数理解析研究所講究録
1002 巻 1997 年 199-212

199

The generalized Whittaker functions for $SU(2,1)$
Yoshi-hiro Ishikawa
Introduction.
In the theory of automorphic forms, Fourier expansion of modular forms is a fundamental tool for investigation. For example, coefficients of the expansion can be used forconstruction of -functions. In spite of this importance, the theory of Fourier expansionof automorphic forms seems still in very primitive state.
Our concern is to have a theory of fully developed Fourier expansion of modular forms
on $SU(2,1)$ , the real special unitary group of signature $(2+, 1-)$ . To have such theory weneed Whittaker functions and generalized Whittaker functions of the standard representations of $SU(2,1)$ . A quite explicit result is obtained by Koseki-Oda [K-O] for Whittaker
functions. The remaining problem for our purpose is to consider the generalized Whittaker functions. This is the theme of the present paper.
The peculiarity of the case of $SU(2,1)$ , different from the case of, is that the
maximal unipotent subgroup $N$ is not abelian. It is isomorphic to the Heisenberg groupof dimension three, and has infinite dimensional irreducible unitary representations ,which are called Stone von Neumann representations. Together with unitary charactersthey constitute the unitary dual of $N$ . The Fourier expansion of automorphic forms on$SU(2,1)$ is to consider irreducible decomposition of the restrictionof automorphic
representations with respect to $N$ . Therefore we have to handle those terms whichcorresponds to the Stone von Neumann representations.
Naive formulation of the problem is to investigate intertwiners inwhich
is isomorphic to) by Robenius reciprocity. But this fails in general, because the intertwining space in question is infinite dimensional. The right formulation of
the problem is given by introducing a larger group containing $N$ .
Here is the formulation of our main result. Let be the minimal parabolic subgroup$SU(2,1)$ofwith Levi decomposition. And let be the maximal closed subgroupof which acts trivially on the center $Z(N)$ of $N$ . The group is the semidirect productand $N$ . We want to investigate the intertwining spacefor certain irreducible unitary representation of , and the images of intertwiners: these are the spaceof generalized Whittaker functionals and the space of generalized Whittaker functions,
respectively. Our main results are to obtain an explicit formula for the radial part of suchgeneralized Whittaker functions with special $K$-type, and to show the multiplicity one.
theorem for the intertwining space (Theorem 7.2.1, Theorem 8.2.1).
Our main concern is the theory of automorphic forms. However the author believes
that our results is also interesting for the problem of realization in generalized Gel’fandGraev representations.
$L$$SL_{2}(\mathrm{R})$$\sigma$$\pi|_{N}$$\pi$$\mathrm{H}\mathrm{o}\mathrm{m}_{N}(\pi|_{N}, \sigma)$$\mathrm{H}\mathrm{o}\mathrm{m}_{c(\sigma}\pi,$$\mathrm{I}\mathrm{n}\mathrm{d}_{N}c$

$R$

$P$

$L\ltimes N$

$L$

$S$

$R$

$S$

$\mathrm{H}\mathrm{o}\mathrm{m}_{c}(\pi, \mathrm{I}\mathrm{n}\mathrm{d}_{R}c\eta)$$\eta$

$R$

200

1

The structure of Lie groups and algebras

1.1

The Iwasawa decomposition

We realize the identity component of the stabilizer group $SU(2,1)$ of the Hermitian formof three variables with signature $(2+, 1-)$ as follows$sU(2,1):=\{g\in SL(3, \mathbb{C})|{}^{t}\overline{g}I_{2,1g}=I_{2,1}\}$

where$I_{2,1}:=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(1,1, -1)$,

. We denote the group by . Let$G$$G=NAK$be the Iwasawa decomposition of , then$G$

$K\cong S(U(2)\cross U(1))$,$N\cong H(\mathrm{R}^{2})$,$A=\{a, :=|r\in \mathrm{R}_{>0}\}$ .

Heredenotes the Heisenberg group of dimension 3.
Denote the Lie algebra The Lie algebraof by$H(\mathrm{R}^{2})$$B\mathrm{U}(2,1)$

$G$

$g$

and let$\mathfrak{g}=\mathrm{f}\oplus \mathfrak{p}$be the Cartan decomposition corresponding to the Cartan involution :.
Since $G/K$ is Hermitian, we have a decomposition Pc
that is identified
with the holomorphic tangent space at the origin 1 $\cdot K\in G/K$ , corresponding to thecomplex structure of $G/K$ . Put$\theta$$=\mathfrak{p}_{+}\oplus \mathfrak{p}_{-\mathrm{S}\mathrm{u}}\mathrm{C}\mathrm{h}$,$a=\mathrm{R}\cdot H$$\mathfrak{n}=\mathrm{R}E_{1}\oplus \mathrm{R}E2,+\oplus \mathrm{R}E_{2,-}$

,

where
$H$

$:=$

,

$E_{1}:=\dot{i}$ ,$E_{2,+}:=,$ $E_{2,-:=}$ ,then$9=\mathfrak{n}\oplus a\oplus \mathrm{f}$gives the Ivasawa decomposition of .
$\mathfrak{g}$$X\vdash*I_{2,1}xI_{2,1}^{-1}$$\mathfrak{p}_{+}$

201
1.2

Root system

We fix a compact Cartan subalgebra in$\mathrm{t}$$\mathrm{g}$and its basis by$\mathrm{t}=\{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\sqrt{-1}h_{1}, \sqrt{-1}h2, \sqrt{-1}h_{3})|h_{i}\in \mathbb{R}, h_{1}+h_{2}+h_{3}=0\}$$H_{12}’=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(1$Define linear forms

on$\beta_{ij}$. $-1,0)$ ,$H_{13}’=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(1.0, -1)$$\mathrm{t}_{\mathrm{C}}(i\neq j, 1\leq i,j\leq 3)$$\beta_{ij}$

,

.

by

.

:

$\mathrm{t}_{\mathbb{C}}\ni \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(t_{1}, t_{2}, t_{3})rightarrow t_{i}-t_{j}\in \mathbb{C}$is given by $\Sigma:=\{\beta_{ij}|i\neq j, 1\leq i,j\leq 3\}$ .
Then the root system associated to
We fix a positive root system $\Sigma_{+}:=\{\beta_{ij}|i<j\}$ .
be the root space associated to $\beta\in\Sigma:\mathfrak{g}_{\beta}:=\{X\in \mathfrak{g}|[H, X]=\beta(H)x,$Let
the sets of compact and noncompact roots, respectively. In ourand
We denotechoice of coordinate,$\Sigma$$(9\mathrm{c}’ \mathrm{t}_{\mathrm{C}})$$\forall H\in$$\mathfrak{g}_{\beta}$$\Sigma_{n}$$\Sigma_{c}$$\mathrm{t}_{\mathbb{C}}\}$$\Sigma_{\mathrm{c}}:=\{\beta\in\Sigma|\mathfrak{g}_{\beta \mathbb{C}}\subset t\}$,$\Sigma_{n}:=\{\beta\in\Sigma|\mathfrak{g}_{\beta}\subset \mathfrak{p}_{\mathrm{c}}\}$and matrix element $E_{ij}(1\leq i,j\leq 3)$ generates the root space$E_{ij}$$X_{\beta_{j}}\dot{.}=\{$$-E$;$\mathfrak{g}_{\beta_{j}}.\cdot$.

