数理解析研究所講究録 1421 巻 2005 年 72-84

72

Confluence from Siegel-Whittaker functions to Whittaker
functions on$Sp(2, \mathrm{R})$

Miki Hirano ( 平野 幹、 愛媛大学理学部), Taku Ishii( 石井卓、 学術振興会特別研究員)
Takayuki Oda( 織田 孝幸、東京大学数理科学)
平成 17 年 2 月 14 日

概要
上の Siegel-Whittaker
この小論では、それぞれの球関数の明示公式を利用して、良い関数から Whittaker functions への合流を論ずる。証明には、Whittaker (resp. SiegelWhittaker) functions の、 第 2 種の Whittaker (resp. Siegel-Whittake]:) functions による展開定理を用いる.
Abstract
We discuss a confluence from Siegel-Whittaker functions to Whittaker functions onby using their explicit formulae. In our proof, we use expansion theorems of thegood Whittaker functions by the secondary Whittaker functions.
$Sp(2, \mathrm{R})$$Sp(2, \mathrm{R})$序によって、 ある半単純実 Lie 群 $G$ の既約認容表現を表わす.

の閉部分群 $R$ とそにの既約ユニタリ表現 の対 $(R, \eta)$ が与えられたとき、 平滑に誘導された GR 表現
$K$
ここで
は
あるいは
を考える.
値を持つ絡空聞$G$ の極大コンパクト部分明で$G$
即ちはLie 環の複素化を表わす. この空間の非零元から表現への絡作用素 (intertwining operator)
を一つ選ぶとき、 その像のはの中での一般化されら球模型 (generalized spherical model) あるいは球実現(generalzied spherical realization) である.
群 $G$ が両側剰余類分分解 $G=RA_{R}K$ を有つような $G$ の最大 RR 分拘引ルタン部分群の分裂成分である閉部分群の適切な K-typeがあるとし、 それを一つ固定する. そのとき1を選び、(通常は重複度 のもの)
また絡空間
は有限次元と仮定する.
のそのとき上の実現の \mbox{\boldmath $\tau$}\mbox{\boldmath $\tau$}等質成分
内の零化元
of(annihilator) の
記号

$\pi$

$G$

$\mathrm{I}\mathrm{n}\mathrm{d}_{R}^{G}(\eta)$$\eta$$\mathrm{H}\mathrm{o}\mathrm{m}G(?\mathrm{r}, \mathrm{I}\mathrm{n}\mathrm{d}_{R}^{G}(\eta))$$\mathrm{H}\mathrm{o}\mathrm{m}(\alpha \mathrm{c},h’)(\pi, \mathrm{I}\mathrm{n}\mathrm{d}_{R}^{G}(\eta))$$I_{\text{、}}$$\mathrm{g}_{\mathrm{C}}$

$\pi$

$I$

$\mathrm{I}\mathrm{n}\mathrm{d}_{R}^{G}(\eta)$$\pi$${\rm Im}(I)$$\mathrm{I}\mathrm{n}\mathrm{d}_{R}^{G}(\eta)$$A_{R}$$\pi$$\tau$$\mathrm{H}\mathrm{o}\mathrm{m}(\mathfrak{g}_{\mathrm{C}},K)(\pi, \mathrm{I}\mathrm{n}\mathrm{d}_{R}^{G}(\eta))$${\rm Im}(I)[\tau]$${\rm Im}(I)$$A_{R^{-}}$$U(\mathrm{g}\mathrm{c})$

動径成分からなるホロノミック系を計算する.
例えば $G=Sp(2, \mathrm{R})$ のとき、 保型形式研究に用いるという動機から、 我々はいろいろな種

類の表現

$\pi$

と対 $(R, \eta)$ に対して、 これらのホロノミック系とその明示的な解 (即ち $G$ 上の一般化された球関数) を求めてきた (cf. [12], [13], [14], [16], [17], [22], [23], [26]).
一度同じに対して別の相異なる二つの対に対して、 二つの異なるand実現を得たとき、 当然次はこれらの二つの模型の関係を知りたくなる. 例えば、 直感的な説明$(R_{1}, \eta_{1})$$\pi$をすると、 $G$ の球部分群$R_{t}$$(R_{\mathrm{O}}, \eta 0)$とその表現孕の対の変形族$(R_{t}, \eta_{t})$で、$t\neq 0$のときには各対は内部自己同型による捻りで、 極限においては 「縮約」 対になるものを考える. これは物理学者が考え始め、 例えば Dooley [3] などによってきちんと取り扱われた Lie 群とその表現の contraction あるいは deformation という手法の変奏版である.
この論説では長いルートに付随するによる parabolic inductionのヤコビ極大部分群$(Rt, \eta t)$は$(R_{1}, \eta_{1})$$tarrow \mathrm{O}$$G$$(R_{0}, \eta_{0})$$P_{J}$

で得られる一般化主系列表現の二つの模型間の合流を論ずる. 我々の設定では、$(R_{0}, \eta_{0})$は極73の対で、 それは付随する球模型が通常と島の非退化指標のはの Whittaker 模型(cf. [18], [30], [19]) であることを意味する. そして対Siegel-Whittaker 模型を導く. これは宮崎琢也 [21] や石井卓 [16] によって調べられた. Lie 群とその表現の縮約や変形族というアイデアを厳密に展開するのは、 それによってここで得られる結果に比べると重い準備が必要となる. この高いコストを回避するため、 ここでは単に関連の球関数の ARR 動径部分のみを直接に計算する. 元の直感的なアイデする対の変形族
アは第 5 節で、 謂わば「後知恵」 として記述される.
主結果は第 4 節にある. それぞれの球関数のホロノミック系の合流 (Theorem 4.1), 「第 2 種」
(secondary) 球関数の合流 (Theorem 42), そして最後に良い増大条件を有つ一意に定まる球関数の合流 (Theorem 43) という順で証明される. ここで言う secondary spherical functions とは、大ユニポテント部分群$R_{0}$$\eta 0$$(R_{1}, \eta_{1})$

