数理解析研究所講究録
第 792 巻 1992 年 151-171

151

AN EXPLICIT INTEGRAL REPRESENTATION
OF WHITTAKER FUNCTIONSFOR THE REPRESENTATIONS OF THE DISCRETE SERIES.
THE CASE OF$Sp(2;\mathbb{R})$--

TAKAYUKI ODA
l-st draft ( \alpha test)

Introduction.
We shall prove an explicit integral formula for the Whittaker function associatedto the highest weight vector in the representation space of the minimal K-type of adiscrete series representation with the maximal Gelfand-Kirillov dimension for thereal symplectic groupof rank 2.
Let us explain the basic idea of this paper. Consider the case $G=SL_{2}$ (IR).
$Sp(2;\mathbb{R})$Put$N=\{(\begin{array}{ll}l x0 1\end{array})|x\in \mathbb{R}\}$, and let$\eta$:$(\begin{array}{ll}1 x0 l\end{array})\mapsto\exp(2\pi icx)(c\in \mathbb{R})$be anon-trivial unitary character of $N$ . Let
be the space of-functions$\varphi(ng)=\eta(n)\varphi(g)(\forall(n,g)\in
N\crossG)$.
satisfying
For an irreducible unitary representation
theof , we denote byspace of smooth vectors in . Whenis a principal series representa, the image of a vector intion ofwith respect to a unique continuous intertwining operator fromtois represented by the modified
Bessel function, i.e. the Whittaker function, if it is restricted to the split torus$c’\infty$$C_{\eta}^{\infty}(N\backslash G)$$(\pi, H_{\pi})$$G$$\varphi$$G$$H_{\pi}^{\infty}$$(\pi, H_{\pi})$$SL_{2}(\mathbb{R})$$H_{\pi}^{\infty}$$H_{\pi}^{\infty}$$A=\{(\begin{array}{lll}a 00 a -1\end{array})|a\in \mathbb{R},$$C_{\eta}^{\infty}(N\backslash G)$$a>0\}$ .

However whenis a discrete series representation of formal degree $k-1$, then the image of minimal K-type vector ofofwith respect to the(if it exists), is written by const.
tointertwining operator fromon (cf. Jacquet-Langlands [J-L]).
Thus as special functions on , the functions realizing the Whittaker model ofthe discrete series representations ofare $degenerate’$ elementary functions,much simpler than those of the principal series representations.
We hope similar phenomena occur in higher rank groups. The purpose of thispaper is to confirm this for the case $G=Sp(2;\mathbb{R})$ . Let us explain the contents ofthis paper.
In the first place, we shall compute explicitly the partial differentail equation
for the radial part of the above $Whi\cdot ttaker$ function. We follow the method of
Yamashita [Y-I] [Y-II] who discussed the case $G=SU(2,2)$ .
In \S 1, we recall basic notation for the structure ofand associated Liealgebras.
$(\pi, H_{\pi})$$SL_{2}(\mathbb{R})$$H_{\pi}$$H_{\pi}^{\infty}$$a^{k}e^{-2\pi|c|\cdot a^{2}}$$C_{\eta}^{\infty}(N\backslash G)$

$A$

$A$

((

$SL_{2}(\mathbb{R})$$Sp(2;\mathbb{R})$Typeset by$A_{\Lambda\Lambda}S- TB^{\backslash }\{$

152
$TAI\{AYUI\{I$ODA\S 2 reviews the Harish-Chandra parametrization of the representations of discrete
series for. In \S 3, we recall the representation of $U(2)$ , and in \S 4 thecharacters of the maximal unipotent subgroup of. In \S 5-\S 8, we write down
explicitly the system of partial differential equations characterizing the Whittaker
functions of the minimal K-type of a dicrete series representation.
New parts different from [Y-I], [Y-II] are Proposition (8.1) and \S 9. \S 9 contains themain result of this paper: an explicit integral expression of the Whittaker functionof the highest weight vector of the minimal K-type of a dicrete series representationof.
The author thanks to Professors T. Oshima and N. Wallach for educational conversations on the representation theory of real reductive groups in various occasions,to Professor H. Matsumoto for communications on the theory of Whittaker models,
and to Professor T. Miwa for an assist to solve partial differential equations.
$Sp(2;\mathbb{R})$$S\dot{p}(2;\mathbb{R})$$Sp(2;\mathbb{R})$153
INTEGRAL REPRESENTATION OF $WHITTAI\langle ER$ FUNCTION\S 1 Basic Notations, and the structure of Lie groups and algebras.

In this section, we determine basic notations on the symplectic group of degree 2,
its maximal compact subgroup and associated Lie algebras.

\langle Lie groups}Let$M_{4}(\mathbb{R})$be the space of real

$4\cross 4$

matrices. Put$J=(\begin{array}{ll}0 l_{2}-1_{2} 0\end{array})\in M_{4}(\mathbb{R})$is a unit matrix of size 2. The symplectic group
where
given by$1_{2}$$Sp(2;\mathbb{R})$$Sp(2:\mathbb{R})=\{g\in M_{4}(\mathbb{R})|{}^{t}gJg=J, \det(g)=1\}$,

of degree 2 is.

Heredenotes the transpose of the matrix , and $\det(g)$ the determinant of . A
maximal compact group $K$ of $G=Sp(2;\mathbb{R})$ is given by${}^{t}g$

$g$

$g$

$K=\{(\begin{array}{ll}A B-B A\end{array})\in Sp(2;\mathbb{R})|A,$ $B\in M_{2}(\mathbb{R})\}$,

which is isomorphic to the unitary group$U(2)=\{g\in GL(2;\mathbb{C})|^{t}\overline{g}\cdot g=1_{2}\}$of size 2 via a homomorphism$(\begin{array}{ll}A B-B A\end{array})\in K A+\sqrt{-1}B\in U(2)$\langle Lie algebras}
The Lie algebra of$G$.

is given by$g=\epsilon p(2;\mathbb{R})=\{X\in M_{4}(\mathbb{R})|JX+{}^{t}XJ=0\}$,

and that of $K$ is given by$t=\{X=(\begin{array}{ll}A B-B A\end{array})|A,$ $B\in M_{2}(\mathbb{R});{}^{t}A=-A,{}^{t}B=B\}$

The Cartan involution for$\not\in$.

is given by$\theta(X)=-{}^{t}X$for$X\in g$.

Hence the subspace’$p=\{X\in g|\theta(X)=X\}=\{(\begin{array}{ll}A BB -A\end{array})$$A=A,{}^{t}B=B;A,$ $B\in M_{2}(\mathbb{R})\}$

given a Cartan decomposition$g=t\oplus p$.

The linear map$(\begin{array}{ll}A B-B A\end{array})\in\epsilon A+\sqrt{-1}B\in u(2)$154TAKAYUKI ODA

defines an isomorphism of Lie algebras from$k$to the unitary Lie algebras$u(2)=\{C\in M_{2}(\mathbb{C})|{}^{t}\overline{C}+C=0\}$of degree 2.
An R-basis of $n(2)$ is given by$\sqrt{-1}(\begin{array}{ll}1 00 -l\end{array})Y=(\begin{array}{ll}0 1-1 0\end{array})Y’=\sqrt{-1}(\begin{array}{ll}0 11 0\end{array})$$\sqrt{-1}(\begin{array}{ll}1 00 1\end{array}),$Letgiven by$n(2)_{\mathbb{C}}=\iota\downarrow(2)\otimes_{\mathbb{R}}\mathbb{C}$$Z=(\begin{array}{ll}1 00 l\end{array})$,be the complexification of $n(2)$ . Then a basis of

Then$\{H’, X,\overline{X}\}$is a$51_{2}$,$\overline{X}=\frac{1}{2}(-Y-\sqrt{-1}Y’)=(\begin{array}{ll}0 01 0\end{array})$is.

-triple, i.e.

$[H’, X]=2X;[H’,\overline{X}]=-2\overline{X}$Via the isomorphism$u(2)_{\mathbb{C}}$,$H’=(\begin{array}{ll}1 00 -1\end{array})$$X= \frac{1}{2}(Y-\sqrt{-1}Y’)=(\begin{array}{ll}0 10 0\end{array})$.

