{h}(a_{2})\partial 2+(s-l_{1}-l2)_{\mathrm{C}}\mathrm{t}\mathrm{h}^{2}(a_{2})$$+ \frac{1}{2}(u-k1+k_{2}+l_{1^{-l)2}}2-(r-k2+1)\frac{\mathrm{c}\mathrm{h}^{2}(a_{2})}{D}\}ck_{1},l1;k_{2},l_{2}$$-2(r-k_{2}+1)(k_{1}+1) \frac{\mathrm{c}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})}{D}c_{k_{1+}1,l_{1}};k2-1,l_{2}$$=2(r-k_{2}+1)(l2+1) \frac{\mathrm{s}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{1})\mathrm{c}\mathrm{t}\mathrm{h}(a_{2})}{D}ck_{1},l1;k2-1,l_{2+}1$

+2
(16)

$(r-k_{2}+1)(l1+1) \frac{\mathrm{s}\mathrm{h}(a_{1})\mathrm{c}\mathrm{h}(a_{2})\mathrm{c}\mathrm{t}\mathrm{h}(a_{2})}{D}Ck1,l_{1+}1;k2-1,l_{2}$,$k_{1}\{\mathrm{c}\mathrm{t}\mathrm{h}(a1)\partial_{1}+(s-l_{1}-l2)_{\mathrm{C}}\mathrm{t}\mathrm{h}^{2}(a_{1})$$+ \frac{1}{2}(u+k_{1}-k_{2}-l_{1}+l_{2})+2(r-k_{1}+1)\frac{\mathrm{c}\mathrm{h}^{2}(a_{1})}{D}\}c_{kl_{1;k_{2}}}1,,l_{2}$$+2(r-k_{1}+1)(k_{2}+1) \frac{\mathrm{c}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})}{D}c_{k-1,l_{1}k}1;2+1,l_{2}$$=-2(r-k_{1}+1)(l_{1}+1) \frac{\mathrm{c}\mathrm{t}\mathrm{h}(a_{1})\mathrm{S}\mathrm{h}(a2)\mathrm{c}\mathrm{h}(a_{2})}{D}Ck1-1,l1+1;k2,l_{2}$$-2(r-k_{1}+1)(l_{2}+1) \frac{\mathrm{c}\mathrm{t}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a1)\mathrm{s}\mathrm{h}(a_{2})}{D}ck_{1}-1,l_{1};k2,l_{2}+1$

(17)

,

$k_{1}\{\mathrm{c}\mathrm{t}\mathrm{h}(a2)\partial_{2}+(s-l_{1}-l2)_{\mathrm{C}}\mathrm{t}\mathrm{h}^{2}(a_{2})$$+ \frac{1}{2}(-u+k_{1}-k_{2}-l_{1}+l_{2})-2(r-k_{1}+1)\frac{\mathrm{c}\mathrm{h}^{2}(a_{2})}{D}\}c_{k,l_{1;k_{2}}}1,l_{2}$$-2(r-k_{1}+1)(k_{2}+1) \frac{\mathrm{c}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})}{D}c_{k-1,l_{1}k1,l}1;2+2$$=2(r-k_{1}+1)(l_{1}+1) \frac{\mathrm{s}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{1})\mathrm{c}\mathrm{t}\mathrm{h}(a_{2})}{D}C_{k1}1-,l_{1}+1;k_{2},l2$

122

+2

.

$(r-k_{1}+1)(l2+1) \frac{\mathrm{s}\mathrm{h}(a_{1})\mathrm{C}\mathrm{h}(a_{2})\mathrm{C}\mathrm{t}\mathrm{h}(a_{2})}{D}ck_{1}-1,l_{1};k2,l_{2}+1$

To make equations more “symmetric”, we consider (10) $\pm(11)$ , etc, and rewrite themusing and . Put
$p$

$t$

$C_{k_{1},l_{1;}k},l_{2}(2a)=(\mathrm{s}\mathrm{h}(a_{1})\mathrm{s}\mathrm{h}(a_{2}))|s-l_{1}-l_{2}|(\mathrm{c}\mathrm{h}(a1)\mathrm{c}\mathrm{h}(a_{2}))-(r+S+2)/2\tilde{c}_{k,l_{1}k},l_{2}(1;2a)$.