. We put;when $(i,j)=(2,1)$ ,$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}(i,j)\neq(2,1)$, because this is natural in the meaning that complex
and take it as a root vector inconverts two root vectors
conjugation with respect to our choice of real form ofandmutually. Put
and$\mathfrak{g}_{\beta_{ij}}$$z1(3, \mathbb{C})$$X_{\beta_{j}}.\cdot$$\Sigma_{n,+}:=\Sigma_{n}\cap\Sigma+\cdot$$\Sigma_{\mathrm{c},+}:=\Sigma_{\mathrm{c}}\cap\Sigma_{+}$$X_{\beta_{ji}}$2

Representation theory of the group

2.1

Representations of

$R$

$R$

with nontrivial central charactersof $N$ is exhausted by uniBy the Stone von Neumann theorem, the unitary dualtary characters and irreducible unitary representations which is determined by its central
withcharacter , up to unitary equivalence. Consider an infinite dimensional one$N$ . Then$P$ ,$Z(N)$
ofcenterthecentral character . Let be the Levi subgroup of andof in is the centralstabilizer
the$Z(N)$
Hence
conjugation.
byacts both on andizer of $Z(N)$ . In particular is independent of . We define the group $R$ by semidirect
product as$\overline{N}$$\sigma\in\overline{N}$$\psi$$\psi$

$L$

$L$

$S$

$\overline{N}$$S$$\sigma$$L$$\sigma$$R=S,\langle N$.

We extend to an irreducible unitary representation of .
can be regarded as a
Because the action of on $N$ by conjugation is faith.
$N$subgroup $N^{ab}=$abelianizedthesubgroup of the automorphism group of . Passing to,
is identified with
$N/[N : N]$ of $N$ , we have $S\sim*t$ Aut. Since. Composing all these identifications, we get an isomorphism between
Aut
and $SO(2)$$R$$\sigma$$S$$\mathrm{f}\mathrm{u}\mathrm{l},$$Narrow \mathrm{A}\mathrm{u}\mathrm{t}N^{ab}$$S$$N^{ab}$$N^{ab}\cong cL_{2}(\mathrm{R})$

$S$

$S$

diag

$(\alpha, \beta,\overline{\alpha}-1)$$\mapsto$$\mathrm{A}\mathrm{u}\mathrm{t}N$$arrow$$rightarrow$$GL_{2}(\mathrm{R})$$\supset$$SO(2)$ ,$R(3\theta)$$\mathrm{R}^{\oplus 2}$

202
by putting$\beta=e^{-2*\theta},$ $\theta\in[0,2\pi)$$\alpha=e^{i\theta},$

We realize$\sigma$on$L^{2}(\mathrm{R})$, wheremeans the rotation of angle$R(3\theta)$$3\theta$.

as$\rho\psi((x, y;t)).\Phi(\xi)=\psi(t+\langle x+2\xi, y\rangle)\cdot\Phi(\xi+X)$,

where $(x, y;t)\in H(\mathrm{R}),$andis the natural symplectic form on
called themodel of .
By the theory of Weil representations, we have the canonical extension$\Phi\in L^{2}(\mathbb{R})$$Sh_{\Gamma\ddot{\mathit{0}}}dinger$$\langle , \rangle$$\mathrm{R}^{2}$. This is$\sigma$:$\omega_{\psi}\cross\rho_{\psi}$.

$\overline{Sp}_{1}(\mathrm{R})\ltimes H(\mathbb{R}^{2})arrow \mathrm{A}\mathrm{u}\mathrm{t}(L^{2}(\mathrm{R}))$Here
is the two-fold covering of
its Weil representation. Identi$N$
$S,$
fying
with
respectively, the semidirect product $R=S\ltimes N$ is regarded
as a subgroup of. Let be the pullbackofby the covering$\overline{Sp}_{1}(\mathrm{R})$$SL_{2}(\mathrm{R}),$$S\underline{O}(2),$$(\omega_{\psi}, L2(\mathrm{R}))$$H(\mathrm{R}^{2})$$\tilde{R}$$Sp_{1}(\mathrm{R})\ltimes H(\mathrm{R}^{2})$$\tilde{R}:=\tilde{S}\ltimes N\cong\overline{SO}(2)\ltimes H(\mathbb{R}^{2})$$R$$pr\cross id$:$\overline{Sp}_{1}(\mathrm{R})\ltimes H(\mathrm{R}^{2})arrow SL_{2}(\mathbb{R})\kappa H(\mathrm{R}^{2})$Then tensoring an odd characterof$\overline{\chi}$of$\overline{So}(2)$to$(\omega_{\psi}\cross\rho_{\psi})|$.

-finally, we have a representation$R$:$\overline{\chi}\otimes(\omega_{\psi}\cross\rho_{\psi})|_{\tilde{R}}$$R=S\ltimes Narrow$$\mathrm{A}\mathrm{u}\mathrm{t}(L^{2}(\mathbb{R}))$.

We denote this representation. A character ofis called odd, if it does
not factors through the covering $SO(2)arrow SO(2)$ , which is usually parameterized by someelements $\mu=m+\frac{1}{2}$ in.
$\mathrm{b}\mathrm{y}\perp\eta,$$\overline{SO}(2)$$L^{2}(\mathbb{R}))$$\frac{1}{2}\mathbb{Z}\backslash \mathbb{Z}(m\in \mathbb{Z})$Here is a diagram explaining the above construction$\tilde{R}=\tilde{S}\ltimes N$$R=S\ltimes N\downarrow$

A basis of$\eta$$\mathrm{A}\mathrm{u}\mathrm{t}(L^{2}(\mathrm{R}))$$SL_{2}(\mathbb{R})\ltimes H(p\mathrm{r}\mathrm{X}ia\downarrow \mathrm{R}2)$.

$arrow$

2.2$\frac{\omega_{\psi}\cross\rho_{4}}{r}$$\overline{Sp}_{1}(\mathbb{R})\ltimes H(\mathrm{R}^{2})$and the action of Lie $N$

It is well known that Hermite functions
an orthogonal Hilbert basis of.
We normalize tha acton of the generators$h_{n}(\xi):=(-1)^{n\epsilon^{2}}e/2$.

$\frac{d^{n}}{d\xi^{n}}e^{-\xi^{2}},$$n=1,2,3,$

form$\ldots$$L^{2}(\mathrm{R})$$E_{1},$$E_{2,\pm}0\mathrm{f}\mathfrak{n}$through$\rho_{\psi_{S}}$as$\rho_{\psi_{\epsilon}}(E_{1}).\Phi(\xi)=2\sqrt{-1}s\cdot\Phi(\xi)$$\rho_{\psi_{s}}(E_{2,+}).\Phi(\xi)=-\sqrt{2}\Phi’(\xi),$on$\Phi\in L^{2}(\mathrm{R})$, where$\psi_{s}$$\rho_{\psi_{S}}(E_{2,-}).\Phi(\xi)=-2\sqrt{2}\dot{i}s\xi\cdot\Phi(\xi)$

is an additive character of$\mathbb{R}\ni t\text{ト}\Rightarrow e\in UiSt(1)$. Then easily seen,

Proposition 2.2.1 The subspace of smooth vectors inis the Schwartz space,
and the action of root vectors
onthrough the underlining HarishChandra module ofare as follows:$L^{2}(\mathrm{R})$$E_{1},$$E_{2,+},$$E_{2,-}$$S(\mathrm{R})$$S(\mathrm{R})$$\eta_{\mu,\psi_{s}}=\tilde{\chi}_{\mu}\otimes\eta_{\psi s}\sim$$\eta(E_{1}).hn=2\sqrt{-1}s\cdot hn$$\eta(E_{2,+}).h_{nn+1}=\frac{1}{\sqrt{2}}h-\sqrt{2}n\cdot hn-1,$ $\eta(E_{2,-}).hn-\sqrt{2}\dot{i}s\cdot hn+1-2=\sqrt{2}isn\cdot h_{n}-1$.
$\square$203