$t_{\mathbb{C}}arrow u_{\mathbb{C}}\sim$$\overline{X}$] $=H’$ .

, the preimage of the above basis of$Z=(-\sqrt{-1})(\begin{array}{llll} 1 l-1 -1 \end{array})$$Y=(\begin{array}{llll}0 1 -1 0 0 1 -1 0\end{array})$

; [X,

;

$u_{\mathbb{C}}$is given by$H’=(-\sqrt{-1})(\begin{array}{llll} 1 -1-l 1 \end{array})$;$Y’=(\begin{array}{llll} 1 1 -1 -1 \end{array})$;.

From now on we use the convention that unwritten components of a matrix are
zero. Now we fix a compact Cartan subalgebra ) of by
$|$

$g$

$\mathfrak{h}=\mathbb{R}(\sqrt{-1}Z)+\mathbb{R}(\sqrt{-1}H’)$.

Write $\tau_{+}=\sqrt{-1}Z$ and $T_{-}=\sqrt{-1}H’$ , and set$T_{1}= \frac{1}{2}(\tau_{+}+\tau_{-})$

and $T_{2}= \frac{1}{2}(T_{+}-T_{-})$ .

Put$H_{1}’= \frac{1}{2}(Z+H’)$

Then $T_{1}=\sqrt{-1}H_{1}’$ ,$T_{2}=\sqrt{-1}H_{2}’$,$H_{2}’= \frac{1}{2}(Z-H’)$

.

, and

$T_{1}=(\begin{array}{llll} 1 0-1 0 \end{array})$,$T_{2}=(\begin{array}{llll} 0 10 -1 \end{array})\in \mathfrak{h}$.

155
INTEGRAL REPRESENTATION OF WHITTAKER FUNCTION{Root$system\rangle$

We consider a root space decomposition of with respect to . For a linear form
, we write, set:. For each$g$$\beta$$\beta(T_{i})=\beta_{i}\in \mathbb{C}$$\mathfrak{h}arrow \mathbb{C}$$\mathfrak{h}$$\beta\in \mathfrak{h}^{*}=Hom(\mathfrak{h}, \mathbb{C})$$g_{\beta}=\{X\in 9\mathbb{C}=g\otimes_{\mathbb{R}}\mathbb{C}|[H, X]=\beta(H)X, \forall_{H}\in\})\}$

Then the roots of$(g, \mathfrak{h})$.

is given by$\sum=\{\beta=(\beta_{1}, \beta_{2})|g_{\beta}\neq 0, \beta\neq 0\}$$=\sqrt{-1}\{\pm(2,0), \pm(0,2), \pm(1,1), \pm(1, -1)\}$

We determine a root vector$X_{\beta}$in$g_{\beta}$, i.e. a generator of$g_{\beta}$by the following table.

Then$f_{\mathbb{C}}=\mathfrak{l})_{\mathbb{C}}+\mathbb{C}X_{(1,-1)}+\mathbb{C}X_{(-1,1)}$,and set$P+=\mathbb{C}X_{(2,0)}+\mathbb{C}X_{(1,1)}+\mathbb{C}X_{(0,2)}$$=\{X=(\begin{array}{ll}X_{1} iX_{1}iX_{I} -X_{1}\end{array})|X_{1}\in M_{2}(\mathbb{C})\}$,

and$P-=\mathbb{C}X_{-(2,0)}+\mathbb{C}X_{-(1,1)}+\mathbb{C}X_{-(0,2)}$$=\{X=(\begin{array}{ll}X_{1} -iX_{1}-iX_{1} -X_{1}\end{array})|X_{1}\in M_{2}(\mathbb{C})\}$

Then$g_{\mathbb{C}}=t_{\mathbb{C}}\oplus p_{+}\oplus p_{-}$For each root$\beta=(\beta_{I}, \beta_{2})$.

, we put$\Vert\beta||=\sqrt{|\beta_{1}|^{2}+|\beta_{2}|^{2}}$

.

.

.

156
TAKAYUKI ODA

Then
Then set$\Vert\beta\Vert^{2}=4$

or $=2$ .

$\{c\cdot\Vert\beta\Vert(X_{\beta}+X_{-\beta}), c\cdot\sqrt{-1}\Vert\beta\Vert(X_{\beta}-X_{-\beta}) (\beta\in\Sigma_{n}^{+})\}$

forms an orthonormal basis of $p=p_{R}$ with respect to the Killing form for somepoints constant . Here $\Sigma_{n}^{+}=\{(2,0), (1,1), (0,2)\}$ is the set of non-compact positive
androots. $\Sigma_{c}^{+}=\{(1, -1)\}$ is the set of compact positive roots.
are the set of compact roots and the set of non-compact roots,
respectively.
$c$$\Sigma_{c}=\Sigma_{c}^{+}\cup(-\Sigma_{c}^{+})$$\Sigma_{n}=\Sigma_{n}^{+}\cup(-\Sigma_{n}^{+})${Iwasawa decomposition}

We choose a maximal abelian subalgebra$\mathfrak{a}$

of

$p$

given by$a= \{(\frac{A|0}{0|-A})|A=diag(t_{1}, t_{2})$Here diag$(t_{1}, t_{2})$is a diagonal matrix with $(1, 1)$ -entry$H_{1}=(\begin{array}{llll}1 0 -1 0\end{array})$

Then$\{H_{1}, H_{2}\}$$(t_{1}, t_{2}\in \mathbb{R})\}$

and.

and $(2, 2)$ -entry$t_{1}$$t_{2}$. Set.

$H_{2}=(\begin{array}{llll}0 1 0 -1\end{array})$

forms a basis of .
$\alpha$lRoot system of}
Let $\{e_{1}=(1,0), e_{2}=(0,1)\}$ be a standard basis of the 2-dimensional Euclidean. Then the root systemofis given by
plane$(g, \alpha)$$\mathbb{R}^{2}$$(g, \alpha)$$\Psi$$\Psi=\{\pm 2e_{1}, \pm 2e_{2}, \pm e_{1}\pm e_{2}\}$A positive root system $\Psi+is$ fixed by$\Psi+=\{2e_{1},2e_{2}, e_{1}+e_{2}, e_{1}-e_{2}\}$.

Put$\mathfrak{n}=\sum_{\alpha\in\Psi+}g_{\alpha}$.

Then it is a nilradical of a minimal parabolic subalgebra. We choose generatorsofas follows.

$E_{\alpha}$$9\alpha(\alpha\in\Psi_{+})$$E_{e_{1}-e_{2}}=(\begin{array}{llll}0 1 0 0 0 0 -1 0\end{array})$.

The Iwasawa decomposition associated to $(a, n)$ is given by$g=t\oplus \mathfrak{a}\oplus \mathfrak{n}$.

In, the Iwasawa decomposition of the root vectors
follows.
$9\mathbb{C}$$\{X_{\beta)}\cdot\beta\in\Sigma\}$are given as

157
INTEGRAL REPRESENTATION OF WHITTAKER FUNCTION

Lemma (1.1).
$X_{(2,0)}=H_{1}’+H_{1}+2\sqrt{-1}E_{2e_{1}}$;$X_{(-2,0)}=-H_{1}’+H_{1}-2^{-}\sqrt{-1}E_{2e_{1}}$$X_{(1,1)}=2\cdot\overline{X}+2\cdot E_{e_{1}-e_{2}}+2\sqrt{-1}E_{e_{1}+e_{2}}$;$X_{(-1}-1)=-2\cdot X+2\cdot E_{c_{1}-e_{2}}-2\sqrt{-1}E_{e_{1}+e_{2}}$$X_{(0,2)}=H_{2}’+H_{2}+2\sqrt{-1}E_{2e_{2}}$

;

;

;

$X_{(0,-2)}=-H_{2}’+H_{2}-2\sqrt{-1}E_{2e_{2}}$$\bullet$

Proof)A direct computa tion.