In the following, we assume that $0\leq l_{1}+l_{2}\leq s$ . We remark that$2\partial_{p}=\mathrm{c}\mathrm{t}\mathrm{h}(a_{1})\partial_{1}+\mathrm{c}\mathrm{t}\mathrm{h}(a_{2})\partial_{2}$$4\partial_{t}=\mathrm{C}\mathrm{t}\mathrm{h}(a_{1})\partial_{1^{-\mathrm{c}\mathrm{t}}}\mathrm{h}(a_{2})\partial 2$,.

Then, we have

Lemma 4.4.
(18)$(r-k_{2})(\partial_{p}-l_{1})\tilde{C}_{k_{1},l_{1;}}k_{2},l2=(k_{2}+1)(s-l2+1)p\tilde{c}k1,l_{1;k+}21,l_{2}-1$$+(k_{2}+1)(s-l_{1}+1)\tilde{c}k_{1},l_{1}-1;k_{2}+1,l_{2}$

(19)

,

$(r-k_{2})(2 \partial_{t}+\frac{u}{2}+(k_{2}+1)\frac{t+1}{t-1})\tilde{c}_{k_{1},l_{1;}k_{2},l_{2}}$+2 $(k_{2}+1)(r-k_{1}+1)z_{-}(t)^{-}1\tilde{C}_{k_{1}}-1,l_{1};k2+1,l_{2}$$=(k_{2}+1)(S-l2+1)Z-(t)^{-}1(2-pz_{+})\tilde{c}k_{1},l1;k_{2+}1,l_{2}-1$$+(k_{2}+1)(_{S}-l_{1}+1)Z-(t)^{-}1(z+-2p)\tilde{C}_{k}1,l1-1;k_{2+1,l_{2}}$

(20)

,

$k_{2}\{(p^{2}-z_{+}(t)p+1)(\partial_{p}+l_{1^{-S}})$$+(s-l1-l2)(2p-z_{+}(t))p\}\tilde{C}_{k_{1}},l1;k_{2},l_{2}$$=(r-k2+1)(l2+1)p\tilde{C}_{k_{1},l}k2-1,l2+11$;$+(r-k_{2}+1)(l1+1)\tilde{c}k_{1},l_{1}+1;k_{2}-1,l_{2}$(21)$(r-k_{1})(\partial p-l_{2})\tilde{C}k_{1},l1;k_{2},l_{2}=(k_{1}+1)(s-l_{1}+1)p\tilde{c}k_{1+}1,l_{1^{-}}1;k2,l_{2}$$+(k_{1}+1)(s-l2+1)\tilde{C}_{k}1+1,l1;k2,l_{2}-1$

(22)

,

,

$(r-k_{1})(2 \partial_{t}-\frac{u}{2}+(k_{1}+1)\frac{t+1}{t-1})\tilde{c}_{k_{1},l_{1;k_{2}}},l_{2}$$+2(k_{1}+1)(r-k_{2}+1)z_{-}(t)^{-}1\tilde{C}_{k_{1}+l_{1;}}1,k2-1,l2$$=(k_{1}+1)(s-l1+1)_{Z}-(t)-1(2-pZ+)\tilde{c}_{k_{1}}+1,l_{1}-1;k_{2},l2$$+(k_{1}+1)(s-l2+1)Z-(t)^{-}1(z+-2p)_{\tilde{C}}k_{1}+1,l1;k2,l_{2}-1$

(23)

,

$k_{1}\{(p^{2}-z_{+}(t)p+1)(\partial_{p}+l2-S)$$+(s-l1^{-}l_{2})(2p-Z+(t))p\}\tilde{c}_{k_{1}},l1;k_{2},l_{2}$$=(r-k_{1}+1)(l_{1}+1)p\tilde{c}k1-1,l_{1}+1;k2,l_{2}$$+(r-k_{1}+1)(l2+1)\tilde{c}k_{1}-1,l_{1;}k2,l_{2}+1$.

As we know, the equations (21), (22) and (23) can be obtained by flipping indices 1 and 2:

Remark 4.5. We have similar equations when $s\leq l_{1}+l_{2}\leq 2s$ .