3

Representations of maximal compact subgroup

3.1

Parameterization of irreducible K-modules

The setof
For each-action as$L_{T}^{+}$$\Sigma_{c,+}$-dominant -integral weights is given by, the vector space
spaned by$T$$L_{T}^{+}=\{(m, n)\in \mathbb{Z}^{\oplus 2}|m\geq n\}$$\mu=(\mu_{1}, \mu_{2})\in L_{T}^{+}$$V_{\mu}$.

with$\{v_{k}^{\mu}|0\leq k\leq d_{\mu}\}$$\mathrm{g}_{\mathrm{C}}$$\tau_{\mu}(Z)v_{k}^{\mu}=$$\tau_{\mu}(H_{12}’)v^{\mu}k$$(\mu_{1}+\mu_{2})v_{k}^{\mu}$$=$$\{\mu-(d_{\mu}-k)\beta 12\}(H_{1}\prime 2)v_{k}^{\mu}=(2k-d\mu)v_{k}^{\mu}$$\tau_{\mu}(H_{1}’3)v_{k}^{\mu}=$,$\tau_{\mu}(x_{\beta 2})1v_{k}\mu$$=$$(k+1)vk+1\mu$$\tau_{\mu}(x_{\beta 1})2v_{k}\mu$$=$$(k-d_{\mu}-1)vk-1\mu$$(\tau_{\mu}, V_{\mu})$Tensor products with,,$\{\mu-(d_{\mu}-k)\beta_{12}\}(H_{1}’3)v_{k}^{\mu}=(k+\mu_{2})v_{k}^{\mu}$gives an irreducible $K$-module

3.2

,

.

via the highest weight theory.

$\mathfrak{p}_{\mathrm{c}}$We regard the -dimensional vector space as a -module via the adjoint representation. Thenandare invariant subspaces, and$\mathrm{g}_{\mathrm{C}}$$\mathfrak{p}_{\mathrm{c}}$$\mathrm{a}\mathrm{d}$$\mathfrak{p}_{-}$$\mathfrak{p}_{+}$$\mathfrak{p}_{+}=\mathbb{C}X_{\beta_{1}}\oplus sX\mathbb{C}\beta 23\cong V_{\beta_{13}}$,Given an irreducible $K$-module we have
Gordan’s theorem tells us the following decomposition of$V_{\mu}$$V_{\mu}\otimes \mathfrak{p}_{\mathrm{C}}=(V_{\mu}\otimes \mathfrak{p}_{+})\oplus(V_{\mu}\otimes \mathfrak{p}_{-})$$V_{\mu}\otimes \mathfrak{p}_{+}\cong V_{\mu+\beta_{13^{\oplus}}\beta}V_{\mu}+2s$

Here we understand$V_{\nu}=(\mathrm{O})$if$\nu\in L_{T}$.

$\mathfrak{p}_{-}=\mathbb{C}X_{\beta_{32^{\oplus \mathbb{C}}}}X\rho s1\cong V_{\beta_{32}}$$V_{\mu}\otimes \mathfrak{p}_{\pm}$:, and Clebsch-$V_{\mu}\otimes \mathfrak{p}_{-}\cong V_{\mu+\beta s2^{\oplus}}V_{\mu\beta_{31}}+$’

is not dominant. We hence have

$V_{\mu}\otimes \mathfrak{p}_{\mathrm{C}}\cong V_{\mu}^{+-}\oplus V_{\mu};$,$V_{\mu}^{+}:=V_{\mu+\beta_{1}}\oplus 3V_{\mu+}\beta_{32}$under the above convention.
The decompositions of$V_{\mu}\otimes \mathfrak{p}_{\mathrm{c}}$In terms of$V_{\mu}^{-}:=V_{\mu-\beta_{1}}\oplus 3V_{\mu-}\beta_{32}$induce the following projectors:$p_{\beta_{13}}^{+}(\mu):V_{\mu}\otimes \mathfrak{p}\mathrm{c}arrow V_{\mu+\beta_{13}}$,$p_{\beta_{2}}^{-}\mathrm{s}(\mu):V\mu^{\otimes}\mathfrak{p}\mathrm{c}arrow V_{\mu+\beta_{32}}$,$\{v_{k}^{\mu}\}$,$p_{\beta_{23}}^{+}(\mu):V\mu^{\otimes}\mathfrak{p}\mathrm{c}arrow V_{\mu-\beta_{32}}$,$p_{\beta_{13}}^{-}(\mu):V\otimes\mu \mathfrak{p}\mathrm{c}arrow V_{\mu-\beta_{13}}$,, they are expressed as follows:

Proposition 3.2.1 ([K-O] Prop 2-3)$p_{\beta_{13}}^{+}(\mu)(v^{\mu}k\otimes X\beta 13)=(k+1)v_{k}^{\mu\beta 3}+1+1$,$p_{\beta_{23}}^{-}(\mu)(v_{k^{\otimes}\beta_{3}2}x\mu)=-(k+1)v_{k}^{\mu+\beta 32}+1$$p_{\beta_{23}}^{+}(\mu)(v_{k}^{\mu}\otimes X\beta 13)=-v_{k^{-}}^{\mu\beta 32}$

$p_{\beta_{13}}^{-}(\mu)(v^{\mu}k\otimes^{x}\beta 32)=v_{k^{-\rho 1}}\mu 3$

for $k=1,$$d_{\lambda}$$\ldots,$$p_{\beta_{13}}^{+}(\mu)(v^{\mu}\otimes x_{\beta 2})k3=(d_{\mu}-k+1)v_{h^{+\beta 13}}\mu$’,$p_{\beta_{23}}^{-}(\mu)(v^{\mu}\otimes kX\beta 31)=(d-k+1)\mu v_{k^{+\beta 32}}^{\mu}$,,. Here one should note that$p_{\rho_{2}}^{+}\mathrm{s}(\mu)(v^{\mu}k\otimes X\beta 23)=vk-\mu-1\beta 32$$p_{\beta_{13}}^{-}(\mu)(v^{\mu}k^{\otimes}X\beta \mathrm{s}1)=v_{k}\mu-\rho-113$$d_{\mu\pm\beta_{13}}=d_{\mu\pm\rho_{32}}=d_{\mu}\pm 1$

.

,

,

,
$\square$204

4

Generalized Whittaker models

4.1

The space of the generalized Whittaker functionals

Let be a unitary representation of with representation space defined in section 2.
of from to with representation space-induced representation
We call the-function satisfying,
is a
:
$S$

$R$

$\eta$$C^{\infty}$$R$$\mathrm{I}\mathrm{n}\mathrm{d}_{R}^{G}\eta$$\eta$$C_{\eta}^{\infty}(R\backslash c):=\{f$$Garrow S^{\infty}$

$|$

$G$

$C^{\infty}$$f$$f(rg)=\eta(r).f(g),$ $\forall r\in R,\forall g\in G\}$

acts via right translation, the reduced generalized Gelfand-Graev represenmeaning the subspace consisting of alltation. Here we used standard notation bysmooth vectors in .
We can now define the space of the generalized Whittaker functionals as the space ofintertwining operators.

on which$G$$S^{\infty}$$S$of , we identify the
Definition For an irreducible admissible representation-module of with itself, and call the space of intertwiners
underlining$G$$(\pi, \mathcal{H}_{\pi})$

$\pi$

$\pi$

$(\mathfrak{g}_{\mathrm{C}}, K)$$I_{\pi,\eta}:=\mathrm{H}_{\mathrm{o}\mathrm{m}_{(\iota \mathrm{c}^{K)}}},(\pi^{*}, \mathrm{I}\mathrm{n}\mathrm{d}_{R}^{G}\eta)$of$(\mathfrak{g}_{\mathrm{C}}, K)$

4.2-modules the space of the algebraic generalized Whittaker functionals.