$\pi$

$\pi$

$(R_{t}, \eta_{t})$行列係数の漸近挙動を調べるために Harish-Chandra [7] によって考えられた、球関数のホロノミック系の無限遠における確定特異点でのべき級数解のことである (その後、 Heckman-Opdam[11], [28] によってこの手法は一般化された) . さらに, この種の関数は、 Miatello-Wallach[20]や Oda-Tsuzuki [27] の論文に見るように、 Poincare 級数の構成で基本的な役割を果たす. 「良い球関数」 の合流の証明で基本的なことは, 「第 2 種」球関数合流の結果と、 良い球関数の 「第2 種」 球関数による展開定理をもちいることである (Theorems 24and 34). 我々の結果は、それ自身興味深い点があると思うが (表現の変形は少なくとも最近はほとんど考えられていない)さらに一型形式研究に、 例えば無限素点での局所ゼータ関数の研究に、 役に立つことを期待している (cf. [25]).
、

関連する周辺の問題はいろいろあるが、 また別の機会に.
注意用語 secondary spherical function の secondary であるが、「第 2 種」 (of the second kind)の意味もあるが、 元はボクシングの 「セコンド」 のような意味で使っている. Latin 語の語源
は、 secundus は元は、 to follow の語から来ていると言う. 当面は、 素直に 「補助的な」 の意

味で受け取って下さい.

and記号 For$a\in \mathrm{C}$

numbers$a_{i}(1\leq \mathrm{i}\leq r)$$n\in \mathrm{Z},$$(a)_{n}=\Gamma(a+n)/\Gamma(a)$and $b_{j}(1\leq j\leq s)$ , set$\Gamma[a_{1}, \ldots, a_{f}]=\prod_{i=1}^{r}\Gamma(a_{i})_{\uparrow}$

1

Preliminaries

1.I

Basic notions

Let

$G$

the Pochhammer symbol. For complex.

$\Gamma\{\begin{array}{llll}a_{1} \cdots \cdots a_{r}b_{1} \cdots ’ b_{s}\end{array}\}= \prod_{i=1}^{r}\Gamma(a_{i})/\prod_{i=1}^{s}\Gamma(b_{i})$be the real symplectic group of degree two:
$G=Sp(2, \mathrm{R})=\{g\in SL(4, \mathrm{R})|{}^{t}gJ_{2}g=J_{2}=(\begin{array}{ll}0 1_{2}-1_{2} 0\end{array})\}$

with 12 the unit matrix of degree two.
Fix a maximal compact subgroup $K$ of

$G$

by

$K=\{k(A, B)=(\begin{array}{ll}A B-B A\end{array})\in G|A,$ $B\in M(2,\mathrm{R})\}$

It is isomorphic to the unitary group $U(2)$ via the homomorphism$K\ni k(A, B)-A+\sqrt{-1}B\in U(2)$ .

.

,

74
Then the set of irreducibie representations of $K$ is parameterized by {( , A2)
the representation corresponding toA2} and we denote by)$\in \mathrm{Z}\oplus \mathrm{Z}|$$\lambda_{1}$$=\mathrm{S}\mathrm{y}\mathrm{m}^{\lambda_{1}-\lambda_{2}}\otimes\det^{\lambda_{2}}$$\lambda_{1}\geq$$\tau\{\lambda_{1},\lambda_{2}$$(\lambda_{1)}\lambda_{2})$.

We deftne two spherical subgroupsofmaximal unipotent radical of given by$R_{i}$$G$and their representationes. The first one is a

$G$

1

1

$n_{1}$$n_{2}$$n_{2}$$n_{3}$$R_{1}=\{n(n_{0}, n_{1}, n_{2}, n_{3})=(^{1n_{1}0}+_{-n_{0}1}1)(\begin{array}{llll}1 n_{1} n_{2} 1 n_{2} n_{3} 1 1\end{array})|ni\in \mathrm{R}\}$1.

1
Any unitary character$\eta_{1}$of$R_{1}$

can be written as$\eta_{1}\langle n(n_{0}, n_{1}, n_{2}, n_{3}))=\exp(2\pi\sqrt{-1}(c_{0}n_{0}+c_{3}n_{3}))$

with someR. In this paper we assume
a maximal split torus of by$c_{0},$$c_{3}\in$

$A$

$G$

$A=$

{ ( , a2)$a$$a_{1}$$=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(a_{1}$$\eta_{1}$is non-degenerate, that is,, a2,$a_{1}^{-1}$$c_{0}c_{3}\neq 0$. Taking, $a_{2}^{-1})|a_{i}>0$ },we have the Iwasawa decomposition $G=R_{1}AK$ .
The second spherical subgroupis defined as follows. Let $Ps=Ls\mathrm{K}Ns$ be the Siegel
parabolic subgroup with the Levi partand the abelian unipotent radical $Ns$ given by$R_{2}$$L_{s}$$L_{S}=\{(\begin{array}{ll}A 00 {}^{t}A^{-1}\end{array})|A\in GL(2, \mathrm{R})\}$,$Ns=\{n(0, n_{1}, n_{2}, n_{3})|n_{1}, n_{2}, n_{3}\in \mathrm{R}\}$Fix a non-degenerate unitary character

$\xi$

.

of $Ns$ by$\xi(n(0, n_{1}, n_{2}, n_{3}))=\exp(2\pi\sqrt{-1}\mathrm{T}\mathrm{r}(H\xi T))$

with

.

