数理解析研究所講究録
第 1826 巻 2013 年 110-128

110

Spherical functions on the space of p-adic unitary
hermitian matrices
Yumiko Hironaka
Department of Mathematics,
Faculty of Education and Integrated Sciences, Waseda University
Nishi-Waseda, Tokyo, 169-8050, JAPAN,

andYasushi Komori
Department of Mathematics,

Faculty of Science, Rikkyo University
Nishi-Ikebukururo, Tokyo, 171-8501, JAPAN\S 0

Introduction

Let be an unramified quadratic extension of a -adic field , and we consider hermitianand unitary matrices with respect to $k’/k$ . For a matrix $A=(r\iota_{ij})\in M_{mn}(k’)$ , we d\‘enote
by $A^{*}\in M_{nm}(k’)$ the conjugate transpose with respect to $k’/k$ , and say is hermitian if$A^{*}=A$ . We introduce the unitary group and the space of unitary hermitian matrices:
$k’$

$k$

$\mathfrak{p}$$A$$G=U(j_{2n})=\{g\in GL_{2n}(k’)|g^{*}j_{2n}g=j_{2n}\},$$j_{2n}=[Matrix]\in GL_{2n}(k’)$ ,$X=\{x\in G|x^{*}=x, \Phi_{xj_{2n}}(t)=(t^{2}-1)^{n}\},$

where$\Phi_{y}(t)$is the characteristic function of the matrix . The group
$y$

$G$

acts on $X$ by$g\cdot x=gxg^{*}, (g\in G, x\in X)$ .

We take the maximal compact subgroup $K=G\cap GL_{2n}(\mathcal{O}_{k’})$ and the Borel subgroupconsisting of all upper triangular matrices in , then $G=BK=KB$ . In the following,

$B$

$G$

2010 Mathematics Subject Classification: $Pri_{1}nary$ ] $1F85$ ; secondly llE95, llF70, $22E50.$
Key words and phrases: spherical functions, unitary groups, hermitian matrices.
-mail: hironaka@waseda.jp, komori@rikkyo.ac.jp, 24540031.
This research is paltially supported by Grant-in-Aid for scientific ResearchThis paper is based on a talk at RIMS (January, 2012) and continued collaboration.
full version ofthis paper will appear elsewhere.
$E$$(C):2254004_{\overline{i}})$
$A$

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we fix a prime element in and the absolute value on normalized by $|\pi|^{-1}=q=$.
In \S 1, we study $K$ -orbits and -orbits in $X$ and obtain (cf. Theorem 1.2, Theorem 1.3,
and Theorem 1.4)
$k$

$\pi$

$k$

$||$

$\#(\mathcal{O}_{k}/(\pi))$$G$

Theorem 1 (1) If$k$has odd residual characteristic, one has,$X=\lambda\in\Lambda_{n}^{+}\sqcup K\cdot x_{\lambda}, x_{\lambda}=Diag(\pi^{\lambda_{1}}, \ldots, \pi^{\lambda_{n}}, \pi^{-\lambda_{n}}, \ldots, \pi^{-\lambda_{1}})\in X$

(0.1)

where
$\Lambda_{n}^{+}=\{\lambda\in \mathbb{Z}^{n}|\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}\geq 0\}$(2) If has even residual characteristic, there are $K$ -orbits not of type(3) There are precisely two -orbits in $X$ , independent of the residual characteristic of$k$$K\cdot x_{\lambda},$$G$

In \S 2, we introduce a spherical function$\omega(x;s)$for $x\in X$ and$\omega(x;s)=\int_{K}\prod_{i=1}^{n}|d_{i}(k\cdot x)|^{s_{i}+\epsilon_{i}}dk$$\lambda\in\Lambda_{n}^{+}.$

$k.$

:

$s\in \mathbb{C}^{n}$

(0.2)

,

is a certainis the determinant of the lower right by block of
where
$dk$
$K$
fixed number,is the Haar measure on . The above integral is absolutely convergent if$1\leq i\leq n$ , continued to a rational function of, and becomes$K$$X$. It is $co$ nvenient
for eacha -invariant function on , hencebyto introduce a new variablewhich is related to$d_{i}(y)$

$i$

$i$

$y,$

${\rm Re}(s_{i})\geq{\rm Re}(\epsilon_{i}),$$q^{s_{1}},$$\ldots,$$q^{s_{n}}$$s\in \mathbb{C}^{n}$$\omega(x;s)\in C^{\infty}(K\backslash X)$$z\in \mathbb{C}^{n}$$\epsilon\in \mathbb{C}^{n}$

$s$

(0.3)

$s_{i}=-z_{i}+z_{i+1}, (1\leq t\leq n) , s_{n}=-z_{n}$ ,and we write $\omega(x;z)=\omega(x;s)$ .
Hereafter, we assume that has odd residual characteristic, i.e. is odd. Let $W$ be theWeyl group of with respect to the maximal -split torus in . Then $W$ acts on rational
characters on so does on and , and we obtain (cf. Theorem 2.5, Theorem 2.6)
$k$

$q$

$G$

$B$

Theorem 2 (1) For every

$B$

$k$

$z$

$s$

$\sigma\in W$, one has$\omega(x;z)=\Gamma_{\sigma}(z)\cdot\omega(x;\sigma(z))$

,

where
$\Gamma_{\sigma}(z)=\prod_{\alpha\in\Sigma_{8}^{+}(\sigma)}\frac{1-q^{\langle\alpha,z\rangle-1}}{q^{\langle\alpha,z\rangle}-q^{-1}},$$\Sigma_{S}^{+}$

is the set of positive roots and$\Sigma_{s}^{+}(\sigma)=\Sigma_{s}^{+}\cap\sigma(-\Sigma_{s}^{+})$.