Generalized Whittaker functions with fixed K-type, we study the
In order to investigate algebraic generalized Whittaker functionals
by . To describe: the image of vectors belonging tofunctions$K$
tothese functions explicitly, we specify a -type of and consider vectors
K-type.
this, and pullback a:of , choose a $K$-equivalent injection
For a $K$-type
generalized Whittaker functional by this injection ,$l\in I_{\pi,\eta}$$v^{*}$$l(v^{*})\in \mathrm{I}\mathrm{n}\mathrm{d}_{R}^{G}\eta$$l$$(\pi^{\mathrm{s}}, \mathcal{H}_{\pi}^{*})$$v^{*}\mathrm{b}\mathrm{e}1_{\mathrm{o}\mathrm{n}}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{g}$$\pi$$(\tau, V_{\tau})$$\pi$$\tau\mapsto\pi$$\iota_{\tau}$$l$$\iota_{\tau}$$\mathrm{H}_{\mathrm{o}\mathrm{m}_{(\mathrm{w}}},)(\mathrm{c}^{K}\mathrm{n}\pi \mathrm{I}*,\mathrm{d}_{R}^{G}\eta)\ni l\vdasharrow\iota^{*}\tau(l)\in \mathrm{H}_{0}\mathrm{m}K(\mathcal{T}^{*}, \mathrm{I}\mathrm{n}\mathrm{d}^{c}\eta R|K)$.

Here we note the isomorphism$\mathrm{H}_{\mathrm{o}\mathrm{m}_{\kappa}}(\tau^{*}, \mathrm{I}\mathrm{n}\mathrm{d}_{R}c\eta|K)\cong(\mathrm{I}\mathrm{n}\mathrm{d}_{R}^{G}\eta|K\otimes\tau)^{K}$.

The latter space is$C_{\eta,\tau}^{\infty}(R\backslash c/K):=(\varphi:carrow S(\mathbb{R})\otimes_{\mathrm{C}}V\mathcal{T}|\varphi \mathrm{i}\mathrm{S}\mathrm{a}C,\infty-\varphi(rgk\forall r\in R)=\eta \mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}_{0}\mathrm{n}\mathrm{s}\forall g\in c,\forall(r)\mathcal{T}(k)-1.\varphi k\in K\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{S}6^{r\mathrm{i}}(g),\mathrm{n}\mathrm{g},$We study functions$F\in C_{\eta,\tau}^{\infty}(R\backslash G/K)$

representing$b_{\mathcal{T}}^{*}(l)$$l(v^{*})(g)=\langle v^{*}, F(\mathit{9})\rangle K$$v^{*}\in V_{\tau}^{*}$. Here ,$\langle$$\rangle_{K}$$\}\cdot$. By definition,’means the canonical pairing of -modules$K$$V_{\tau}^{*}$

and$V_{\tau}$.

the algebraic
Definition We call the above function corresponding to$K$generalized Whittaker function associated to representation with -type . Moreover if
we impose the slowly increasing condition for the -radial part of , such a function iscalled the generalized Whittaker function.
$F$$b_{\mathcal{T}}^{*}(l),$$l\in I_{\pi,\eta}$

$\pi$

$A$

$\tau$

$F$

205

5

Differential equations for generalized Whittaker functions of the discrete series

5.1

The Schmid operators

We denote the space of$V_{\lambda}$$C_{\tau_{\lambda}}^{\infty}(G/K):=\{\varphi$:-valued functions on$Garrow V_{\lambda}$$|$$\varphi$

is a

$G$

$C^{\infty}$

with the$\tau_{\lambda}$-equivalence by-function satisfying,$\varphi(rgk)=\mathcal{T}\lambda(k)-1.\varphi(g),\forall g\in G,\forall k\in K\}$

We can regard Pc as a $K$-module through the adjoint representation
operator:$\nabla_{\tau_{\lambda}}$$\mathrm{A}\mathrm{d}\mathfrak{p}_{\mathrm{c}}$.

. The differential$C_{\tau_{\lambda}}^{\infty}(G/K)arrow C_{\tau_{\lambda}\otimes \mathrm{d}}^{\infty}\mathrm{A}(G\mathfrak{p}\mathbb{C}/K)$,$\nabla_{\tau_{\lambda}}\varphi:=\sum_{i=1}^{4}RX_{i}\varphi\otimes X_{i}$is a $K$-homomorphism. Here $\{X_{i}(i=1\sim 4)\}$ is an orthonormal
is of with respect
to the Killing form on andmeans the right differential of function by.
The operator
is called the Schmid operator. We take as orthonormal
is of$\mathrm{b}\mathrm{a}s$$\mathfrak{p}$$R_{X}\varphi$$\mathfrak{g}$$\varphi$$X\in \mathfrak{g}$$\mathrm{b}\mathrm{a}s$$\nabla_{\tau_{\lambda}}$$\mathfrak{p}_{\mathrm{C}}$$C(X_{\beta}+X_{-\beta})$,$c\sqrt{-1}(X_{\beta}-x_{-\beta})$,where isorand is a positive constant depending on the normalization of the
fixed Killing form. Then, using this basis, the Schmid operator
can be written as$\beta$$\beta_{13}$$C$$\beta_{23}$$\nabla_{\tau_{\lambda}}$$\nabla_{\tau_{\lambda}}\varphi=2c^{2}\sum_{\beta=\beta 13\beta 23},RX\beta\varphi-\otimes X_{\beta}+2C^{2}\sum_{1\beta=\beta 3\beta 23},Rx_{\beta}\varphi\otimes x_{-^{\rho}}$.

Here we note thatis the set of root vectors corresponding to positive noncompact roots. The above description of
in two terms corresponds to the decompositionofNow we define two differential operators as following$\{X_{\beta 13}, X_{\beta 23}\}$$\nabla_{\tau_{\lambda}}$$\mathfrak{p}_{\mathrm{c}}=\mathfrak{p}_{+}\oplus \mathfrak{p}-\cdot$$\nabla_{\tau_{\lambda}}^{\pm}$:$C_{\tau_{\lambda}}^{\infty}(G/K)arrow C_{\tau_{\lambda^{\otimes \mathrm{A}\mathrm{d}_{\mathfrak{p}}}}\pm}^{\infty}(G/K)$$\nabla_{\tau_{\lambda}}^{+}\varphi:=R_{x_{-\beta_{13}}}\varphi\otimes x_{\beta\varphi\otimes}13+Rx-\beta_{2}3x_{\beta}23$’$\nabla_{\tau_{\lambda}}^{-}\varphi:=R_{x_{\beta_{1}}3}\varphi\otimes X-\beta_{1}3+RX_{\beta}\varphi\otimes 23X_{-}\beta 23^{\cdot}$

For later use, we prepare the.
and$\beta\in\Sigma_{n,+}$$\pm\beta$-shift operators for every positive noncompact root$\lambda\in L_{\tau}^{+}$$D_{\tau_{\lambda}}^{\pm\beta}$:$C_{\tau_{\lambda}}^{\infty}(c/K)arrow C_{\tau_{\lambda\pm\beta}}^{\infty}(G/K)$$D_{\tau_{\lambda}}^{\pm\beta}\varphi(g):=p_{\beta}^{\pm}(\nabla_{\mathcal{T}_{\lambda}}^{\pm_{\varphi(g))}}$

Here$p_{\beta}^{\pm}$

are the projectors$\tau_{\lambda}\otimes \mathrm{A}\mathrm{d}_{\mathfrak{p}\pm}arrow\tau_{\lambda\pm\beta}$

defined in subsection 3.2.