$T$

$=(\begin{array}{ll}n_{1} n_{2}n_{2} n_{3}\end{array})$,$H_{\xi}=(\begin{array}{ll}h_{1} h_{3}/2h_{3}/2 h_{2}\end{array})\in M(2, \mathrm{R})$and $\det H\xi\neq 0$ . Consider the actionof $Ls$ on $Ns$ by conjugation and the induced action on the character group
to be the identity component of the subgroup of $Ls$ which stabilize :$I\overline{\mathrm{V}s}$. Define SO$(\xi)$$\xi$

SO$(\xi):=\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}Ls(\xi)^{\mathrm{o}}=\{(\begin{array}{ll}A 00 t_{A^{-1}}\end{array})|{}^{t}AH_{\xi}A=H_{\xi}\}$.

Then SO(\mbox{\boldmath $\xi$}) is isomorphic to SO (2) if $\det H\xi>0$ and to $SO_{O}(1,1)$ if
H4 $<0$ . In thispaper we treat the case that 4 is a ‘definite’ character, that is, $\det H\xi>0$ . So we may assume$h_{2}>0$ and $h_{3}=0$ without loss of generality.
We sometimes identify the element of SO; SO (2)with its upper left
component. Fix a unitary characterof SO
by$\det$$h_{1},$$(\xi)$$2\cross 2$$\chi_{m_{0}}(m_{0}\in \mathrm{Z})$$(\xi)$$\chi_{m0}((\sqrt{h_{1}} \sqrt{h_{2}})(\begin{array}{ll}\mathrm{c}\mathrm{o}\mathrm{s}\theta \mathrm{s}\mathrm{i}\mathrm{n}\theta-\mathrm{s}\mathrm{i}\mathrm{n}\theta \mathrm{c}\mathrm{o}\mathrm{s}\theta\end{array})(\sqrt{h_{1}} \sqrt{h_{2}}))=\exp(\sqrt{-1}m_{0}\theta)$

We define $R_{2}=SO(\xi)\triangleright;Ns$ and$G=R_{2}AK$.

$\eta_{2}=\chi_{m\mathrm{o}}\otimes\xi$.

. Note that we also have the decomposition

75
1.2

Spherical functions

For the pairfunctions on$f$defined as above, consider the spacesatisfying$(R_{i}, \eta_{i})$$G$$C_{\eta i}^{\infty}(R_{f}\backslash G)$of complex valued$C^{\infty}$

for ali $(r,g)\in R_{i}\mathrm{x}G$ .

$f(rg)=\eta_{i}(r)f(g)$By the right translation,
is a smooth -module and we denote by the same symbol
its underlying-module (gc is the complexification of the Lie algebra of ). For anirreducible admissible representationof and the subspaceof $K$ -finite vectors,
the intertwining space$G$$C_{\eta_{\tau}}^{\infty}(R_{\dot{\mathrm{q}}}\backslash G)$$G$$(\mathrm{B}\mathrm{c}, K)$$G$$(\pi, H_{\pi})$$H_{\pi,K}$$\mathrm{I}_{\eta_{i},\pi}=\mathrm{H}o\mathrm{m}(\mathfrak{g}_{\mathrm{C}},K)(H_{\pi},,{}_{K}C_{\eta_{\mathrm{t}}}^{\infty}(R_{i}\backslash G))$

between the-modules is called the space of algebraic Whittaker functionals for. For a finite-dimensional $K$ -moduleor algebraic Siegel- Whittaker functionals forthe spacedenote by$(\mathfrak{g}\mathrm{c}, K)$$\mathrm{i}=2$$\mathrm{i}=1_{\}}$$(\tau, V_{\Gamma}’)$,$C_{\eta i,\mathcal{T}}^{\infty}(R_{i}\backslash G/K)${$\phi:Garrow V_{\tau},$$C^{\infty}|\phi(rgk)=\eta:(r)\tau(k^{-1})\phi(g)$be a -type of and :Let
contragredient representation of . Then for$I\acute{4}$$(\tau^{*}, V_{\tau^{\mathrm{r}}})$$\pi$$\iota$, for all $(r,g,$$V_{\tau^{n}}arrow H_{\pi}$$\tau$$\Phi\in \mathrm{I}_{\eta i,\pi}$$k)\in R_{i}\mathrm{x}G\mathrm{x}K$

be an injection. Here
, we can find an element$\tau^{*}$$\phi_{\iota}$}.

means thein

$C_{\eta_{\dot{\mathrm{t}}},\tau}^{\infty}(R_{i}\backslash G/K)=C_{\eta i}^{\infty}(R_{i}\backslash G)\otimes V_{\tau^{*}}\cong \mathrm{H}\mathrm{o}\mathrm{m}_{K}(V_{\tau^{*}},C_{\eta i}^{\infty}(R_{i}\backslash G))$.
with , the canonical paring onvia
isSince there is the decomposition $G=R_{i}AK$ , our (generalized) spherical function
to , which we call the radial part of . For a subspacedetermined by its restriction$X|_{A}=\{\phi|_{A}\inC^{\infty}(A)|\phi\in X\}$ .
$X$ of, we denoteof spherical functions and their
Let us define two spaces Wh(\pi , , r) and SW
a follows:
and SW
subspaces Wh$\Phi(\iota(v^{*}))(g)=\langle v^{*}, \phi_{\iota}(g)\rangle$$\langle$$V_{\tau}\cdot \mathrm{x}V_{\tau}$$\rangle$$\phi_{\iota}$$A$$\phi_{\iota}|_{A}$$\phi_{\iota}$$C_{\eta\tau}^{\infty}(\mathrm{i},R_{\iota}\backslash G/K)$$(\pi, \eta_{2}, \tau)$$\eta_{1}$$(\pi, \eta_{1}, \tau)^{\mathrm{m}\mathrm{o}\mathrm{d}}$$(\pi, \mathrm{y}72, \tau)^{\mathrm{r}\mathrm{a}\mathrm{p}}$WhWh$(\pi, \eta_{1}, \tau)^{\mathrm{m}\mathrm{o}\mathrm{d}}=${$\mathrm{s}$’$(\pi, \eta_{1}, \tau)=\mathrm{U}_{r(\tau\pi)}.,\{\phi_{\iota}|\Phi\in \mathrm{I}_{\eta_{1\prime}\pi}\}\iota\in \mathrm{H}\circ \mathrm{m}_{\mathrm{A}}$$\phi_{\iota}\in \mathrm{W}\mathrm{h}(\pi,$$\eta_{1},$$\tau)|\phi_{\iota}|_{A}$is of moderate growth as$\mathrm{S}\mathrm{W}(\pi, \eta_{2}, \tau)=\cdot\cup\{\phi_{\iota}|\Phi\in \mathrm{I}_{\eta_{2},\pi}\}\iota\in \mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{A}’}(\tau^{*},\pi)$$a_{1}$,$a\mathit{2}arrow\infty$