158
TAKAYUKI ODA\S 2 Parametrization of the representation of the discrete series.
Consider a compact Cartan subgroup of$G$$\exp(\mathfrak{h})=\{(\begin{array}{llll}cos\theta_{1} sin\theta_{1} cos\theta_{2} sin\theta_{2}-sin\theta_{1} -sin\theta_{2} cos\theta_{1} cos\theta_{2}\end{array})|\theta_{1},$ $\theta_{2}\in \mathbb{R}\}$

corresponding to . Then the characters are given by$\mathfrak{h}$$(\begin{array}{llll}cos\theta_{1} sin\theta_{1} cos\theta_{2} sin\theta_{2}-sin\theta_{1} -sin\theta_{2} cos\theta_{1} cos\theta_{2}\end{array})\exp\{\sqrt{-1}(m_{1}\theta_{1}+m_{2}\theta_{2})\}\in \mathbb{C}^{*}$.

Hereare some integers. The derivation of these characters determines an, the weight lattice.
integral structure of
The set of compact positive roots is given by $\Sigma_{c}^{+}=\{(1, -1)\}$ . Hence the set of. In order to parametrize thedominant weight are given byof, we first enumerate all the positive
representation of the discrete seriesroot systems compatible to. There are four such positive root systems:$m_{1},$ $m_{2}$$\mathfrak{h}^{*}=Hom(\mathfrak{h}, \mathbb{C})$$\{(\lambda_{1}, \lambda_{2})\in Z^{\oplus 2}|\lambda_{1}\geqq\lambda_{2}\}$$Sp(2;\mathbb{R})$$\Sigma_{c}^{+}$:(I)$\Sigma_{I}^{+}=\{(1, -1), (2,0), (1,1), (0,2)\}$;;(II) :$\Sigma_{II}^{+}=\{(1, -1), (1,1), (2,0), (0, -2)\}$(III) :$\Sigma_{III}^{+}=\{(1, -1), (2,0), (0, -2), (-1, -1)\}$(IV) :$\Sigma_{IV}^{+}=\{(1, -1), (-2,0), (-1, -1), (0, -2)\}$;.

be a variable running over the set of indices {I, II, III, IV}. Then we write
for the set of non-compact positive roots for each index .
Define a subsetof dominant weights by

Let

$J$

$J$

$\Sigma_{J,n}^{+}=\Sigma_{J}^{+}-\Sigma_{c}^{+}$$\Xi_{J}$$\Xi_{J}=\{\Lambda=(\Lambda_{1}, \Lambda_{2})$Then the set

dominant w.r. .
$t$.

$\Sigma_{c}^{+}|\{\Lambda, \beta\rangle>0, \forall\beta\in\Sigma_{J,n}^{+}\}$

gives the Harish-Chandra parametrization of the representationof the discrete series for. Let
be the associated representation of for$\bigcup_{J=I}^{IV}\Xi_{J}$$Sp(2,\cdot \mathbb{R})$$\Lambda\in\bigcup_{J=I}^{IV}\Xi_{J}$. The K-types of$\pi_{\Lambda}|_{K}$$G$$\pi_{\Lambda}$

is described by the formula of Blattner proved

finally by Hecht-Schmid. Among others the minimal K-type ofis given by
Aand. Hereare the half of the sum of compact positiveroots and non-compact positive roots. Here is a table of.
$[$$\lambda_{\min}=$$\rho_{c}+\rho_{n}$$]$$p_{c}$$\pi_{\Lambda}$$\rho_{n}$$\lambda_{\min}$$\frac{typeJ|I,II,III,IV}{\lambda_{n\iota in}|(A_{1}+1,\Lambda_{2}+2),(\Lambda_{1}+1,\Lambda_{2}),(\Lambda_{1},\Lambda_{2}-1),(\Lambda_{1}-2,\Lambda_{2}-1)}$

159
INTEGRAL REPRESENTATION OF WHITTAKER FUNCTION\S 3 Representations of the maximal compact subgroup.

For our later computation, we recall some basic facts about the representation of. Since $K$ is identified
the maximal compact subgroup $K$ or its complexification
is isomorphic towith the unitary group of degree 2 $U(2),$. Recall agiven in Section 1:
basis of
$K_{\mathbb{C}}$$GL(2, \mathbb{C})$$K_{\mathbb{C}}$$u(2)_{\mathbb{C}}$$Z=(\begin{array}{ll}l 00 l\end{array})H’=(\begin{array}{ll}1 00 -l\end{array})X=(\begin{array}{ll}0 l0 0\end{array})\overline{X}=(\begin{array}{ll}0 01 0\end{array})$

The irreducible finite-climensional representations of the Lie algebraparametrized by a set{A$=(\lambda_{1},$ $\lambda_{2})\in Z^{\oplus 2}|\lambda_{1}\geqq\lambda_{2}$, i.e.

$\lambda$.

$g((2, \mathbb{C})$areis dominant}.

For each dominant weight , we set. Then the dimension of therepresentation spaceassociated to A is $d+1$ . We can choose a basis $\{v_{k}|0\leqq$$k\leqq d\}$ insuch that the associated representation
is given by$\lambda$$d=\lambda_{1}-\lambda_{2}\geqq 0$$V_{\lambda}$$V_{\lambda}$$\tau_{\lambda}$$\{\begin{array}{l}\tau_{\lambda}(Z)v_{k}=(\lambda_{1}+\lambda_{2})v_{k}.\cdot\tau_{\lambda}(H’)v_{k}=(2k-d)v_{k}\cdot\tau_{\lambda}(X)v_{k}=(k+l)v_{k+1}\cdot\tau_{\lambda}(\overline{X})v_{k}=(d+l-k)v_{k-1}=\{d-(k-1)\}v_{k-1}\end{array}$Since $H_{1}’= \frac{1}{2}(Z+H’)$ and $H_{2}’= \frac{1}{2}(Z-H’)$ , we have$\tau_{\lambda}(H_{1}’)v_{k}=(k+\lambda_{2})v_{k}$and$\tau_{\lambda}(H_{2}’)v_{k}=(-k+\lambda_{1})v_{k}$.

If it is necessary to refer explicitly to the dominant weight , we denote.
by
For the adjoint representation of $K$ on $P+$ we have an isomorphism $P+\cong V_{(2,0)}$ ,and the correspondence of the basis is given by$\lambda$$v_{k}$$(X_{(0,2)}, X_{(1,1)}, X_{(2,0)})(v_{0}, v_{1}, v_{2})$$v_{\lambda,k}$.

Similarly for $P-$ , we have $P-\cong V_{(0,-2)}$ , and the identification of the basis is$(X_{(-2,0)}, X_{(-1,-1)}, X_{(0,-2)})=(v_{0}, -v_{1}, v_{2})$Let us consider a tensor productirreducible factors:$V_{\lambda}\otimes P+\cdot$.

Then it has a decomposition into

$V_{\lambda}\otimes P+\cong V_{\lambda}\otimes V_{(2,0)}=V_{(\lambda_{1}+2,\lambda_{2})}\oplus V_{(\lambda_{1}+1,\lambda_{2}+1)}\oplus V_{(\lambda_{1},\lambda_{2}+2)}$

Let

, and
, and

$P^{(2,0)},$ $P^{(1,1)}$$V_{(\lambda_{1}+1,\lambda_{2}+1)}$$0,1,2)$ .

be the projectors fromto the factors,, respectively. We denote $v_{(2,0),k}(k=0,1,2)$ by $w_{k}(k=$$P^{(0,2)}$$V_{(\lambda_{1},\lambda_{2}+2)}$.

$V_{\lambda}\otimes p_{+}$$V_{(\lambda_{1}+2,\lambda_{2})}$

160
TAKAYUKI ODA

Lemma (3.1). Se $t\mu=(\lambda_{1}+2, \lambda_{2})$ . Then up to scalars, the projector
given by(i)$P^{(2,0)}(v_{\lambda,k} \otimes w_{2})=\frac{(k+1)\cdot(k+2)}{2}v_{\mu,k+2;}$(ii)$P^{(2,())}(v_{\lambda,k}\otimes w_{1})=(k+1)(d+1-k)\iota)\mu,k+1$(iii)$P^{(2,0)}(v_{\lambda,k} \otimes w_{0})=\frac{(d+1-k)(d+2-k)}{2}v_{\mu,k}$$P^{(2,0)}$;.