123

5. SOLUTION FOR THE HOLONOMIC SYSTEM: THE MAIN THEOREM

5.1. Separation of variables. We treat the case when $l_{1}+l_{2}\leq s$ .
Proposition 5.1. Write $M=(k_{1}, l_{1}; k2, l_{2})$ . Then
can be written in the form of “sepa$\tilde{c}_{M}$ration of variables”:$\tilde{c}_{M}(a)=\sum^{+2}l_{1}:=0l(-1)r-k1-l1p^{l_{1+}l_{2}}-isM,i(t)$We can prove it by induction on$l_{1}+l_{2}$. Assume that $l_{1}=l_{2}=0$ . By (18), we have,$\partial_{p}\tilde{c}_{k_{1},0;k_{2},0}=0$

so that actually we can put.

,. Next assume that $l_{1}+l_{2}>0$ .
and $k_{1}<r,$ $k_{2}<r$ , then (18) $/(r-k_{2})-(21)/(r-k_{1})$ shows the
Otherwise, we can assume. Consulting (18), we readily prove the formula.
According to Proposition 5.1, we can rewrite the difference equations of Lemma 4.4 interms of and . Comparing the coefficients as a polynomial of , we have the following.
If$S_{(0}k_{1},0;k_{2},$), $0(t)$$:=\tilde{c}_{k_{1},0;k0}(2,a)$assef.tion.

$l_{1}\neq l_{2}$$k_{2}\neq r$

$t$

$p$

Lemma 5.2.
(24)

$p$

1. If $0\leq k_{2}<r$, then,$(l_{2}-i)s_{(}k1,l1;k2,l_{2}),i=(s-l_{2}+1)S_{(}k_{1},l1;k2+1,l_{2}-1),i$$-(s-l_{1}+1).S_{(}k_{1},l1-1;.k_{2+}1,l_{2}),i-1$

(25)

,

$(2 \partial_{t}+\frac{u}{2}+(k_{2}+1)\frac{t+1}{t-1})s(k_{1},l_{1;}k_{2},l_{2}),i-\frac{2k_{1}}{z_{-}(t)}s(k_{1}-1,l1;k2+1,l_{2}),i$$=(s-l_{2}+1)( \frac{2}{z_{-}(t)}s_{(),-1}k_{1_{)}}l1;k_{2}+1,l_{2}-1i-\frac{t+1}{t-1}s_{(,1),i)}k1l_{1;}k2+1,l_{2}-$$-(s-l_{1}+1)( \frac{t+1}{t-1}s_{(k_{1}},l_{1}-1;k_{2}+1,l_{2}),i-1-\frac{2}{z_{-}(t)}s_{(1;}k_{1},l_{1}-k_{2}+1,l2),i)$(26)$l_{2^{S_{(l)}}}k1,l_{1;k_{2}},2,i+1^{-}(l_{1}+1)s_{(}k1,l_{1+}1;k2,l2-1),$,:$=(s-l_{2}-i)S(k1,l1;k2+1,l2-1),i+1-(l_{1}-i)Z+(t)s(k1,l_{1};k2+1,l_{2}-1),i$$+(2l_{1}+l_{2^{-}}S-i)S_{(,),i-}k1l_{1;}k_{2}+1,l2-11$.

2. If $0\leq k_{1}<r$, then,(27)$(l_{1}-i)S_{(}k_{1},l_{1};k_{2},l_{2}),i=(s-l_{1}+1)_{S_{(}}k1+1,l1-1;k_{2},l_{2}),i$$-(s-l_{2}+1)_{S_{(,1),-1}}k1+1,l1;k2l2-:$ ,(28)$(2 \partial_{t}-\frac{u}{2}+(k_{1}+1)\frac{t+1}{t-1})s(k1,l_{1};k2,l2),i-\frac{2k_{2}}{z_{-}(t)}S_{(k}k1+1,l_{1;}2-1,l2),i$$=(s-l_{1}+1)( \frac{2}{z_{-}(t)}s_{(,)}k1+1,l_{1}-1;k2l_{2},i-1-\frac{t+1}{t-1}s_{(}k_{1+1},l_{1}-1;k_{2},l_{2}),i)$$-(s-l_{2}+1)( \frac{\mathrm{t}+1}{t-1}s_{()}k_{1+}1,l1;k_{2},l_{2}-1,i-1-\frac{2}{z_{-}(t)}s_{(1}k1+1,l1;k_{2},l2-),i)$(29)$l_{1}s_{(k_{1}},l1;k_{2},l2),i+1-(l2+1)_{S}(k1,l_{1}-1;k_{2},l2+1),i$$=(s-l_{1^{-}}i)s_{(k_{1+}1},l1-1;k2,l_{2}),i+1-(l_{2}-i)z_{+(t})s_{(k+}1,l1-1;k2l2)1,,i$$+(2l_{2}+l_{1}-s-i)s(k_{1+1},l1^{-1;}k_{2},l_{2}),i-1$

.