},

’

and
$\mathrm{S}\mathrm{W}(\pi, \eta_{2}, \tau)^{\mathrm{r}\mathrm{a}\mathrm{p}}=$

We call an element in Wh
Whittaker function) for{$\phi_{\iota}\in \mathrm{S}\mathrm{W}(\pi,$$(\pi, \eta_{1)}\tau)$$(\pi, \eta_{i}, \tau)$.

$\eta_{2},$$\tau)|\phi_{\iota}|_{A}$decays rapidly as$a_{1\}}$$a\mathit{2}arrow\infty$}.

(resp. SW( , ep2 , r)) a Whittaker function (resp. Siegel$\pi$

As we shall see in the next two sections, radial parts of spherical functions satisfy certain
holonomic systems of regular singular type. We call the power series solutions at the regular
singularities of the systems secondary spherical functions, and the elements of Wh
good spherical functions.
and SW$(\pi, \eta_{1} , \tau)^{\mathrm{m}\mathrm{o}\mathrm{d}}$$(\pi, \eta_{2}, \tau)^{\mathrm{r}\mathrm{a}\mathrm{p}}$

78
1.3

$P_{J}$-principal series representations

In this section we recall the generalized principal series representations of associated withof corresponding to the long root. A Langlands
the Jacobi maximal parabolic subgroupdecomposition $P_{J}=M_{J}AJN_{J}$ is given by
$G$

$G$

$P_{J}$$\epsilon$

$b$

$a$

$(\begin{array}{ll}a bc d\end{array})\in SL(2, \mathrm{R})\}$$M_{J}=\{(\begin{array}{llll}\epsilon a b c \epsilon d\end{array})|\epsilon\in\{\pm 1\},$,$\epsilon$

$d$

$c$

$A_{J}=\{a(a_{1},1)=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(a_{1},1_{\backslash }a_{1}^{-1},1)\in A|a_{1}>0\}$,.

$N_{J}=\{n(n_{0}, n_{1}, n_{2}, \mathrm{O})\in N=R_{1}|n_{i}\in \mathrm{R}\}$of the semisimple part $M_{J}\cong\{\pm 1\}\cross SL(2, \mathrm{R})$ of $PJ$
A discrete series representation)(resp.
is a character and, where :is of the form
with Blattner parameter A (resp. -&). For
is the discrete series representation ofby $\exp(\iota/)(a(a_{1} , 1))=a_{1}^{l/}$ . We call an induced
define a quasi-character $\exp(\nu)$ ofrepresentation$(\sigma, V_{\sigma})$$\sigma=\epsilon \mathbb{H}D_{k}^{\pm}(k\geq 2)$$\epsilon$$D_{k}^{+}$$\{\pm 1\}arrow \mathrm{C}^{*}$$D_{k}^{-}$$SL(2, \mathrm{R})$$A_{J}$$lJ\in \mathrm{C}_{\}}$$I(PJ;\sigma, \nu)=C^{\infty}- \mathrm{I}\mathrm{n}\mathrm{d}_{P_{J}}^{G}(\sigma\otimes\exp(\nu+1)\otimes 1_{N_{J}})$

the$P_{J}$-principal series representation of$G$.

is fully described in [23, Proposition 2.1] and [13, Proposition
The $K$-types of(even2.3]. In particuiar, if
withoccurs in with multiplicity one.
principal series), then the corner $K$ -type$I(P_{J};\sigma, \nu)$$\pi$$\tau^{*}=\tau(k,k)$

2

Whittaker functions

2.1

Basic results$P_{J}-$$\epsilon(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(-1,1, -1,1))=(-1)^{k}$$\pi=I(P_{J};\epsilon \mathrm{H}D_{k}^{+}, \nu)$p) be an irreducible even -principal series representation of with, and $\tau^{*}=\tau(k,k)$ is the corner A-type of . We first prepare
some basic facts on the Whittaker functions for. Throughout this section we use acoordin ate $x=$
on defined by

Let $\pi=I$ ($G$$P_{J}$$P_{J};\epsilon \mathbb{H}D_{k}^{+},$$\epsilon(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(-1,1, -1,1))=(-1)^{k}$$\pi$$(\pi, \eta_{1}, \tau)$$A$$(x_{1} , x_{2})$$x_{1}=( \pi c_{0}\frac{a_{1}}{a_{2}})^{2}$,$x_{2}=4\pi c_{3}a_{2}^{2}$.