Lemma (3.2). Se $t\iota/=(/\backslash 1+1, \lambda_{2}+1)$ . Then up to scalars, the projector
give by$is$is$P^{(1,1)}$

$n$

(0)

$P^{(1,1)}(v_{\lambda,d}\otimes w_{2})=0$(i)$P^{(1,1)}(v_{\lambda,k}\otimes w_{2})=(k+1)v_{\nu,k+1}$$(0\leqq k\leqq d-1)$(ii)$P^{(1,1)}(v_{\lambda,k}\otimes w_{1})=(d-2k)v_{\nu,k}$$(0\leqq k\leqq d)$

;

(iii)

$P^{(1,1)}(v_{\lambda,k}\otimes w_{0})=-(d+1-k)v_{\nu,k-1}$$(1 \leqq k\leqq d)$.

Lemma (3.3). Set $\pi=(/\backslash , \lambda+2)$ . Then up to scalars, the projector$giv$ en by;$P^{(0,2)}$(i)$P^{(0,2)}(v_{\lambda,k}\otimes w_{2})=v_{\pi,k}$$(0\leqq k\leqq d-2)$

;

(ii)

$P^{(0,2)}(v_{\lambda,k}\otimes w_{1})=-2\cdot v_{\pi,k-1}$$(1 \leqq k\leqq d-1)$

;

(iii)

$P^{(0,2)}(u_{\lambda,k}\otimes w_{0})=v_{\pi,k-2}$$(2\leqq k\leqq d)$(iv)$P^{(02)}\rangle$is;$(v_{d}\otimes w_{2})=P^{(0,2)}(v_{d}\otimes w_{1})=P^{(0,2)}(v_{d-1}\otimes w_{2})=0$.

Proofs of the above lemmas are easy. It is enough to find the highest weightvectors in, and, respectively. Other
corresponding to the factors
steps of proofs are settled by induction.
$V_{\lambda}\otimes p_{+}$$V_{\mu},$$V_{\nu}$$V_{\pi}$161
INTEGRAL REPRESENTATION OF WHITTAKER FUNCTION\S 4 Characters of the unipotent radical.
Put $N=\exp(\mathfrak{n})$ . Then $N$ is written as$(\begin{array}{llll}l_{2} n_{1} n_{2} n_{2} l_{2} n_{3}\end{array})|n_{0},$ $n_{1},$ $n_{2},$ $n_{3}\in \mathbb{R}\}$$N=\{(\begin{array}{llll}1 \gamma\gamma_{0} 0 l 1 0 -n_{0} 1\end{array})$.

The commutator group $[N, N]$ of $N$ is given by$[N, N]=\{(\begin{array}{llll}1_{2} n_{1} n_{2} n_{2} l_{2} 0\end{array})|n_{1},$Hence a unitary character$\eta$$(\begin{array}{llll}1 ??o 0 1 l 0 -n_{0} l\end{array})$.

$(\begin{array}{llll}1_{2} n_{1} n_{2} n_{2} l_{2} n_{3}\end{array})\exp\{2\pi i(c_{0}n_{0}+c_{3}n_{3})\}$$c_{3}\in \mathbb{R}$$\eta$$\eta$

Since$\in \mathbb{R}\}$of $N$ is written as.
for some real numbers
We denote by the same letter , the derivative of$c_{0},$$n_{2},$$[\mathfrak{n}, \mathfrak{n}]=\mathbb{R}E_{(e_{1}-e_{2})}\oplus \mathbb{R}E_{(2e_{2}),\eta}$:$\mathfrak{n}arrow \mathbb{C}$$\eta$.

is determined by the purely imaginary numbers$\eta_{0}=\eta(E_{e_{1}-e_{2}})$and$\eta_{3}=\eta(E_{2e_{2}})$.

162
TAKAYUKI ODA\S 5 Characterization of the minimal K-type.

Let :the space$N=\exp(\mathfrak{n})arrow \mathbb{C}^{*}$$\eta$$C_{\eta}^{\infty}(N\backslash G)=${$\phi$:

be a unitary character. Then we denote by$Garrow \mathbb{C},$$C^{\infty}$$C_{\eta}^{\infty}(N\backslash G)$-function $|\phi(ng)=\eta(n)\phi(g),$ $(n,$ $g)\in N\cross G$ }.

has structures of smooth G-module,
By the right regular action of $G,$
and-module.
For any finite-dimensional K-module $(\tau, V)$ , we put$C_{\eta}^{\infty}(N\backslash G)$$(g_{\mathbb{C}}, K)$$C_{\eta,\tau}^{\infty}(N\backslash G/K)${ $F;Garrow V,$$=$$C^{\infty}$-function $|F(ngk^{-1})=\eta(n)\tau(k)F(g),$ $(n,$$g,$$k)\in N\cross G\cross K$}.

Let
be the representation of discrete series with Harish-Chandra parameter, and letbe its contragradient representation.
Assume that there exists a continuous homomorphism $W$ :ofof smooth G-modules. Then the restriction of $W$ to the minimal K-typesuch thatgives an element$(\pi_{\Lambda}, E_{\Lambda})$$\Lambda$$(\pi_{\Lambda}^{*}, E_{\Lambda}^{*})$$(\pi_{\Lambda}^{*}, E_{\Lambda}^{*})arrow C_{\eta}^{\infty}(N\backslash G)1$$\pi_{\Lambda}^{*}$$\tau_{\lambda}^{*}$$F_{W}\in C_{\eta)}^{\infty_{\tau_{\lambda}}}(N\backslash G/K)$$W(v^{*})=\{v^{*}, F_{W}(\cdot)\}$for all$v^{*}\in V_{\lambda}^{*}$.

There is a characterization of the minimal K-type function $F$ by means of a.
differential operator acting onLetdecompositionof , andthe adjoint
be a Cartan$K$
representation of on . Then we have a canonical covariant differential operator
from
to
:
$c_{\eta}\infty_{\tau_{\lambda}}(N\backslash G/K)$$g=\mathfrak{k}\oplus p$$g$$Ad=Ad_{P\mathfrak{c}}$$p_{\mathbb{C}}$$c_{\eta}\infty_{\tau_{\lambda}}(N\backslash G/K)$$\nabla_{\lambda,\eta}$$C_{\eta}^{\infty_{\tau_{\lambda}\otimes Ad}}(N\backslash G/K)$$\nabla_{\eta,\lambda}F=\sum_{i}L_{X}.F(\cdot)\otimes X_{i},$$F\in c_{\eta}\infty_{\tau_{\lambda}}(N\backslash G/K)$

,

where
is any fixed orthonormal basis of with respect to the Killing form of$L_{X_{i}}F(g)=(\frac{d}{dt}F(g\cdot\exp(tX_{i})))|_{t=0}(g\in G)$ .
, and$(X_{i})_{i}$

$p$

$g$

Let
be the sum of irreducible K-submodules ofweights of the form
being a non-compact root insurjective K-homomorphism fromto
. We define
of
with
:
$(\tau_{\lambda^{-}}, V_{\lambda^{-}})$$\lambda-\beta,$$\beta$$V_{\lambda}\otimes p_{\mathbb{C}}$$V_{\lambda^{-}}$

with highesta. Denote byas the composite$V_{\lambda}\otimes p_{\mathbb{C}}$$\Sigma^{+}$$P_{\lambda}$$D_{\eta,\lambda}$$P_{\lambda}$$\nabla_{\eta,\lambda}$$\mathcal{D}_{\eta,\lambda}$:$c_{\eta,\tau_{\lambda}}^{\infty}(N\backslash G/K)arrow C_{\eta,\tau_{\lambda^{-}}}^{\infty}(N\backslash G/K)$,$D_{\eta,\lambda}F=P_{\lambda}(\nabla_{\eta,\lambda}F(\cdot))$$(F\in C_{\eta,\tau_{\lambda}}^{\infty}(N\backslash G/K))$.