,

124
5.2. Expression of peripheral entries using Gaussian hypergeometric functions. First
assume that $l_{1}=l_{2}=0$ . We simply write
By (19) and (22), we have$s_{k_{1},k_{2}}=s_{(k_{1},0k0),0};2,\cdot$$(r-k_{2})( \partial_{t}+\frac{u}{4}+\frac{k_{2}+1}{2}\frac{t+1}{t-1})sk1;k2+\frac{(k_{2}+1)(r-k_{1}+1)}{z_{-}(t)}S_{k}1;k2+1=1^{-}0$,$(r-k_{1}+1)( \partial_{t}-\frac{u}{4}+\frac{k_{1}}{2}\frac{t+1}{t-1})s_{k}-1;k_{2}+1+1\frac{k_{1}(r-k_{2})}{z_{-}(t)}s_{kk}=1;20$.

Eliminating$s_{k_{1}-1k_{2+1}};$ ’

we have$\{(\partial_{t}-\frac{u}{4}+\frac{k_{1}+1}{2}\frac{t+1}{t-1}\mathrm{I}(\partial_{t}+\frac{u}{4}+\frac{k_{2}+1}{2}\frac{t+1}{t-1}\mathrm{I}-k_{1}(k_{2}+1)z-(t)^{-}2\}Sk1;k_{2}=0$.

Considering $r+s=k_{1}+k_{2}$ , we have$( \partial_{t}^{2}+\frac{r+s+2}{2}\frac{t+1}{t-1}\partial_{t}+\frac{u(k_{1^{-}}k_{2})}{8}\frac{t+1}{t-1}+\frac{(r+s+2)2-(k1-k_{2})^{2}-u2}{16})Sk_{1};k2=0$

and its$\dot{\mathrm{R}}$emann’s $P$ scheme is:$P[_{\frac{\frac{r+s+2}{r+s+24}}{4}+\frac{\frac{k_{1}-k_{2}+u}{k_{1}-\not\in_{2+u}}}{4}}-0$$-(r+S+101)$$\frac{\frac{r+s+2}{r+s+24}}{4}-+\frac{k_{1}-k_{2}-u}{\frac{k_{1}-\not\in_{2}-u}{4}}]\infty$

In general, let $\Phi(m_{1}, m_{2})=\Phi(m_{1}, m_{2};u;t)$ be a regular function around 1 having the Pscheme$P[ \frac{\frac{m_{1}+m_{2+2}}{m_{1}+^{4}m_{2}+2}}{4}+-\frac{\frac{m_{1}-m2+u}{m_{1}-m_{2}+u4}}{4}0$$-(m_{1}+m_{2}01+1)$$\frac{\frac{m_{1}+m_{2}+2}{m_{1}+^{4}m2+2}}{4}+\frac{\frac{m_{1}-m_{2}-u}{m_{1}-m2-4u}}{4}]-\infty$with condition $\Phi(m_{1}, m_{2;}u;1)=$ We also write $\Phi(m)=\Phi(m, r+s-m)$ for
simplicity. Then it follows), $0=c0\Phi(k_{1}, k_{2})$ .
$s_{(k_{1},0;}k_{2},0$.

5.3. Reduction of general coefficients
To describe general solutions, we introduce
the notion of height and bias. Write $M=(k_{1}, l_{1}; k_{2}, l_{2})$ as before. Define $h=h(M, i)=$$s_{M,i}$$\min(i, l_{1}, l_{2}, l_{1}+l_{2}-i)$and$b=b(M, i)=$
Then we have,
Proposition 5.3.
(30)$s_{M,i}= \sum_{hj=b-}^{b+h}Qj(-z+)\Phi(k1+l_{1}+j)$

for a polynomial $Q_{j}(t)=Q_{j}(M, i;t)$ which is actually independent of the choice ofand . The degree of is equal to $h-|j-b|$ and itfollows$k_{2}$$Q_{j}$$Q_{j}(-z+)=(-1)^{\mathrm{d}\mathrm{e}}\mathrm{g}QjQj(z+)$

.