All the operators constructed above can be defined similarly for$\nabla_{\eta,\tau_{\lambda}}$$\nabla_{\eta,\tau_{\lambda}}^{\pm}$$D_{\eta,\tau_{\lambda}}^{\pm\beta}$:::$C_{\eta,\tau_{\lambda}}^{\infty}(R\backslash G/K)$,$C_{\eta,\tau_{\lambda}}^{\infty}(R\backslash G/K)$$arrow$$C_{\eta,\tau_{\lambda}}^{\infty}(R\backslash G/K)$$arrow$$C_{\eta,\tau_{\lambda^{\otimes \mathrm{A}\mathrm{d}}}\mathfrak{p}\pm}^{\infty}(R\backslash G/K)$$C_{\eta,\tau_{\lambda}}^{\infty}(R\backslash G/K)$$arrow$$C_{\eta,\tau_{\lambda\pm}}^{\infty}(\beta R\backslash G/K)$$C_{\eta,\mathrm{A}\mathrm{d}}^{\infty}\tau_{\lambda\otimes}\mathfrak{p}\mathrm{c}(R\backslash G/K)$

.

,

.

206

5.2

Yamashita’s characterization

Here is a variant of a result of Yamashita which characterize the space of the algebraic
minimal $K$-type generalized Whittaker functionals for discrete series representations. Thisis fundamental for our purpose. Let$\Sigma_{I}^{+}:=\{\beta_{12}, \beta_{13}, \beta_{23}\},$$\Sigma_{I\tau}^{+}:=\{\beta_{12}, \beta_{3}2, \beta 13\},$$\Sigma_{t\tau\tau}^{+}:=\{\beta_{1}2, \beta_{3}2, \beta_{31}\}$.

and $–j-$ be the set of Harish-Chandra parameters correspond to positive root systems$\Sigma_{J}^{+}(J=I, II, III)$ compatible with the positive compact root system. We underthe half-sum of the compact positive roots andof the noncompact ones in$\Sigma_{c,+}$$\rho_{n}^{J}$$\mathrm{s}\mathrm{t}\Sigma_{J}^{+}\mathrm{a}.\mathrm{n}\mathrm{d}\rho_{c}$Proposition 5.2.1 ([Ya] Theorem 2.4) Let
be a discrete series representation ofwith Harish-Chandra parameter A $\in--J-$ , Blattner parameter, andbe the representation constructed in section 2. Assume A is far from walls, then theimage ofby the correspondence of subsection 4.2 in)is characterized by$\pi_{\Lambda}$$G$$\lambda=\Lambda+\rho_{J}-2\rho_{c}$$\eta$$\mathrm{H}\mathrm{o}\mathrm{m}_{(\mathrm{c}}\mathrm{g},K$$(\pi\Lambda’ \mathrm{I}*\mathrm{n}\mathrm{d}_{R}G\eta)$

$(D)$

:

$C_{\eta,\tau_{\lambda}}^{\infty}(R\backslash G/K)$$D_{\eta,\tau\lambda}^{-\beta}.F=0$$(\forall\beta\in\Sigma_{J}^{+}\mathrm{n}\Sigma_{n})$.

In short$\mathrm{H}\mathrm{o}\mathrm{m}_{K}(\mathcal{T}\mathrm{I}*\mathrm{n}\mathrm{d}_{R}G\eta\lambda’)$$\cong\beta\in\Sigma_{J}^{+_{\cap\Sigma}}n\mathrm{n}KerD^{-\beta}\lambda\eta,\mathcal{T}^{\cdot}$.6

Difference-differential equations for coefficients

6.1

Radial part of Schmid operators

For the representation
constructed in section 4 and for any finite dimensional$K$ -module $W$ , wedenote the space of the smooth-valued functions on by$(\eta, L^{2}(\mathrm{R}))$$A$$S(\mathbb{R})\otimes_{\mathrm{C}}W$$C^{\infty}(A;S(\mathrm{R})\otimes_{\mathrm{C}}W):=$Let$\mathrm{r}\mathrm{e}\mathrm{s}_{A}$$\mathrm{r}\mathrm{e}\mathrm{s}_{A,\pm}$::{$\phi:Aarrow S(\mathrm{R})\otimes_{\mathrm{C}}W|C^{\infty}$$C^{\infty}(A;S(\mathrm{R})\otimes_{\mathrm{c}}V_{\lambda})$$arrow$$C_{\eta,\tau_{\lambda}}^{\infty}(R\backslash c/K)$-function}.
,$C_{\eta,\mathcal{T}_{\lambda^{\otimes}}}^{\infty}Ad_{\mathrm{p}}\pm(R\backslash G/K)arrow C^{\infty}(A;S(\mathrm{R})\otimes_{\mathrm{c}}V_{\lambda}\otimes_{\mathrm{c}}\mathfrak{p}_{\pm})$be the restriction maps tobyimage of$A$

. Then we define the radial part$R(\mathrm{v}_{\eta,\lambda}^{\pm})\mathcal{T}$of$\nabla_{\eta,\tau_{\lambda}}^{\pm}$on the$\mathrm{r}\mathrm{e}\mathrm{s}_{A}$.

$R(\nabla_{\eta,\tau_{\lambda}}^{\pm}).(\mathrm{r}\mathrm{e}\mathrm{s}_{A}\varphi)=\mathrm{r}\mathrm{e}\mathrm{s}_{A,\pm}(\nabla^{\pm}.\varphi\eta,\tau_{\lambda})$Let us denote by andof , respectively,$\phi$$a$$\partial$

the restriction to of. We remark$A$$\partial\phi=(H.\varphi)|_{A}$and the generator $H$: the Euler operator in variable .

$\varphi\in C_{\eta,\tau_{\lambda}}^{\infty}(R\backslash G/K)$$\partial=r\frac{d}{d\mathrm{r}}$Proposition 6.1.1 ([K-O] Prop 4-1) Let be the above element in
Then the radial partis given byof$\phi$$R(\nabla_{\eta,\tau}^{+}\lambda)$

(i)

$r$

$C^{\infty}(A;S(\mathbb{R})\otimes_{\mathrm{c}}V\lambda)$$\nabla_{\eta,\tau_{\lambda}}^{+}$$R(\nabla_{\eta,\tau}^{+}\lambda).\phi$$= \frac{1}{2}\{\partial-\sqrt{-1}r^{2}\eta(E_{1})-4\}.(\phi\otimes x_{\beta})+\frac{1}{2}(_{\mathcal{T}\otimes}\lambda Ad\mathfrak{p}+)(H’)1s\cdot(13\emptyset\otimes X\beta 13)$$- \frac{1}{2}r\{\eta(E2,+)-\sqrt{-1}\eta(E2,-)\}.(\phi\otimes x\beta_{23})+(_{\mathcal{T}}\lambda^{\otimes}Ad\mathfrak{p}_{+})(x_{\rho_{1}).(}\emptyset 2\otimes x\beta 23)$

.

.