We have the following

Proposition (5. 1). (Proposition (2.1) of Yamashita [Y-I])Let
be a repre sentation of discre series with Harish-Chan dra param eterof
. Set
. Th en the linear $map$$\cdot$$te$$\pi_{\Lambda}$$Sp(2, \mathbb{R})$$\lambda=\Lambda-p_{c}+\rho_{n}$$W\in Hom_{(g_{C)}K)}(\pi_{\Lambda}^{*}, C_{\eta}^{\prime\infty}(N\backslash G))arrow F_{W}\in Ker(\mathcal{D}_{\eta,\lambda})$$\Lambda$

163
INTEGRAL REPRESENTATION OF WHITTAKER FUNCTION

is injective.
Letbe the continuous homomorphisms of smooth Gmodules, then we have a canonical injection.
$Hom_{G}^{\infty}(\pi_{\Lambda}^{*}, C_{\eta}^{\infty}(N\backslash G))$

$Hom_{G}^{\infty}(\pi_{\Lambda)}^{*}C_{\eta}^{\infty}(N\backslash G))arrow Hom_{(gc,l\backslash ’)}(\pi_{\Lambda}^{*}, C_{\eta}^{\infty}(N\backslash G))$

By the results of Kostant

$]$

$[$

.

, we have$\dim_{\mathbb{C}}Hom_{G}^{\infty}(\pi_{\Lambda}^{*}, C_{\eta}^{\infty}(N\backslash G))+\dim_{\mathbb{C}}Hom_{G}^{\infty}(\pi_{\Lambda}, C_{\eta}^{\infty}(N\backslash G))\leqq 1$

,

and

or

$\dim_{\mathbb{C}}Hom_{(g_{\mathbb{C}},K)}(\pi_{\Lambda}^{*}, C_{\eta}^{\infty}(N\backslash G))+\dim_{\mathbb{C}}Hom_{(g_{\mathbb{C}},K)}(\pi_{\Lambda}, C_{\eta}^{\infty}(N\backslash G))=0$

$|W|$

.

, hence 8.
Here $|W|$ is the order of the Weyl group ofSince holomorphic discrete series and antiholomorphic discrete series are not, if, we havelarge in the sense of Vogan$Sp(2, \mathbb{R})$

$[$

$]$

$\pi_{\Lambda}\in\Xi_{I}^{\cup}\Xi_{VI}$$\dim_{\mathbb{C}}Hom_{G}^{\infty}(\pi_{\Lambda}^{*}, C_{\eta}^{\infty}(N\backslash G))=\dim_{\mathbb{C}}Hom_{(g_{\mathbb{C}},K)}(\pi_{\Lambda}^{*}, C_{\eta}^{\infty}(N\backslash G))=0$

In subsequent sections, we show that if$\Lambda\in\Xi_{II}\cup\Xi_{III}$,,$\dim Hom_{(gc,K)}(\pi_{\Lambda}^{*}, C_{\eta}^{\infty}(N\backslash G))=\dim_{\mathbb{C}}Ker(\mathcal{D}_{\eta,\lambda})=\frac{1}{2}|W|=4$

and accordingly$\dim_{\mathbb{C}}Hom_{G}^{\infty}(\pi_{\Lambda}^{*}, C_{\eta}^{\infty}(N\backslash G))+\dim_{\mathbb{C}}Hom_{G}^{\infty}(\pi_{\Lambda}, C_{\eta}^{\infty}(N\backslash G))=1$

by a theorem of Kostant

$[$

$]$

.

.

164
TAKAYUKI ODA\S 6 Radial part of differential operators.
Put $A=\exp(a)$ , i.e.

$A=\{(\begin{array}{llll}a_{1} a_{2} a_{1}^{-1} a_{2}^{-1}\end{array})|a_{1},$$a_{2}\in \mathbb{R},$$a_{1}>0,$ $a_{2}>0\}$.

: $G=NAK$ . The value ofThen we have the Iwasawa decomposition ofis determined by its restriction $\phi=F|_{A}$ to .
, respectively.
and
of
and
We compute the radial parts
As an orthogonal basis of , we take$Sp(2;\mathbb{R})$$A$$F\in c_{\eta}\infty_{\tau_{\lambda}}(N\backslash G/K)$$R(\nabla_{\eta,\lambda})$$R(\mathcal{D}_{\eta,\lambda})$$\nabla_{\eta,\lambda}$$\mathcal{D}_{\eta,\lambda}$$p$$C\Vert\beta\Vert(X_{\beta}+X_{-\beta})$,$(\beta\in\Sigma^{+})$$\frac{C||\beta\Vert}{\sqrt{-1}}(X_{\beta}-X_{-\beta})$with some $C>0$ depending on the Killing form. Then
$2 \nabla_{\eta,\lambda}F=C\sum_{\beta\in\Sigma_{n}^{+}}\Vert\beta\Vert^{2}L_{X_{-\beta}}F\otimes X_{\beta}+C\sum_{\beta\in\Sigma_{n}^{+}}\Vert\beta\Vert^{2}L_{X_{\beta}}F\otimes X_{-\beta}$.

We write$\{\nabla_{\eta,\lambda}^{+}F=\nabla_{\eta,\lambda}^{-}F=\frac{1}{\frac{1}{4}4}\sum_{\Sigma}|_{1}|_{\beta}^{\beta}\Vert_{2}^{2}.\cdot L_{X^{-\beta}}L^{X_{\beta}}F^{F}\otimes^{\otimes_{X_{-\beta}^{X_{\beta};}}}$.

In order to write, it is better to introduce some “macro” symbols. We set. restricted to A. $(i=1,2)$ , and define linear differential operators
and
on
by
$R(\nabla_{\eta,\lambda}^{\pm})$$\partial_{i}=L_{H_{1}}$$S^{\pm}$$\mathcal{L}_{i}^{\pm}$$C^{\infty}(A, V_{\lambda})$$\{\begin{array}{l}\mathcal{L}_{i}^{\pm}\phi=(\partial_{i}\pm 2\sqrt{-l}a_{i}^{2}\eta(E_{2e_{\mathfrak{i}}}))\phi(i=l,2).\cdot S^{\pm}\phi=\{a_{1}a_{2}^{-1}\eta(E_{e_{1}-e_{2}})\pm\sqrt{-1}a_{1}a_{2}\eta(E_{e_{1}+e_{2}})\}\phi\end{array}$

Proposition (6.1). The opera tors
expressed as(i)$R(\nabla_{\eta,\lambda}^{\pm})=C^{\infty}(A, V_{\lambda})arrow C^{\infty}(A, V_{\lambda}\otimes p_{\pm})$$R(\nabla_{\eta,\lambda}^{+})\phi=(\mathcal{L}_{1}^{-}+\tau_{\lambda}\otimes Ad_{P+}(H_{1}’)-4)(\phi\otimes X_{(2,0)})$$+(S^{-}+\tau_{\lambda}\otimes Ad_{P+}(X))(\phi\otimes X_{(1,1)})$$+(\mathcal{L}_{2}^{-}+\tau_{\lambda}\otimes Ad_{P+}(H_{2}’)-2)(\phi\otimes X_{(0,2)})$(ii)$R(\nabla_{\eta)\lambda}^{-})\phi=(\mathcal{L}_{1}^{+}-\tau_{\lambda}\otimes Ad_{P-}(H_{1}’)-4)(\phi\otimes X_{(-2,0)})$$+(S^{+}-T_{\lambda^{\otimes Ad_{P-}(\overline{X}))(\phi\otimes X_{(-1,-1)})}}$$+(\mathcal{L}_{2}^{+}-T_{\lambda^{\otimes Ad_{P-}(H_{2}’)-2)(\phi\otimes X_{(0,-2)})}}$

Proof).

In order to prove (i), we have to note that$(L_{X_{-(2,0)}}F)_{1_{A}}\otimes X_{(2,0)}=\{-H_{1}’+H_{1}-2\sqrt{-1}E_{2e_{1}}\}F_{1_{A}}\otimes X_{(2,0)}$$=\{\mathcal{L}_{1}^{-}\phi+(\tau_{\lambda}(H_{1}’)\cdot F)_{1_{A}}\}\otimes X_{(2,0)}$

and

,

are

165
INTEG RAL REPRESENTATION OF $WHITTAI\langle ER$ FUNCTION$(\tau_{\lambda}(H_{1}’)\cdot F)_{1_{A}}\otimes X_{(2,0)}=\tau_{\lambda}\otimes Ad_{P+}(H_{1}’)(\phi\otimes X_{(2,0)})-\phi\otimes[H_{1}’, X_{(2,0)}]$$=\tau_{\lambda}\otimes Ad_{P+}(H_{1}’)(\phi\otimes X_{(2,0)})-2(\phi\otimes X_{(2,0)})$The case of (ii) is similar.