By combining the results of Kostant ([18, \S 6]), Wallach (Theorem 8.8]), Matumoto ([19,Theorem 6.2.1]) and Miyazaki and Oda ([23, Proposition 7.1, Theorem 8.1]), we obtain$[30_{\}}$

Proposition 21 Let(i) We have

$\pi$

and

$\dim \mathrm{I}_{\eta_{1},\pi}=\dim$$\tau$

be as above Then we have the following:Wh$(\pi, \eta_{1}, \tau)=4_{f}$and a fundion$\phi w(a)=a_{1}^{k+1}a_{2}^{k+1}\exp(-2\pi c_{3}a_{2}^{2})hw(a)$on

is in the space Wh(\pi , qr , ) if and only if $hw(a)=h_{W}(x)$ is a smooth solution of thefollowing holonomic system of rank 4:
$A$

(2.1)

$\tau$$|_{A}$$\{\partial_{x_{1}}(-\partial_{x_{1}}+\partial_{x_{2}}+\frac{1}{2})+x_{1}\}h_{W}(x)=0$

,

77
(2.2)

$\{(\partial_{x_{2}}+\frac{k+l\mathit{1}}{2})(\partial_{x_{2}}+\frac{k-\nu}{2})-x_{2}(-\partial_{x_{1}}+\partial_{x_{2}}+\frac{1}{2})\}h_{W}(x)=0$

where$\partial_{x:}=xi(\partial/\partial x_{i})(\mathrm{i}=1,2)$(ii) $\dim$ Wh$(\pi, \eta_{1} , \tau)^{\mathrm{m}\mathrm{o}\mathrm{d}}\leq 1$,

is the Euler operator with rcspect to $xi$ .

. Moreover this inequality is an equality if and only if $c_{3}>0$ .

, we need a minor change by using theRemark 1. Since [23] treated the case
explicit formulas of ‘shift operators’ ([22, Proposition 8.3]).
$\sigma=\epsilon \mathbb{H}D_{k}^{-}$

2.2

Explicit formulas of secondary Whittaker functions

In this section we determine the space of smooth solutions of the holonomic system in
Proposition 2.1, therefore the space of the Whittaker functions Wh ( , , r) , explicitly. Set$\pi$$\eta_{1}$$h_{W}(a)=h_{W}(x)= \sum_{m,n\geq 0}a_{m,n}x_{1}^{\sigma_{1}+m}x_{2}^{\sigma_{2}+\tau\iota}$with $a0,0\neq 0$ . Then we have the following difference equations for
(2.3)
(2.4)

$\{a_{m,n}\}$:$( \sigma_{1}+m)\{-(\sigma_{1}+m)+(\sigma_{2}+n)+.\frac{1}{2}\}a_{m,n}+a_{m-1,n}=0$,$( \sigma_{2}+n+\frac{k+\nu}{2})(\sigma_{2}+n+\frac{k-\nu}{2})a_{m,n}+\{(\sigma_{1}+m)-(\sigma_{2}+n)+\frac{1}{2}\}a_{m,n-1}=0$.

Here we promise $a_{m,n}=0$ if $m<0$ or $n<0$ . By putting $m=n=0$ in (2.3) and (2.4), we
can find the characteristic indices$(\sigma_{1}, \sigma_{2})=(0,$$\frac{-k\pm\iota\prime}{2}),$$( \frac{-k\pm\nu+1}{2},$If is not an integer, we can determine the coefficients
thus obtain$\nu$

Proposition 22. (cf. [29, Proposition
on by$h_{W}^{i}(\nu;x)$$\mathit{2}.\mathit{1}J\mathrm{J}$For$\frac{-k\pm\nu}{2})$$a_{m,n}$$\nu\not\in Z_{2}$.

inductively for each case anddefine the functions$h_{W}^{i}(l/;a)=$$A$$h_{W}^{1}( \nu,\cdot x)=\sum_{m,n\geq 0}c_{m,n}^{1}x_{1}^{m}x_{2}^{n+(-k+1/)/2}$,,$h_{W}^{2}( \nu;x)=\sum_{m,n\geq 0}c_{m,n}^{2}x_{1}^{m+n+\langle-k+\nu+1)/2}x_{2}^{n+(-k+\nu)/2}$

with$c_{m,n}^{1}=\Gamma\ovalbox{\tt\small REJECT}_{-\nu}^{-n-\nu},$$c_{m,n}^{2}=\Gamma\ovalbox{\tt\small REJECT}_{-\nu}^{-n-\nu},$

’

’

,

$-m+n+ \underline{-k}\pm_{2}\nu \mathrm{B}1\ovalbox{\tt\small REJECT}\underline{-k}\pm_{2}\nu\Leftrightarrow 1\frac{(-1)^{m+n}}{m!n!}$$-m- \frac{k-\nu-1}{2}\ovalbox{\tt\small REJECT}\frac{k-\nu-1n+}{2}\frac{(-1)^{m+n}}{m!n!}$.

;$and the set {converges for any
Then the power series$1,2,$
forms a basis of the space of solutions of the system in Proposition 2,1.
$h_{W}^{i}(\nu;x)$$\epsilon\in\{\pm 1\}\}$$x\in \mathrm{C}^{2}$$h_{W}^{i}(\epsilon\nu$$)$$|\mathrm{i}=$78Explicit formulas of good Whittaker functions

2.3

When $c_{3}<0$ , Proposition 2.1 tells us that there is no non-zero moderate growth Whittaker
function. Therefore Jet us assume $c_{3}>0$ in the foltowing discussion. The integral expression
for the Whittaker functions of moderate growth was obtained by Miyazaki and Oda.

Proposition 23. ([$\mathit{2}\mathit{3}_{f}$Theoxm 8.1]J Let

$\pi$

and

$\tau$

be as before. Define,$gw(a)=g_{W}(x):=x_{2}^{-1/2} \int_{0}^{\infty}t^{-k+1/2}W_{0,\nu}(t)\exp(-\frac{t^{2}}{16x_{2}}-\frac{16x_{1}x_{2}}{t^{2}})\frac{dt}{t}$with$W_{\kappa,\mu}$the classical Whittaker function. Then the function$\phi w(a)=a_{1}^{k+1}a_{2}^{k+1}\exp(-2\pi c_{3}a_{2}^{2})gw(a)$gives a non-zero elernent in Wh$(\pi, \eta_{1} , \tau)^{\mathrm{m}\mathrm{o}\mathrm{d}}|_{A}$which is unique up to constant multiple.