207

Similarly for the radial part(ii)

$R(\nabla_{\eta_{\mathcal{T}}\lambda}^{-},)$

of$1$$\nabla_{\eta,\tau_{\lambda}}^{-},$

we have$R(\nabla_{\eta_{\mathcal{T}}\lambda}^{-},).\phi$$= \frac{1}{2}\{\partial+\sqrt{-1}^{2}r\eta(E1)-4\}.(\emptyset\otimes x_{\beta 1})3-\frac{1}{2}(\tau\lambda\otimes Ad\mathfrak{p}_{-})(Hr31).(\phi\otimes X_{\beta s})1$$- \frac{1}{2}r\{\eta(E_{2},+)+\sqrt{-1}\eta(E2,-)\}.(\phi\otimes X\beta 32)+(_{\mathcal{T}_{\lambda^{\otimes}}}Ad\mathfrak{p}-\beta_{3}2))(x_{\beta_{21}}).(\emptyset\otimes^{x}$.
$\square$

6.2

Compatibility of -type and K-type$S$

If we write$\phi=\varphi|_{A}\in C^{\infty}(A;s(\mathbb{R})\otimes_{\mathrm{C}}V_{\lambda})$as$\phi(a)--\sum_{n=1k=}^{\infty}\sum_{0}^{d}\lambda c_{n}k(a)(hn\otimes v^{\lambda}k)$in terms of
is
and
of
and
respectively, the$K$
compatibility of -action and -action implies theing of many coefficients.
Here is the precise statement.
Letbe the representation constructed in section 2 as a tensor product. Hereis an odd character ofparameterized by a half integer$\phi(mam^{-1}),$. Culculate$m\in S=M,$ $a\in A$ in two different ways, first, $\phi(mam^{-1})=$since $M=Z_{K}(A)$ , second, $\phi(mam^{-1})=\eta(m)\tau_{\lambda}(m^{-1}).\emptyset(a)$ since, then we have next linear relation between and .
$\mathrm{b}\mathrm{a}s$$\{h_{n}|n\in \mathrm{N}_{0}\}$$\{v_{k}^{\lambda}|k=0, \ldots, d_{\lambda}\}$$S(\mathrm{R})$$V_{\lambda}$

vaniS.h$S$$c_{nk}$$(\eta, S(\mathbb{R}))$$\tilde{\chi}_{\mu}\otimes$$(\omega_{\psi}\cross\rho_{\psi})|_{\overline{R}}$$\overline{S}\cong\overline{So}(2)$$\overline{\chi}_{\mu}$$\mu$$\phi(a)$$\phi=\varphi|_{A},$$C_{\eta,\tau_{\lambda}}^{\infty}(R\backslash G/K)$$n$

Lemma 6.2.1 (1) The image of$\mathrm{r}\mathrm{e}\mathrm{s}_{A}$$\ell_{\lambda}\in \mathbb{Z}$, then the$A$$C^{\infty}(A;S(\mathrm{R})\otimes_{\mathbb{C}}V_{\lambda})$and$\frac{-\lambda_{1}+2\lambda_{2}}{3}\in \mathbb{Z}$(2) Assume

in-radial part$\phi$$\varphi\in C_{\eta,\tau_{\lambda}}^{\infty}(R\backslash G/K)$$\phi(a_{f})=\sum_{k=0}^{d_{\lambda}}Ck(a_{\mathrm{f}})(hn\otimes v^{\lambda}k)$where$c_{k}(a_{f})’ S$

are$C^{\infty}$-functions on
$n=$

$A$

is zero unless$\frac{-\lambda_{1}+2\lambda_{2}}{3}\geq\frac{1}{2}+\mu$

of

and the index$n$$\varphi\in$$k$.

is written as,

is given by$-k+ \frac{2\lambda_{1}-\lambda_{2}}{3}-\frac{1}{2}-\mu$.
$\square$

6.3

Difference-differential equations

We first write down the -radial partcoefficient functions
of .
$A$

$c_{k}(a_{t})’ \mathrm{S}$$R(D_{\eta,\tau_{\lambda}}^{-\beta})$of the -shift operators$\beta$$D_{\eta,\tau_{\lambda}}^{-\beta}$$\phi$

1In [K-O], there is a misprint in the formula (ii). The symbol–after.

$\partial \mathrm{i}\mathrm{s}+\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{l}\mathrm{y}$in terms of208

Proposition 6.3.1 Let
part of$\phi$$\varphi\in C_{\eta,\tau_{\lambda}}^{\infty}(R\backslash c/K)$

be any function in. By using the lemma 6.2.1, we can$C^{\infty}(A;S(\mathrm{R})\otimes_{\mathrm{c}^{V_{\lambda}}})$$\phi(a_{r})=\sum_{k=0}^{d_{\lambda}}C_{k}(ar)(h_{n_{\mu,\lambda()k}}k\otimes v^{\lambda})$which is the A-radial$e\varphi reSs\phi$as,$\beta,theacti_{on}oftheA-rwherewedenote-k+\frac{2\lambda_{1}-\lambda_{2}}{adials}pa-\frac{1}{t2}-\mu,yrR(D^{\frac{b}{\eta}\beta})\tau_{\lambda}oftn_{\mu,\lambda}(k).Thenfo\Gamma ana\Gamma bihe\beta- shifloperatorisgiven\dot{i}ntermtrarynoncompa_{\mathit{0}}cStroo,tfck^{S}$

as follows:$R(D_{\eta,\tau}^{-\beta}) \lambda\phi(a_{f})=\sum_{k=0}^{d_{\lambda-\beta}}Ck(-\beta a_{\Gamma})(h_{n-(\mu,k})^{\otimes v_{k^{-\beta}}^{\lambda}}\lambda\beta)$urith$c_{k}^{-\beta_{23}}(a_{f})$$=$$\frac{1}{2}\{(d_{\lambda^{-}}k+1)(\partial+k-\lambda 2-2r^{2}s).ck(a_{t})+k\frac{1+2s}{\sqrt{2}}r\cdot ck-1(a_{t})\}$$c_{k}^{-\beta_{13}}(a_{r})$$=$$\frac{1}{2}\{(\partial+k-2d\lambda-\lambda_{2}-1-2rS)2.c_{k1}+(a_{f})-\frac{1+2_{S}}{\sqrt{2}}r\cdot C_{k}(a_{\Gamma})\}$$C_{k}^{-\rho_{32}}(a,)$$=$$\frac{-1}{2}\{(\partial-k+\lambda 2-2+2r^{2}s).\mathrm{c}k$

$=$

,

(at)

$-\sqrt{2}(1+2s)(n_{\mu,\lambda}(k)+1)r\cdot C_{k+1}(a_{\Gamma})\}$$.c_{k}^{-\beta_{31}}(a,)$,,$\frac{1}{2}\{k(\partial-k+\lambda_{2}+2d_{\lambda}+1+2r^{2}S).C_{k1}-(a_{\mathrm{f}})$$+(d_{\lambda^{-k}}+1)\sqrt{2}(1+2s)(n_{\mu,\lambda}(k)+1)r\cdot c_{k}(af)\}$.
$\square$Using the above proposition, we can write the differential equations $(D)$ in Proposition
5.2.1 in terms of the coefficient functions of the -radial part of the algebraic gen. As for the, which comes from
eralized Whittaker function, see [I] subsection 7.3.
system of the difference-differential equations satisfied by$c_{k}$$A$$\phi$$l\in I_{\pi_{\Lambda},\eta}$$F\in C_{\eta,\tau_{\lambda}}^{\infty}(R\backslash G/K)$$c_{k}’ \mathrm{s}$An explicit formula and the multiplicity one theo-

7

rem
7.1

An explicit formula for coefficients

Now we are in a position to formulate the generalized Whittaker functions with analytic
condition. Let us define the generalized Whittaker model for the representation ofwith $K$ -type as follow
$G$

$\pi$

$\tau$$Wh_{\eta}^{\tau}(\pi):=\{F\in C_{\eta,\tau}^{\infty}(R\backslash G/K)$$|$$F|_{A}(a_{r})$is of moderate growth when$l(v^{*})=\langle v^{*}, F(\cdot)\rangle_{K},$$l\in I_{\pi,\eta},$$rarrow\infty$$v^{*}\in V_{\tau}^{*}\}$,.