$r,$

$k_{1}$

125
5.4. Polynomials $Q_{j}(z+)$ . The remaining paper deals with the determination of the polynomial . We can deduce the difference equations of equivalent to (26). Proposition 5.3
says that is in the form
$Q_{j}$$Q_{j}$$Q_{j}$$Q_{j}(z_{+})=m \geq 0\sum\tilde{\beta}_{m}(M, i, j)z+h-|j|-2m$.

For simplicity, we put $\tilde{\beta}_{m}(M, i, j)=\beta_{m}(M, i,j)$ . Comparing the coefficient of, we see that our difference equations become as follows: If $j\geq 0$ , then,$z_{+}^{h-|j}|-2m$(31)$(s-i+1)i\beta m(M, i,j)=(s-l1)l_{2}\beta m(M+(0,1;\mathrm{o}, -1), i-1,j-1)$$+l_{1}(s-l2-i+1)\beta_{m}-1(M+(\mathrm{O}, -1;1,0), i-1,j+1)$$+(l_{1}-i+1)l_{2}\beta m(M+(\mathrm{O}, 0;1, -1), i-1,j)$$+(2l_{1}+l_{2^{-s-}}i+1)l_{2}\beta_{m}-1(M+(\mathrm{O}, 0;1, -1), i-2,j)$.

If $j<0$ , then,(32)$(s-i+1)i\beta m(M, i,j)--(S-l1)l_{2}\beta_{m-1}(M+(\mathrm{o}, 1;0, -1), i-1,j-1)$$+l_{1}(s-l_{2^{-i1}}+)\beta m(M+(\mathrm{o}, -1;1,0), i-1,j+1)$$+(l_{1}-i+1)l_{2}\beta_{m}(M+(\mathrm{O}, 0;1, -1), i-1,j)$$+(2l_{1}+l_{2^{-}}s-i+1)l_{2}\beta_{m-}1(M+(\mathrm{O}, 0;1, -1), i-2,j)$.

The solution can be expressed as follows:

Proposition 5.4. Assume that $0\leq l_{1}+l_{2}\leq s$. Then,$\beta_{m}=\alpha(m;i, |j|)\sum_{n=0}^{m}$for$\alpha(m;i,j)=$ ,$j$$(j\geq 0)$$0$$(j<0)$$j_{+}=\{$,

and $j_{-}=(-j)_{+}$ .
We can check that each $\beta_{m}(M, i,j)$ fits the definition of as $\beta_{m}(M, i,j)$ is nonzero if
and only if $i-|j|-2m\geq 0,$ $l_{1}-i+j_{+}+m\geq 0$ and $l_{2}-i-j_{-}+m\geq 0$ .
$h$

Main

. Let

$\mathrm{T}\mathrm{h}\underline{\underline{\mathrm{e}}}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}5.5$$1;u]\in\cup \mathrm{I}\mathrm{I}\mathrm{I}$coefficient,and$\Phi_{\pi,\tau}$$\tau_{d}$, putbe a middle discrete series representation with $\Lambda=[r-1,$ $s+$
the minimal $K$ -type ofwith $d=[r, s;u]$ . For a-matrix$\pi_{\Lambda}$$(\tau_{d}, \tau_{d^{*}})$$\pi_{\Lambda}$$\Phi_{\pi,\mathcal{T}}.(a)=\sum_{2k_{1},l1;kl2},c‘ k_{1}.’l_{1;k}2,l2.(a)f_{kl_{1;}k_{2},l}1,2^{\cdot}$

Then, $c_{M}(a)(M=(k_{1}, l_{1} ; k_{2}, l_{2}))$ is given by the following:

126

1. Suppose that $l_{1}+l_{2}\leq s$ . The matrix coefficients$c_{k_{1},l_{1;k_{2}}},l_{2}(a_{1}, a_{2})$can be expressed asfollows:$c_{M}(a_{1,2}a)=c_{0}(-1)^{r}-k_{1}-l1(\mathrm{s}\mathrm{h}(a1)\mathrm{s}\mathrm{h}(a_{2}))S-l1-l_{2}$$l_{1}+l_{2}$$\cross\sum(\mathrm{c}\mathrm{h}(a_{1})\mathrm{c}\mathrm{h}(a_{2}))^{-(r+S}+2)/2+l1+l2-i$$i=0$$\mathrm{x}\sum_{hj=b-}^{bh}(-1)^{h|j-}-b|\beta+[\frac{i-|j-b|}{\sum_{\mu=0}^{2}}]\mu(M, i,j)(\frac{\mathrm{c}\mathrm{h}(a_{1})}{\mathrm{c}\mathrm{h}(a_{2})}+\frac{\mathrm{c}\mathrm{h}(a_{2})}{\mathrm{c}\mathrm{h}(a_{1})})^{h}-|j-b|-2\mu$$\cross\Phi(k_{1}+l1+j,$ $k_{2}+l2-j;u;( \frac{\mathrm{c}\mathrm{h}(a_{1})}{\mathrm{c}\mathrm{h}(a_{2})})^{2})$.