.

(q.e. )
$d.$

, let
in
For a non-compact positive root
be the projector $fro\ln$.
, andthe projector from
[I],
Then, similarly as Lemma (5.2) of Yamashita
we can show the following.
$\beta=(\beta_{1}, \beta_{2})$$P^{-\beta}$$V_{\lambda}\otimes P+toV_{\lambda+\beta}$$\Sigma^{+}$$P^{\beta}$$V_{\lambda}\otimes P-toV_{\lambda-\beta}$

Lemma (6.2). Let be th $e \min$ imal K-type of a $dis$ crete eries representation$wi$ th Harish-Chan dra
a rameter A.
$\lambda$

$s$

$p$

(i)

When

$\Lambda\in\Xi_{II},$$R(D_{\eta,\lambda})\phi=0$

if and on $ly$ if$\{\begin{array}{l}P^{(0,2)}(R(\nabla_{\eta,\lambda}^{+})\phi)=0\cdot P^{(-1,-1)}(R(\nabla_{\eta,\lambda}^{-})\phi)=0\cdot P^{(-2,0)}(R(\nabla_{\eta,\lambda}^{-})\phi)=0\end{array}$

(ii)

When

$\Lambda\in\Xi_{III},$$R(D_{\eta,\lambda})\phi=0$

if an only if$\{\begin{array}{l}P^{(11)})(R(\nabla_{\eta,\lambda}^{+})\phi)=0\cdot P^{(0,2)}(R(\nabla_{\eta,\lambda}^{+})\phi)=0\cdot P^{(-2,0)}(R(\nabla_{\eta,\lambda}^{-})\phi)=0\end{array}$$d$$\pi_{\Lambda}$

166
ODA

$TAICAYUI\{I$\S 7 Difference-differential equations.

In this section, we write the system of differential equations in the last lemmaof the previous section explicitely in terms of components of .
Let A
be the minimal K-type of the discrete series representation, we choose a basis $\{v_{k}|0\leq k\leq d\}$ defined in Section 3. Here. Then in. Then :is written as$\phi$$=(\lambda_{1}, \lambda_{2})$$V_{\lambda}$$\pi_{\Lambda}$$d=/\backslash _{1}-\lambda_{2}$$\phi$$Aarrow V_{\lambda}$$\phi(a)=\sum_{k=0}^{d}c_{k}(a)v_{k}$with coefficients$c_{k}(a)$:$Aarrow \mathbb{C}$.

Lemma (7.1).
The condition$(?)$$(C_{2}^{+})_{k}$$P^{(1,1)}(R(\nabla_{\eta,\lambda}^{+})\phi)=0$

is equival $ent$ to the system:: $k(\mathcal{L}_{1}^{-}+\lambda_{2}+d-k-1)c_{k-1}(a)+(d-2k)S^{-}c_{k}(a)$$+(k-d)(\mathcal{L}_{2}^{-}+\lambda_{1}-k-1)c_{k+1}(a)=0$(ii)

The onditi on$c$$(C_{2^{-}})_{k}$is equivalen to the $system$ :$P^{(-1,-1)}(R(\nabla_{\eta,\lambda}^{+})\phi)=0$: $(k-d)(\mathcal{L}_{1}^{+}-\lambda_{2}+k-d-1)c_{k+1}(a)+(2k-d)S^{+}c_{k}(a)$(iii) The ondition$c$$P^{(0,2)}(R(\nabla_{\eta,\lambda}^{+})\phi)=0$$(0\leq k\leq d)$.

is equivalen to the $system$ :$t$: $(\mathcal{L}_{1}^{-}+/\backslash _{2}-k-2)c_{k}(a)-2S^{-}c_{k+1}(a)$$(0\leq k\leq d-2)$ .

$+(\mathcal{L}_{2}^{-}+\lambda_{1}-k-2)c_{k+2}(a)=0$(iv) The condition$(C_{3}^{\prime-})_{k}$.

$t$$+k(\mathcal{L}_{2}^{+}-\lambda_{1}+k-1)c_{k-1}(a)=0$$(C_{3}^{+})_{k}$$(0\leq k\leq d)$$P^{(0,-2)}(R(\nabla_{\eta,\lambda}^{-})\phi)=0$

is equivalent to the system:: $(\mathcal{L}_{I}^{+}-/\backslash 2-2d+k)c_{k+2}(a)+2S^{+}c_{k+1}(a)$$(0\leq k\leq d-2)$ .

$+(\mathcal{L}_{2}^{+}-\lambda_{I}+k)c_{k}(a)=0$Here we in terprete, in the above formul $ae,$ $c_{-1}(a)=c_{d+1}(a)=0$ .
Proof) The proof is a direct computation and easy. We omit it.
Since is trivial on the commutator group $[N, N]$ , we have$\eta$$\{\begin{array}{l}\mathcal{L}_{1}^{+}=\mathcal{L}_{1}^{-}=\partial_{1}=a_{1}\frac{\partial}{\partial a_{1}}S^{+}=S^{-}=\frac{a}{a}\perp_{2}\eta(E_{(1,-1)})\end{array}$Thus we have the following
Proposition (7.2). Under the arne assumption as in Lemma (6.2) (i), $\phi(a)=$satisfies the follo wing system of arti differen ti equations:$s$$\sum_{k=0}^{d}c_{k}(a)v_{k}$$(C_{3}^{+})_{k}$

$p$

$al$

: $(\mathcal{L}_{1}+\lambda_{2}-k-2)c_{k}(a)-2S\cdot c_{k+1}(a)$$+(\mathcal{L}_{2}^{-}+\lambda_{1}-k-2)c_{k+2}(a)=0$$(C_{3}^{-})_{k}$$al$$(0\leq k\leq d-2)$: $(\mathcal{L}_{I}-\lambda_{2}-2d+k)c_{k+2}(a)+2S\cdot c_{k+1}(a)$$+(\mathcal{L}_{2}^{+}-\lambda_{1}+k)c_{k}(a)=0$$(C_{2^{-}})_{k+I}$$(0\leq k\leq d-2)$: $(k+1-cl)(\mathcal{L}_{1}-/\backslash _{2}-d+k)c_{t+2}(a)+(2k+2-d)S\cdot c_{k+1}(a)$$+(k+1)(\mathcal{L}_{2}^{+}-/\backslash _{1}+k)c_{k}(a)=0$$(-1\leq k\leq d-1)$$(q.e.d.)$

167
INTEGRAL REPRESENTATION OF WHITTAKER FUNCTION\S 8 Reduction of the system of partial differential equations.

In this section we reduce the system of partial differential equations of the previous proposition to a simpler holonomic system, when is generic.
is determined by the coefficient
In the first place, we see that the functionsof the highest weight vector.
In fact, when $k=0$ , or $k=d$$\eta$$c_{k}(a)$$c_{d}(a)$$(C_{2^{-}})_{0}$:$(\mathcal{L}_{1}-\lambda_{2}-d-1)c_{1}(a)+Sc_{0}(a)=0,\cdot$$(C_{2^{-}})_{d}$:$Sc_{d}(a)+(\mathcal{L}_{2}^{+}-\lambda_{1}+d-1)c_{d-1}(a)=0$.

Moreover for $1\leqq k\leqq d-1$ , the computation of $(k+1)(C_{3}^{-})_{k}-(C_{2}^{-})_{k+1}$ yields$(\mathcal{L}_{1}-\lambda_{2}-d-1)c_{k+2}(a)+Sc_{k+1}(a)=0$

Noting$(E)_{k}$$\lambda_{2}+d=\lambda_{1}$:together with$(C_{2^{-}})_{0}$.

, we have$(-1\leqq k\leqq d-1)$$(\mathcal{L}_{1}-\lambda_{1}-1)c_{k+2}(a)+Sc_{k+1}(a)=0$.