We call the elements in the space above the generalized Whittaker functions associated to
the representation with $K$ -type .
$\pi$$\tau$

209

which comes from
Proposition 7.1.1 Let $\phi=F|_{A}$ be a function in
, A
. And
are the coefficient functions of expanded
and , respectively. Then each $c_{k}(0\leq$andwith respect to the basisofsatisfies the following differential equation.
$C^{\infty}(A;S(\mathrm{R})\otimes_{\mathrm{C}}V_{\lambda})$$\in\Sigma_{II}$$l\in \mathrm{H}\mathrm{o}\mathrm{m}_{(\mathfrak{g}\mathrm{c}},K)(\pi_{\Lambda}, \mathrm{I}\mathrm{n}\mathrm{d}_{R}^{G}\eta)$$\{h_{n}\}$$c_{k}’ s$$\phi$$S(\mathrm{R})$$\{v_{k}^{\lambda}\}$$V_{\lambda}$$k\leq d_{\lambda}-1)$$\{\partial^{2}-(2d_{\lambda}+4)\partial+G_{k}(r)\}.ck(ar)=0$

where$G_{k}(r)=-4_{S^{24}}r-\{4(\lambda_{2^{-}}k+d_{\lambda^{-}}1)s+(n_{k}+1)(1+2s)2\}r^{2}-(k-2d_{\lambda}-\lambda 2^{-}2)(k-\lambda 2+2)$

Here we abbreviated$n_{\mu\lambda}(k)=-k+ \frac{2\lambda_{1}-\lambda_{2}}{3}-\mu-\frac{1}{2}$as$n_{k}$.

.

$\square$of the minimal
As a result, we obtain an explicit formula of the coefficient functions$K$ -type generalized Whittaker functions for large discrete series representations.
$c_{k}’ \mathrm{s}$Theorem 7.1.2 The coefficient functionsof the -radial part of the minimal K-type
representageneralized Whittaker functions
for the large discretetions
of $SU(2,1)$ are of the form
$A$$c_{k}’ s$$ser\dot{i}eS$$F’ s\in Wh_{\eta}^{\tau_{\lambda}}(\pi\Lambda)$$\pi_{\Lambda^{\mathrm{Z}}}s(\Lambda\in\Sigma_{II})$(const.)$c_{k}(a_{r})=$$\cross r^{d_{\lambda}+1}W_{\hslash,(}k-\lambda)/2(2|s|r^{2})$with parameters$\kappa=\{-(\lambda_{2}-k+d_{\lambda}-1)_{S}-(n_{k}+1)(2s+1)^{2}/4\}/2|s|$. Hereis the Blattner parameter ofthe classical Whittaker function.

$k=0,$$\lambda=(\lambda_{1}, \lambda_{2})$$d_{\lambda}$$\ldots,$$\pi_{\Lambda}$,. Function$W_{\kappa,m}(X)\square is$In the cases of the holomorphic and the antiholomorphic discrete series repr.esentations, the differential equations satisfied by the coefficient functions are of the first order.
Consequently the solutions are essentially exponential functions.

which comes from
Proposition 7.1.3 Let $\phi=F|A$ be a function in. And
are the coefficient functions of expanded with respect$c_{k}(0\leq k\leq d_{\lambda}+1)$
.
Then
each
and
to the basis
satisfies the following differential
equation$C^{\infty}(A;S(\mathbb{R})\otimes_{\mathrm{C}}V_{\lambda})$$c_{k}’ s$$l\in \mathrm{H}\mathrm{o}\mathrm{m}_{(K)}\mathfrak{g}_{\mathbb{C}},(\pi\Lambda, \mathrm{I}\mathrm{n}\mathrm{d}_{R}^{G}\eta)$$\{h_{n}\}$$\phi$$\{v_{k}^{\lambda}\}$$\{\partial+G_{k}(r)\}.C_{k}(a_{f})=0$

where$-2_{S}r^{2}-(\lambda_{2}+k)$$G_{k}(r)=\{$$2sr^{2}+(\lambda_{2}+k)$

when
when

$\Lambda\in\Sigma_{t}$;$\Lambda\in\Sigma_{III}$.
$\square$Hence we have an explicit formula of$c_{k}’ \mathrm{s}$.

Theorem 7.1.4 The coefficient functions
generalized Whittaker functions(A
discrete series representations$c_{k}’ s$of the -radial part of the minimal K-type
for the holomorphic (resp. antiholomorphic)
of $SU(2,1)$ are of the form$A$$F’ s\in Wh_{\eta}^{\tau_{\lambda}}(\pi_{\Lambda})$$\in\Sigma_{I}(resp.\Sigma t\Pi)$$\pi_{\Lambda}’ s$$c_{k}$$k=0,$$\ldots$,$d_{\lambda}$(at)$d_{\lambda}$$\ldots,$$\cross r^{\lambda_{2}+k}e^{s\Gamma^{2}}$,with $s<0$ , (resp.
$c_{k}(a_{r})=$$k=0,$(const.)$=$(const.)with $s>0$ ). Here variable$a_{f}$$\cross r^{-\lambda_{2}k-s\mathrm{r}}-e2$,

is an element of .
$A$$\square$210

Remark These explicit formulae of generalized Whittaker functions for the holomorphic
and the antiholomolphic discrete series representations are compatible with the classicaltheory of Fourier-Jacobi expansion of holomolphic modular forms on $SU(2,1)$ , or on theassociated symmetric domain $SU(2,1)/K$ . We just remark here that the conditions onparameter of the central character ofin Theorem 7.1.4 is the consequence of themoderate growth condition on.
$s$$\rho_{\psi_{S}}$$Wh_{\eta^{\lambda}}^{\mathcal{T}}(\pi_{\Lambda})$7.2

The multiplicity one theorem f-or the discrete series

Assemble the parts prepared in previous sections, then we obtain simultaneously the multiplicity one theorem and an explicit form of elements in the minimal $K$-type generalized
Whittaker modelfor the discrete series representations.
$Wh_{\eta}^{\tau_{\lambda}}(\pi)$$\pi’ \mathrm{s}\cdot \mathrm{o}\mathrm{f}SU(2,1)$

In order to formulate the multiplicity one theorem we have to introduce a
submoduleof.
$A_{\eta}(R\backslash G)$$A_{\eta}(R\backslash G):=\{f\in C_{\eta}^{\infty}(R\backslash G)|\mathrm{W}\mathrm{h}’ \mathrm{e}\mathrm{n}rarrow\infty,\forall C_{f,h}|_{A}c_{f}h\mathrm{i}\mathrm{S}\mathrm{r}\mathrm{i}(a_{r}\mathrm{g}\mathrm{h}\mathrm{t}K-\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}_{h(\eta,S}\mathrm{a}\mathrm{n}\mathrm{d})\mathrm{i}\mathrm{S}\mathrm{o}\mathrm{f}\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}0\mathrm{w}\mathrm{t}\mathrm{h}\in(\mathrm{R}))\}$

whereis a -valued function on defined as $c_{f,h}(g):=(f(g), h)_{\eta}$ and-radial part of. It is easy to see thatis a-submodule of$G$$\mathbb{C}$$c_{f,h}$$A$$(\mathfrak{g}_{\mathrm{C}}, K)-$

$C_{\eta}^{\infty}(R\backslash G)$$c_{f,h}|_{A}$$A_{\eta}(R\backslash G)$$c_{f,h}$,Theorem 7.2.1 The discrete series representation
parameter A $\in---and$ Blattner parameter$(\mathfrak{g}_{\mathrm{C}}, K)$$\pi_{\Lambda}$

of $SU(2,1)$ with Harish-Chandra$\lambda=(\lambda_{1}, \lambda_{2})\in L_{T}^{+}$

i.e.

is the.