Expansion theorem for Whittaker functions

2.4

Now we express the moderate growth Whittaker function $gW$ as a linear combination of$h_{W}^{i}$.

Theorem 2.4. For ( , letand 2.3 respectively. Then
$\nu$

$Z$

$gw(a)=c_{W} \sum_{\epsilon\in\{\pm 1\}}(\Gamma[-\epsilon\nu,$$h_{W}^{i}(\nu; a)$and $gw(a)$ be the function defined in Proposition 2.2$\frac{-k+\epsilon\nu+1}{2}]h_{W}^{1}(\epsilon\nu;a)+\Gamma[-\epsilon\nu,$$\frac{k-\epsilon\nu-1}{2}]h_{W}^{2}(\epsilon\nu;a))$with CW=21-2k エー 1/2.

3

Siegel-Whittaker functions

3.1

Basic results

Miyazaki ([21]) studied the Siegel-Whittaker functions for -principal series and obtained
the multiplicity one property and the explicit integral representation for rapidly decreasing
function. As in the previous section, we introduce the coordinate $y=(y_{1}, y_{2})$ on by$P_{J}$$A$$y_{1}= \frac{h_{1}a_{1}^{2}}{h_{2}a_{2}^{2}}$,$y_{2}=4\pi h_{2}a_{2}^{2}$.

We remark on a com patibility condition. For a non-zero element
we have$\phi$

of$C_{\eta_{i},\tau_{\langle-}}^{\infty},,-k$$\phi(a)=\phi(mam^{-1})=(\chi_{m\mathrm{o}}\otimes\xi)(m)\tau_{(-k,-k)}(m)\phi(a)$)$(R_{i}\backslash G/K)$,,

where $a\in A$ and $m\in SO(\xi)\cap Z_{I\acute{\backslash }}(A)=\{\pm 1_{4}\}$ . If we take $m=-1_{4},$$\chi_{m_{0}}(m)=\exp(\pi\sqrt{-1}m_{0})$ and $\tau(-k,-k\}(m)=1$
imply thatis an even integer.

$(\chi_{m\mathrm{o}}\otimes\xi)(m)=$$m_{0}$

78
Proposition 31. ([21, Proposition$7.\mathit{2}\mathrm{J}^{\cdot}$

Let

$\pi$

and

$\tau$

be as in \S 2.1,

Then we have thefollowing:(i) We have$\dim \mathrm{I}_{\eta_{2},\pi}=\dim \mathrm{S}\mathrm{W}(\pi, \eta_{2}, \tau)\leq 4$

and a function$\phi sw(a)=a_{1}^{k+1}a_{2}^{k+1}\exp(-2\pi(h_{1}a_{1}^{2}+h_{2}a_{2}^{2}))hsw(a)$is in the spacesysterre:$\mathrm{S}\mathrm{W}(\pi, \eta_{2}, \tau)|A$

if and only if $hsw(a)=hsW(y)$ is a smooth solution of following
,

(3.1)

$\{\partial_{y1}(-\partial_{y1}+\partial_{y_{2}}+\frac{1}{2})+\frac{y_{1}}{y_{1}-1}(-\partial_{y1}+\frac{1}{2}\partial_{y2})+\frac{m_{0}^{2}}{4}\frac{y_{1}}{(y_{1}-1)^{2}}\}h_{SW}(y)=0$(3.2)$\{(\partial_{y2}+\frac{k+\nu}{2})(\partial_{y2}+\frac{k-\nu}{2})-y_{1}y_{2}(\partial_{y1}+\frac{1}{2})-y_{2}(-\partial_{y1}+\partial_{y\mathrm{z}}+\frac{1}{2})\}h_{SW}(y)=0$

with
(ii)

$\partial_{yi}=y_{i}(\partial/\partial y_{i})$.

$\dim \mathrm{S}\mathrm{W}(\pi, \eta_{2}, \tau)^{\mathrm{r}\mathrm{a}\mathrm{p}}\leq 1$

3.2

,

.

Explicit formulas of secondary Siegel-Whittaker functions

We consider the power series solution of the system in Proposition 3.1 around $(y_{1}, y_{2})=$.
$(0,0)$ . In the notation in [16], this is the solution at$Q_{\infty}$

Proposition 32. For$\nu\not\in Z$

, set

by

$h_{SW}^{i}(\nu;a)=h_{SW}^{i}(\nu \mathrm{i}y)$’$h_{SW}^{1}( \nu;y)=(1-y_{1})^{|m\mathrm{o}|/2}\sum_{m,n\geq 0}c_{m,n}^{1}\Gamma\ovalbox{\tt\small REJECT}^{m-n+}-n+\frac{k+\frac{k+|m_{0}|-\nu}{|m\mathrm{o}|-\nu 2}}{2},$$\ovalbox{\tt\small REJECT}_{W(\nu;y)}$$=(1-y_{1})^{|m0|/2} \sum_{m,n\geq 0}ccm,n2$,$m \frac{+\frac{|m0|+1}{|+12}1m_{0}}{2}\ovalbox{\tt\small REJECT} y_{1}^{m}y_{2}^{n+(-k+\nu)/2}$$+1 \frac{-k+|mn+}{}$$m+ \frac{-k+|m_{0}|+\nu}{20|+\nu 2+1},$,,$m+ \frac{|J7\iota_{0}|+1}{2|+12}\mapsto m\mathrm{o}$$n \frac{k+|m_{0}|-\nu}{+\frac{k+|m_{0}|-\nu 2}{2}}||$. $y_{1}^{m+n+(-k+\nu+1)/2}y_{2}^{n+(-k+\nu)/2}$ .
are the coefficients defined in Proposition 2.2. Then the power series
andand the set$h_{SW}^{l}(\nu;y)$ converges $|y_{1}|<1$ andforrnsbasis of the space of solutions of the system in Proposition 3. 1 .
Here$c_{m,n}^{1}$$c_{m,n}^{2}$$y_{2}\in \mathrm{C}$