2. Suppose that $s<l_{1}+l_{2}\leq 2s$ . Define $M^{\wedge}=(r-k_{1}, s-l1;r-k_{2}, s-l_{2}),$ $b\wedge=b(M^{\wedge}, i)$and. Then,$h^{\wedge}=h(M^{\wedge}, i)$$c_{M}(a_{1,2}a)=c_{0}(-1)r-k1-l1$

$($

sh

$(a_{1})$sh$(a_{2}))^{-}s+l_{1+}l_{2}$$l_{1}+l_{2}$$\cross\sum(\mathrm{c}\mathrm{h}(a_{1})\mathrm{c}\mathrm{h}(a_{2}))^{-(+}r+s2)/2+2s-l1-l2-i$$i=0$$\sum_{\Lambda,j=b\wedge-h}^{b^{\wedge\wedge}}(-1)h\wedge-|j-b\wedge|\beta_{m}(+h[\frac{i-|\mathrm{j}-b^{\wedge}|}{\sum_{m=0}^{2}}]\wedge M, i,j)(\frac{\mathrm{c}\mathrm{h}(a_{1})}{\mathrm{c}\mathrm{h}(a_{2})}+\frac{\mathrm{c}\mathrm{h}(a_{2})}{\mathrm{c}\mathrm{h}(a_{1})})^{h^{\wedge}}-|j-b\wedge|-2m$$\cross$$\mathrm{x}\Phi(k_{2}+l_{2}+j,$ $k1+l_{1^{-j;-u}};( \frac{\mathrm{c}\mathrm{h}(a_{1})}{\mathrm{c}\mathrm{h}(a_{2})})^{2})$.

Remark 5.6. We can determine the unique unknown constant by using the normalization
condition, $i.e.$ , by specification of the value of at the identity of .
$c_{0}$$\Phi$$G$GRADUATE SCHOOL OF MATHEMATICAL SCIENCES,
THE UNIVERSITY OF TOKYO,
KOMABA, TOKYO 153, JAPANDEPARTMENT OF MATHEMATICS, FACULTY OF EDUCATION,MIE UNIVERSITY,
TSU, MIE 514, JAPANGRADUATE SCHOOL OF MATHEMATICAL SCIENCES,
THE UNIVERSITY OF TOKYO,
KOMABA, TOKYO 153, JAPAN127
REFERENCES[1] Y. Gon, The

Dr. Thesis.

generalized. Whittakerfunctions on $SU(2,2)$ with respect to the Siegel parabolic subgroup,[2] T. Hayata, Differential equations of principal series Whittakerfunctions on $SU(2,2)$ , Indag. Math. 8(1997), No. 4, 493-528.
[3], Whittakerfunctions of generalized principal series $on\cdot SU(2,2)$ , to appea..r in J. Math. Kyoto$\overline{\mathrm{U}\mathrm{n}\mathrm{i}\mathrm{v}.}$[4] T. Hayata and T. Oda, An explicit integral representation of Whittakerfunctions for the representationsof the discrete series–The case of $SU(2,2)-$ , to appear in J. Math. Kyoto Univ.
[5] H. Koseki and T. Oda, Matrix coefficients of-principal series of $SU(2,2)$ , preprint, 1998.
[6] E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed., Cambridge University Press,$P_{J}$

London, 1927.

[7] H. Yamashita, Embeddings of discrete series into induced representations of semisimple Lie groups ,general theory and the case of $SU(2,2)-$ , Japan. J. Math.
(1990), No. 1, 31-95.
[8] –, Embeddings of discrete series into induced representations of semisimple Lie groups II, -generalized Whittaker models for $SU(2,2)-$ , J. Math. Kyoto Univ. 31-2 (1991), 543-571.
$I$$(\mathrm{N}.\mathrm{s}.)16$-