$C_{\eta}^{\infty}(R\backslash G)$

has multiplicity one property$\dim_{\mathrm{C}}\mathrm{H}\mathrm{o}\mathrm{m}(\mathfrak{g}\mathrm{c},K)(\mathcal{H}_{\pi_{\mathrm{A}}’\eta\psi}^{*}A(\mu,\backslash RG))=1$

if and only if$\frac{2\lambda_{1}-\lambda_{2}}{3}\in \mathbb{Z}$,$\frac{1}{2}+\mu\leq\frac{-\lambda_{1}+2\lambda_{2}}{3}$.

Under this condition, the minimal $K$ -type generalized Whittaker model
has a basis
whose -radial part is given as follows.
$\in---II$
1) When A(i.e.
is a large discrete series representation),$Wh_{\eta^{\lambda}}^{\tau}(\pi_{\Lambda})$$A$$F_{\eta}^{\tau_{\lambda}}$$\pi_{\Lambda}$$F_{\eta}^{\tau_{\lambda}}(a_{\Gamma})= \sum_{k=0}^{d_{\lambda}}r^{d_{\lambda}}+1W_{\kappa},\frac{k-\lambda}{2}(2|_{S|}r)2$.

$(hn\lambda(k)^{\otimes}\mu v^{\lambda}k)$

,

where
.
is a holomorphic discrete series represemtation),$\kappa=\{-(\lambda_{2}-k+d\lambda^{-}1)_{S}-(n_{k}+1)(2s+1)^{2}/4\}/2|s|$

2) When$\Lambda\in--I-$(i.e.

$\pi_{\Lambda}$$F_{\eta}^{\tau_{\lambda}}(a_{r})=k= \sum_{0}^{d}\lambda r^{\lambda_{2}+k}e^{st^{2}}\cdot(h\otimes n_{\mu\lambda}(k)v^{\lambda}k)$where $s<0$ .
3) When A$\in---t\Pi$(i.e.

$\pi_{\Lambda}$,

is an antiholomorphic discrete series representation),$F^{\mathcal{T}_{\lambda}}( \eta a’)=\sum_{k=1}^{d_{\lambda}}r^{-}-e-sf^{2}(\lambda_{2}k.h_{n(\mu k)k}\lambda\otimes v^{\lambda})$,of$\pi_{\Lambda}$

211
where $s>0$ .
Here, and the index of each base$r\in \mathrm{R}_{>0}$$h_{n}$

of$\eta$

is.

$n_{\mu\lambda}(k)=-k+ \frac{2\lambda_{1}-\lambda_{2}}{3}-\frac{1}{2}-\mu$

口The case of principal series representations

8

The case of the principal series representation$e^{\nu}$:$\pi_{k,\nu}=\mathrm{I}\mathrm{n}\mathrm{d}_{P}c(1_{N}\otimes e^{\nu}\otimes\chi_{k})$)$A\ni a\vdash\Rightarrow e^{(\beta}$$\nu+(\log a)\in \mathbb{C}^{*}$, where,:,we can obtain an explicit formula for the corner $K$-type generalized Whittaker function bysolving the differential equation arose from the Casimir action. For details and notation,see [I] \S 9, [K-O] \S 7.
$\chi_{k}$8.1$M\ni \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(e, e^{-}i\theta 2i\theta, e)i\theta e\vdash*\in ik\theta U(1)$Radial part of the Casimir operator

Proposition 8.1.1 Let
Then the radial part$\phi\in C^{\infty}(A;s(\mathbb{R})\otimes \mathrm{c}V\mathcal{T})$$R(\Omega)$of$\Omega$

is given by

be the -radial part of$A$$\varphi\in C_{\eta,\tau}^{\infty}(R\backslash G/K)$.

$R( \Omega).\phi=\frac{1}{2}\{\partial^{2}-4\partial+\frac{1}{3}k^{2}-r^{4}\eta(E1)^{2}$$-2\sqrt{-1}r^{2}\eta(E1)\tau(H_{1s}’)+r^{2}(\eta(E2,+)^{2}+\eta(E_{2},-)^{2})$$+2r\eta(E2+)\mathcal{T}(X_{\beta}X12^{+}\beta 21)+2\sqrt{-1}r\eta(E2-)\tau(X_{\beta}x\beta 21)12^{-}\}\phi$

.
口

8.2

An explicit formula and the multiplicity one theorem

Theorem 8.2.1 The irreducible principal series representation
tiplicity one property i.e.

$\pi_{k,\nu}$of $SU(2,1)$ has mul-$\dim_{\mathrm{c}}\mathrm{H}_{0}\mathrm{m}_{(}\mathfrak{g}\mathrm{c},K)(\pi_{k,\nu}^{*}, A(\eta_{\mu},\psi\backslash RG))=1$

if and only if$\frac{k}{3}-\mu-\frac{1}{2}\in \mathbb{Z}0\geq$.

Under this condition, the corner $K$ -type generalized Whittaker model)a basiswhose -radial part is given by$F_{\eta}^{\tau_{(-}}k,-k$$A$$F_{\eta}^{\tau_{(-k}}’-k)(a_{f})=rW_{\kappa,\frac{\nu}{2}}(2|S|r^{2})\cdot(h_{n_{0}}\otimes v_{0})$where$\kappa=\{-ks-(2n_{0}+1)(4_{S^{2}}+1)/4\}/2|s|$Here$r\in \mathrm{R}_{>0}$, and the index of the base$h_{n_{0}}$of$\eta$

is$n_{0}= \frac{k}{3}-\mu-\frac{1}{2}$

.

.

,

$Wh_{\eta}^{\tau}(-k,-k)(\pi_{k,\nu})$

has

212

References[I]

Ishikawa, Y. The generalized Whittaker functions for the standard representationsof $SU(2,$ 1), preprint., (1997)

Koseki, H. and Oda, T., Whittaker functions for the large discrete series representations of $SU(2,1)$ and related zeta integral, Publ. RIMS Kyoto Univ., 31 (1995),$.[\mathrm{K}-\mathrm{O}]$959-999.
[Sch] Schmid, W., On realization of the discrete series of a semisimple Lie group, Rice
University Studies, 56 (1970), 99-108.
$\mathrm{J}.\mathrm{A}93.\cdot$$[\mathrm{S}\mathrm{h}\mathrm{S}.]17\mathrm{s}1_{-1}\mathrm{h}\mathrm{a}1\mathrm{i}\mathrm{k}\mathrm{a},$, The multiplicity one theorem for$GL_{n}$, Ann. of Math., 100 (1974),[Ya] Yamashita, H., Embedding of discrete series into induced representations of semisimple Lie groups II: Generalized Whittaker models for $SU(2,2)$ , J. Math. Kyoto Univ.,
31-1 (1991), 543-571.

Graduate School of Mathematical Sciences, University of Tokyo,
3-8-1 Komaba Meguro-ku Tokyo, 153, Japan-mail address: ishikawa@ms406ss5.ms.u-tokyo.ac.jp$E$