$\ldots c_{d-1}(a)$ are determined downward recursively by.
Hence$(E)_{k}$The system of the equations
andare now replaced by the above$c_{0}(a),$ $c_{1}(a),$$c_{d}(a)$$(C_{2^{-}})$$(C_{2^{-}})_{d-1}$:$Sc_{d}(a)+(\mathcal{L}_{2}^{+}-\lambda_{1}+d-1)c_{d-1}(a)=0$.

Thus the system of the equations of Proposition (7.2) in Section 7 is equivalent toa system of equations:(F-1) :
(F-2) :

(F-3) :
(F-4) :$(\mathcal{L}_{1}-\lambda_{1}-1)c_{d}(a)+Sc_{d-1}(a)=0,\cdot$$(\mathcal{L}_{1}-\lambda_{1}-1)c_{d-1}(a)+Sc_{d-2}(a)=0$;$Sc_{d}(a)+(\mathcal{L}_{2}^{+}-\lambda_{1}+d-1)c_{d-1}(a)=0$;$(\mathcal{L}_{1}+\lambda_{2}-d)c_{d-2}(a)-2Sc_{d-1}(a)+(\mathcal{L}_{2}^{-}+\lambda_{1}-d)c_{d}(a)=0$.

In order to make the above equations simpler, we replace unknown functions
by $h_{k}(a)$ defined by relations$c_{k}(a)$$c_{k}(a)=a_{1}^{\lambda_{1}+1-d}a_{2}^{\lambda_{1}}( \frac{a_{1}}{a_{2}})^{k}e^{-i\eta(E_{(0,2)})a_{2}^{2}}h_{k}(a)$Now we introduce Euler operators $\partial_{i}(i=1,2)$ by(F-4) is replaced by
Then the system of equations (F-1)$\partial_{i}=a_{i^{\frac{\partial}{\partial a:}}}$$\sim$;(G-1) :$\partial_{1}h_{d}(a)+\eta(E_{e_{1}-e_{2}})h_{d-1}(a)=0$(G-2) :$(\partial_{1}-1)h_{d-1}(a)+\eta(E_{e}, -e_{2})h_{d-2}(a)=0$(G-3):$S( \frac{a_{1}}{a_{2}})h_{d}(a)+\partial_{2}h_{d-1}(a)=0$(G-4):$( \partial_{1}+2\lambda_{2}-1)h_{d-2}(a)-2S(\frac{a_{1}}{a_{2}})h_{d-1}(a)$;;$+( \frac{a_{1}}{a_{2}})^{2}(\partial_{2}+2\backslash 1-2d-2S’)h_{d}(c\iota)=0$.

for each $i=1,2$ .

168$TAI\{AYUI\{I$

ODA

.
Here
(G-1) and (G-3) are equivalent to a single equation:$S’= \frac{1}{2}(\mathcal{L}_{2}^{+}-\mathcal{L}_{2}^{-})=2\sqrt{-1}\eta(E_{2e_{2}})a_{2}^{2}$(H-1) :$(\partial_{1}\partial_{2}-S^{2})h_{d}(a)=0$.

(G-1), (G-2), and (G-4) is equivalent to a single equation:
$(*)$

:

$( \partial_{1}+2\lambda_{2}-1)(\partial_{1}-1)\partial_{1}\{(\frac{a_{1}}{c\iota_{2}})^{2}h_{d}(a)\}+2(\frac{a_{1}}{a_{2}})^{2}\partial_{1}h_{d}(a)$$+( \frac{a_{1}}{a_{2}})^{2}(\partial_{2}+2\lambda_{2}-2S’)h_{d}(a)=0$Here we used the assumption that$\eta$.

is generic, i.e.

$\eta(E_{e_{1}-e_{2}})=\eta_{0}\neq 0$

,

and

.

$\eta(E_{2e_{2}})=\eta_{3}\neq 0$

to the above equation $(*)$ , and use (H-1) to replace
Apply the operator
by $S^{2}h_{d}(a)$ . Then we have$\partial_{2}$$\partial_{1}\partial_{2}h_{d}(a)$(H-2) :$\{\partial_{1}^{2}+2\partial_{1}\partial_{2}+\partial_{2}^{2}+(2\lambda_{2}-2)(\partial_{1}+\partial_{2})+(-2\lambda_{2}+1)-2S’\partial_{2}\}h_{d}=0$

At last we have the following
Lemma (8.1). The system of equations of Proposition (7.2) is equivalen to$t$(H-1) :
and(H-2) :$(\partial_{1}\partial_{2}-S^{2})h_{d}(a)=0$$\{(\partial_{1}+\partial_{2})^{2}+(2\lambda_{2}-2)(\partial_{1}+\partial_{2})+(-2\lambda_{2}+1)-2S’\partial_{2}\}h_{d}(a)=0$

We can easily check that the system (H-1), (H-2) is a holonomic system of 1 ank.
4 defined over $(\mathbb{R}>0)^{2}=\{(a_{1}, a_{2})\in \mathbb{R}^{2}|a_{1}, a_{2}>0\}$ . Hence
with
The contragradient representationofis written as,some. Using the difference-differential equations
and.
we can similarly show thatoffor the minimal K-type
Since Kostant’s result implies (cf. \S 5)$\dim_{\mathbb{C}}Ker(D_{\eta,\lambda})=4$$\pi_{\Lambda}^{*}$$\pi_{\Lambda}(\Lambda\in\Xi_{II})$$\pi_{\Lambda}^{*}=\pi_{\Lambda’}$$(C_{2}^{+}),$$\Lambda’\in\Xi_{III}$$(C_{3}^{+})$$(C_{3^{-}})$$\lambda’$$\dim_{\mathbb{C}}Ker(D_{\eta,\lambda’})=4$$\pi_{\Lambda’}$$8=\dim Hom_{(g_{C},K)}(\pi_{\Lambda}^{*}, c_{\eta}^{\infty}(N\backslash G))+\dim Hom_{(g_{\mathbb{C}},K)}(\pi_{\Lambda}, c_{\eta}^{\infty}(N\backslash G))$,$\leqq\dim_{\mathbb{C}}Ker(D_{\eta)\lambda})+\dim_{\mathbb{C}}Ker(D_{\eta,\lambda’})=8$

we have the following

Proposition. Assume that$\eta$

is generic, $i.e$ .

$\eta_{0}=\eta(E_{e\iota-e_{2}})\neq 0$

Then for a discrete series representation

and$\eta_{3}=\eta(E_{2e_{2}})\neq 0$$\pi_{\Lambda}(\Lambda\in\Xi_{II}\cup\Xi_{III})$.

$\dim_{\mathbb{C}}Hom_{(g,K)}(\pi_{\Lambda}, c_{\eta}^{\infty}(N\backslash G))=4$

,

.

169
INTEGRAL REPRESENTATION OF WHITTAKER FUNCTION

When$\Lambda\in\Xi_{II}\cup\Xi_{III}$, we have$\dim_{\mathbb{C}}Hom_{G}^{\infty}(\pi_{\Lambda}^{*}, c_{\eta}^{\infty}(N\backslash G))+\dim_{\mathbb{C}}Hom_{G}^{\infty}(\pi_{\Lambda}, c_{\eta}^{\infty}(N\backslash G))=1$.

Hence two cases occur:$(A)$$Hom_{G}^{\infty}(\pi_{\Lambda}^{*}, c_{\eta}^{\infty}(N\backslash G))\cong \mathbb{C}$and,$Hom_{G}^{\infty}(\pi_{\Lambda}, c_{\eta}^{\infty}(N\backslash G))=\{0\}$

or
$(B)$

$Hom_{G}^{\infty}(\pi_{\Lambda}^{*}, c_{\eta}^{\infty}(N\backslash G))=\{0\}$

,

and

.