3.3

$a$

$\{h_{SW}^{i}(\epsilon\nu;y)|\mathrm{i}=1,2, \epsilon\{\pm 1\}\}$

Explicit formulas of good Siegel-Whittaker functions

is given by Miyazaki( , y72 , r)
The integral representation of the unique element in([21, Theorem 7.5]). For our purpose, however, we need another integral expression for thisfunction. Inspired by the work of Debiard and Gaveau $([1],[2])$ , we obtain the following Euler
type integral. See also Iida ([15]) and Gon ([6]).
$\mathrm{S}\mathrm{W}$

Proposition 33$\pi$$\mathrm{r}\mathrm{a}\mathrm{p}|_{A}$Define$gsw(a)=gSW(y):=(1-y_{1})^{|m_{0}|/2}y_{2}^{|m_{0}|/2}$.

$\int_{0}^{1}t^{(|m0\}-1\rangle/2}(1-t)^{\{|m_{0}|-1)/2}F(\frac{y_{2}}{2}\{1-t(1-y_{1})\})dt$

,

80
with
$F(z)=e^{z}(2z)^{(-k-|m\mathrm{o}|-1)/2}W_{(k-|m_{0}|-1)/2,\nu/2}(2z)$.

Then the function$\phi sw(a)=a_{1}^{k+1}a_{2}^{k+1}\exp(-2\pi(h_{1}a_{1}^{2}+h_{2}a_{2}^{2}))gsw(a)$gives a non-zero element in$\mathrm{S}\mathrm{W}(\pi, \eta_{2} , \tau)^{\mathrm{r}\mathrm{a}\mathrm{p}}|_{A}$which is unique up to constant multiple.

Proof. See $[6, 84]$ .
3.4$\square$Expansion theorem for Siegel-Whittaker functions

Theorem 3.4. Forlet
3.2 and 3.3 respectively. Then$\nu\not\in Z_{2}$$h_{SW}^{\dot{f}}(\nu;a)$and $gsw(a)$ be the function defined in Proposition$gsw(a)=c_{SW} \sum_{\epsilon\in\{\pm 1\}}(\Gamma[-\epsilon\nu,$$\frac{-k+\epsilon\nu+1}{2}]h_{SW}^{1}(\epsilon\nu\}.a)$,$+\Gamma[-\epsilon\nu$

with$\frac{|m_{0}|+1}{2}$$c_{SW}=\Gamma[_{\frac{-k+|m_{0}|-\nu}{2}+1},$$\frac{-k+|m\mathrm{o}|+\nu}{2}+1]$

4

Confluences

4.1

Confluence of the differential equations.

Theorem 4.1. If we substitute(4.1)$h_{1}=t^{2}c_{3}$,$h_{2}=c_{3}$,$m_{0}= \frac{2\pi c_{0}}{t}$the system in Proposition 3.1 and take the limit $tarrow 0_{2}$ then we obtain the system in
Proposition 2, 1.
in

4.2

Confluence of the secondary spherical functions

Theorem 42. For
in. Then$\nu\not\in Z$, define the functions$h_{SW}^{i}(\nu, t,\cdot a)(\mathrm{i}=1,2)$$h_{SW}^{i}(\nu,\cdot a)$$\lim_{t\prec 0}h_{SW}^{1}(\nu, t;a)=h_{W}^{1}(\nu;a)$$\lim_{tarrow 0}t^{k--1}’ h_{SW}^{2}(\nu, t;a)=h_{W}^{2}(\nu,\cdot a)$,.

by substituting (4.1)

81
Confluence of the spherical functions

4.3

Theorem 4.3. Define the function $gsw(t;a)$ by substituting$\lim_{tarrow 0}\frac{gSW(t,a)}{c_{SW}}.=\frac{g_{W}(a)}{c_{W}}$

Deformation ffom

5$(R_{2}, \eta_{2})$to$(R_{1}, \eta_{1})$$(\mathit{4}\cdot \mathit{1})$in gsvv (a). Then.

and the confluence

In tlis section we expiain the main results in the previous section from the points of view ofa heuristic backgrounddeformations and contractions of Lie groups (cf. [3]). This is tofor the computations in the previous sections.
$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{l}\mathrm{y}$5.1

From SO(2) to$N_{0}$

We first consider the deformation of two subgroups of

SO (2)$SL(2,\mathrm{R})$$=\{(\begin{array}{ll}\mathrm{c}\mathrm{o}\mathrm{s}\theta \mathrm{s}\mathrm{i}\mathrm{n}\theta-\mathrm{s}\mathrm{i}\mathrm{n}\theta \mathrm{c}\mathrm{o}\mathrm{s}\theta\end{array})|\theta\in \mathrm{R}\}$$N_{0}=\{(\begin{array}{ll}1 c\mathrm{o} 1\end{array})|c\in \mathrm{R}\}$

:

,

.