$Hom_{G}^{\infty}(\pi_{\Lambda}, c_{\eta}^{\infty}(N\backslash G))\cong \mathbb{C}$In the next section, we see that this dichotomy is controlled by the parity of
the imaginary part of the purely imaginary number $\eta_{3}=\eta(E_{2e_{2}})\neq 0$ . And whenis non-zero and generated by $W$ , we have an explicit inteof the intertwining
gral formula for the image
operator $W$ .
$Hom_{G}^{\infty}(\pi_{\Lambda}^{*}, c_{\eta}^{\infty}(N\backslash G))$

$F_{W}\in Ker(D_{\eta,\lambda})\subset c_{\eta,\tau_{\lambda}}^{\infty}(N\backslash G/K)$\S 9. Integral formula for the Whittaker function..

Let us recall the confluent hypergeometric equation given by Whittaker
Chap. 16):$\frac{d^{2}}{dz^{2}}W+\{-\frac{1}{4}+\frac{k}{z}+\frac{\frac{1}{4}-m^{2}}{z^{2}}\}W=0$

$($

$[]$

,

.

When ${\rm Re}(k- \frac{1}{2}-m)\leq 0$ , for $z\not\in(-\infty, 0)$ , a unique solution, which rapidly
decreases if $zarrow+\infty$ , is given by$W_{k,m}(z)= \frac{e^{-\frac{1}{2}z}\cdot z^{k}}{\Gamma(\frac{1}{2}-k+m)}\int_{0}^{\infty}t^{-k-\frac{1}{2}+m}(1+\frac{t}{z})^{k-\frac{1}{2}+m}\cdot e^{-t}dt$.

The following is the main result of this paper.

Theorem (9.1). Assume that$\eta_{3}=\eta(E_{2e_{2}})\neq 0$ .
,(i) For$\eta$:$Narrow \mathbb{C}^{*}$

is generic, $i.e$ .

$\eta_{0}=\eta(E_{e_{1}-e_{2}})\neq 0$

and$\Lambda\in\Xi_{II}$$\{\begin{array}{l}Hom_{G}^{\infty}(\pi_{\Lambda}^{*},c_{\eta}^{\infty}(N\backslash G))\cong \mathbb{C}Hom_{G}^{\infty}(\pi_{\Lambda}^{*},c_{\eta}^{\infty}(N\backslash G))=\{0\}\end{array}$$if{\rm Im}(\eta_{3})<0if{\rm Im}(\eta^{3})>0^{)}$.

(ii) Assume thatand
, and let $W$ be a un ique intert wining operator in
up to scalar $m$ ultiple. Then the fun ction
associated to $\phi(a)=F_{W|A}(a)=\sum_{i=0}^{d}c_{i}(a)v_{i}(F_{W}\in Ker(D_{\eta,\tau_{\lambda}}))$ has an int egral
represen a tion$\Lambda\in\Xi_{II}$${\rm Im}(\eta_{3})<0$$Hom_{G}^{\infty}(\pi_{\Lambda}^{*}, C_{\eta}^{\infty}(N\backslash G))$

$h_{d}(a_{1}, a_{2})$$t$$h_{d}(a_{1}, a_{2})= \int_{0}^{\infty}t^{\lambda_{2}-\frac{3}{2}}W_{0,-\lambda_{2}}(t)\exp(-\frac{t^{2}}{32\sqrt{-1}\cdot\eta_{3}a_{2}^{2}}+\frac{8\sqrt{-1}\eta_{0}^{2}\eta_{3}a_{1}^{2}}{t^{2}})\frac{dt}{t}$Proof). It is easy to check that the integral represents a solution of the differential equations $(H-1)$ and $(H-2)$ , by derivation of integrand and partial
integration.

170
TAKAYUKI ODA

Replace$t$by$a_{1}t$in the above integral expression of$h_{d}(a_{1}, a_{2})$, then$h_{d}(a_{1}, a_{2})= \int_{0}^{\infty}(\frac{a_{1}}{a_{2}}\cdot a_{2}\cdot t)^{\lambda_{2}-\frac{3}{2}}W_{0,-\lambda_{2}}(\frac{a_{1}}{a_{2}}\cdot a_{2}\cdot t)$$\cross\exp\{-\frac{1}{32\sqrt{-1}\eta_{3}}(\frac{a_{1}}{a_{2}})^{2}\cdot t^{2}+8\sqrt{-1}\eta_{0}^{2}\eta_{3}\cdot t^{-2}\}\frac{dt}{t}$.

. Also since A
is aand
If ${\rm Im}(\eta_{3})<0,$
negative integer. Hence the integrand is rapidly decreasing when $tarrow+\infty$ , and,when $tarrow 0$ . Therefore the above integral converges, and as a function in
. Put
and
it is rapidly decreasing when$- \frac{1}{32\sqrt{-1}\eta_{3}}<0$$8\sqrt{-1}\eta_{0}^{2}\eta_{3}<0$$\in\Xi_{II},$$\lambda_{2}$$(a_{1}, a_{2})$$a_{2}arrow\infty$$\frac{a}{a}\perp_{2}arrow\infty$$c_{d}(a)=a_{1}^{\lambda_{1}+1-d}a_{2}^{\lambda_{1}}$.

$( \frac{a_{1}}{a_{2}})^{d}\cdot e^{-i\eta_{3}a_{2}^{2}}$. $h_{d}(a)$ ,of \S 8.
andfor $0\leq k\leq d-1$ by the recurrence relation$c_{k}(a)(0\leq
k\leq
d)$
rapidlyarealso
decreasing
Thenfunctions inWrite $\phi(a)=\sum_{k=0}^{d}c_{k}(a)v_{k}\in C^{\infty}(A, V_{\lambda})$ . Then for any vectorof the dual$(\phi(a),v^{*})$, i.e.
spaceis also a rapidly decreasing function. A fortiori,$F(g)=\eta(n)\tau_{\lambda}(k)^{-1}\phi(a)$ is slowly increasing in $g=nak\in G$ .
This $F$ defines an
by Schmidt’s characterization $([])$ .
element $W$ in
Now Wallach’s version of multiplicity one (cf. [W], \S 8) implies that the operators$W$ insuch that $W(v)$ are slowly increasing on
for any, forms a linear subspace of dimension at most one.
, if
Hence
. If
, by a similarargument, we can show that
Since$(E)_{k}$$c_{k}(a)$$(^{a_{1}}/a_{2)}^{a_{2})}\cdot$$v^{*}$$\phi(a)$$V_{\lambda^{*}},$$Hom_{g,K}(\pi_{\Lambda}\cdot, C^{\infty}(N\backslash G))$

$G$

$Hom_{g)K}(\pi_{\Lambda}^{*}, C^{\infty}(N\backslash G))$$v\in\pi_{\Lambda}$$Hom_{G}^{\infty}(\pi_{\Lambda}^{*}, C_{\eta}^{\infty}(N\backslash G))\neq\{0\}$${\rm Im}(\eta_{3})<0$${\rm Im}(\eta_{3})>0$$Hom_{G}^{\infty}(\pi_{\Lambda}, c_{\eta}^{\infty}(N\backslash G))\neq\{0\}$$\dim_{\mathbb{C}}Hom_{G}^{\infty}(\pi_{\Lambda}^{*}, C_{\eta}^{\infty}(N\backslash G))+\dim_{\mathbb{C}}Hom_{G}^{\infty}(\pi_{\Lambda}, C_{\eta}^{\infty}(N\backslash G))=1$

if is generic, this proves (i). The part (ii) is immediately follows from the uniqueness of the Whittaker model.
$\eta$

Remark. For general cases, the condition of (i) is described in terms of wave
front set by Matsumoto $([]$ , \S .
When $G=SU(2,2)$ , we have a similar integral expression of the Whittalcer
function of the highest weight vector of the minimal K-type of a discrete series
representation. Details are discussed elsewhere about this case.
$)$

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[H-S]

[J-L]
[K]
[M]

Hecht, H. and Schmid, W., A proof of Blattner’s conjecture, Inventiones math. 31 (1975),

129-154.
Jacquet, H. and Langlands, R. P., Automorphic forms on $GL(2)$ , Lecture Notes in Math.,
No. 114, Springer.
Kostant, B., On Whittaker Vectors and Representation Theory, Inventiones Math. 48(1978), 101-184.
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$C^{-\infty}$$rc^{J}c/uct\uparrow ve$

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$1\langle yoto$

RIMS, $Ie_{YOTO}$ UNIV., KYOTO, 606, JAPAN