SO (2)
to the upper half plane
Under the usual action of. Set $zt=\sqrt{-1}/t$ for $t>0.$ Then
fixes
and
is the stabilizer subgroup ofisof inand the stabilizer subgroup StabsL(2,R)$\mathfrak{h}=\{z\in \mathrm{c}|{\rm Im}(z)>0\}_{1}$$SL(2, \mathrm{R})$$\sqrt{-1}$$N_{0}$$\sqrt{-1}\infty$$\lim_{tarrow\infty}z_{t}=\sqrt{-1}\infty$$(z_{t})$$SL(2, \mathrm{R})$$z_{t}$$\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}_{SL(2,\mathrm{R})}(z_{t})=\{r_{\theta}(t)=(\begin{array}{ll}\mathrm{c}\mathrm{o}\mathrm{s}\theta \mathrm{s}\mathrm{i}\mathrm{n}\theta/t-t\mathrm{s}\mathrm{i}\mathrm{n}\theta \mathrm{c}\mathrm{o}\mathrm{s}\theta\end{array})|\theta\in \mathrm{R}\}$$=(\begin{array}{ll}t^{-1}/2 00 t^{1/2}\end{array})$

For our purpose, we have to moveThen
(5. 1)$\theta=\theta(t)$SO (2)$(\begin{array}{ll}t^{1}/2 00 t^{-\mathrm{l}/2}\end{array})$such as $\sin\theta(t)/tarrow c$ as.

$tarrow \mathrm{O}$. Let$\theta(\mathrm{f})=ct$.

$r_{\theta\{t)}(t)=(\begin{array}{ll}\mathrm{c}\mathrm{o}\mathrm{s}\theta(t) \mathrm{s}\mathrm{i}\mathrm{n}\theta(t)/t-t\mathrm{s}\mathrm{i}\mathrm{n}\theta(t) \mathrm{c}\mathrm{o}\mathrm{s}\theta(t)\end{array})arrow(\begin{array}{ll}1 c0 1\end{array})$$\cdot$as desired.

52
Let

From

$R_{2}$$\xi_{t}(t\neq 0)$

to$R_{1}$

be the definite character of $Ns$ associated with

$H\xi \mathrm{r}=(\begin{array}{ll}ht^{2} 00 h\end{array})(h>0)$$\xi_{t}(n(0, n_{1}, n_{2}, n_{3}))=\exp(2\pi\sqrt{-1}(ht^{2}n_{1}+hn_{3}))$:.

is identified with $\{r_{\theta}(t)|t>0\}$ . Therefore if we take
Then the stabilizer subgroup SO$\theta=\theta(t)=tn0$ as in the previous subsection, (5.1) implies$(\xi_{t})$(5.2)$\lim_{tarrow 0}$

SO $(\xi_{t})=\{n(n_{0},0,0,0)|n_{0}\in \mathrm{R}\}$

in $Ls$ makes up $R_{1}=N$ together with $Ns$ , as we expected.

82
5.3

From

$\eta_{2}$to$\eta_{1}$Define the character$\chi_{m\mathrm{o}(t\rangle}$of SO$(\xi_{t})$by$\chi_{m\mathrm{o}(t)}(r_{\theta}(t))=\exp\sqrt{-1}(m_{0}(t)\theta)$

and put$\eta_{2,t}=\chi_{m\mathrm{o}(t)}$図$\xi_{t}$.

.

$\eta 2,t(r\theta(t)(t)\cdot n(0, n_{1}, n_{2}, n_{3}))=\exp 2\pi\sqrt{-1}(ht^{2}n_{1}+hn_{3})\cdot\exp\sqrt{-1}(m_{0}(t)\theta(t))$.

Since $\theta(t)=n0t$ , we should take$m_{0}(t)= \frac{2\pi c_{0}}{t}$.

Then the right hand side goes toand thus combined with (5.2), we obtain$\exp 2\pi\sqrt{-1}(c_{3}n_{3}+c_{0}n\mathrm{c})$(after the replacement $h=c_{3}$ ),.

$\lim_{t\prec 0}\eta_{2,t}(r\theta(t)(t)\cdot n(0, n_{1}, n_{2}, n_{3}))=\eta_{1}(n(n_{0}, n_{1}, n_{2}, n_{3}))$

Remark 2. Our result should be regarded as the investigation of the intertwining spaces:$\mathrm{H}_{0\mathrm{m}_{(\mathrm{B}\mathrm{c},K\rangle}}(H_{\pi},,{}_{K}C_{\eta 2,t}^{\infty}(R_{t}\backslash G))$with
$R_{\mathrm{f}}=SO(\xi t)\ltimes N_{s}$6$(t>0)$ .

Further comments

We only treat the even-principal series, however, we also have the same results for the
odd case, that is,.
In the case of the principal series (induced from minimal parabolic subgroup of ), theholonomic systems of rank 8 for the radial part of Whittaker functions (resp. Siegel-Whittaker
functions) are obtained in [22] (resp. [21], [16]) and we can prove the same assertion as
Theorem 4.1. However, explicit formulas for secondary spherical functions are 1nown only
for Whittaker functions ([17]), we can not say any more.
The other kinds of spherical functions on
are studied by Moriyama ([24]) and by
Hirano. The spherical subgroup of [24] isand of [12], [13] and [14]iswiththe -dimensional Heisenberg group. We hope that similar results
hold between the two spherical functions.
$P_{J}$$\epsilon(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(-1,1, -1,1))=-(-1)^{k}$$G$$Sp(2, \mathrm{R})$$([12], \lfloor \mathrm{r}13], [14])$$SL(2, \mathrm{R})\ltimes H_{3},$$SL(2, \mathrm{C})$$H_{3}$$3$

We finally remark on the case of the special unitary group $SU(2,2)$ , which has the same
restricted root system as. (Siegel-) Whittaker functions on $SU(2,2)$ are studied
by Hayata and Oda ([10], [8], [9]) and by Gon ([6]). Since their differential equations arecompatible to those of, analogous argument seems to be possible.
$Sp(2, \mathrm{R})$$Sp(2, \mathrm{R})$

83
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