(2) One has$\prod_{\alpha\in\Sigma_{S}^{+}}\frac{1+q^{\langle\alpha,z\rangle}}{1-q^{\langle\alpha,z)-1}}\cross\omega(x;z)\in \mathbb{C}[q^{\pm z_{1}}\ldots., q^{\pm z_{n}}]^{W}(=\mathcal{R}, say)$

.

112

(Theorem 3.1) byfor each
In \S 3, we will give the explicit formula forand somea method introduced in [H4], which is based on functional equations of
is -invariant for , it is enough todata depending only on the group . Since
by Theorem 1.
consider the explicit formula for$\omega(x_{\lambda};z)$$\lambda\in\Lambda_{n}^{+}$$\omega(x;z)$$G$$x_{\lambda},$Theorem 3 For each$\lambda\in\Lambda_{n}^{+}$$\omega(x;z)$

$K$

$x$

$\lambda\in\Lambda_{n}^{+}$, one has$\omega(x_{\lambda};z) = \frac{(1-q^{-2})^{n}}{w_{2n}(-q^{-1})}\cdot\prod_{\alpha\in\Sigma_{s}^{+}}\frac{1-q^{\langle\alpha,z\rangle-1}}{1+q^{\langle\alpha,z\rangle}}\cdot c_{\lambda}\cdot Q_{\lambda}(z)$

(0.4)

,

where $G(z)$ is the same as in Theorem 2,$c_{\lambda}=(-1)^{\Sigma_{i}\lambda_{i}(n-i+1)}q^{-\Sigma_{i}\lambda_{t}(n-i+\frac{1}{2})}, w_{m}(t)= \prod_{i=1}^{m}(1-t^{i})$,$Q_{\lambda}(z)= \sum_{\sigma\in W}\sigma(q^{-\langle\lambda,z\rangle}c(z)) , c(z)=\prod_{\alpha\in\Sigma_{s}^{+}}\frac{1+q^{\langle\alpha,z\rangle-1}}{1-q^{\langle\alpha,z\rangle}}\prod_{\alpha\in\Sigma_{\ell}^{+}}\frac{1-q^{\langle\alpha,z\rangle-1}}{1-q^{\langle\alpha,z\rangle}}.$is a polynomial in . On the other hand, this is a
By Theorem 2, we see
specialization of Macdonald polynomial , and it is known that the set
forms a -basis for and $Q_{0}(z)$ is a constant. Hence, we have$\mathcal{R}$$Q_{\lambda}(z)$$\{Q_{\lambda}(z)|\lambda\in\Lambda_{n}^{+}\}$$P_{\lambda}$$\mathcal{R}$$\mathbb{C}$$\omega(1_{2n};z)=\frac{(1-q^{-1})^{n}w_{n}(-q^{-1})^{2}}{w_{2n}(-q^{-1})}\cross\prod_{\alpha\in\Sigma_{s}^{+}}\frac{1-q^{\langle\alpha,z\rangle-1}}{1+q^{\langle\alpha,z\rangle}}$.

In \S 4, we consider the spherical Fourier transform on the Schwartz space$F$:$S(K\backslash X)$$arrow$(0.5)$\mathcal{S}(K\backslash X)$:$\mathcal{R}$$\varphi \mapsto F(\varphi)=\int_{X}\varphi(x)\Psi(x;z)dx,$

where $\Psi(x;z)=\omega(x;z)/\omega(1_{2n};z)$ and $dx$ is a -invariant measure on $X$ . We obtain thefollowing (cf. Theorem 4.1, Theorem 4.2, Theorem 4.5).
$G$

Theorem 4 (1) The spherical Fourier transform $F$ is an-module of rankin particular,
is a free$S(K\backslash X)$$\mathcal{H}(G, K)$$\mathcal{H}(G, K)$-module isomorphism,$2^{n}.$, the set $\{\Psi(x;z+u)|u\in\{0, \frac{\pi\sqrt{-1}}{\log q}\}^{n}\}$ forms a basis for the spherical(2) For eachfunctions on $X$ corresponding to$z\in \mathbb{C}^{n}$$\lambda_{z}.$(3) (Plancherel formula) We give explicitly the normalization of $dx$ on $X$ and a measure$d\mu(z)$on$\mathfrak{a}^{*}=\{\sqrt{-1}(\mathbb{R}/\frac{2\pi}{\log q}\mathbb{Z})\}^{n}$

113
for which$\int_{X}\varphi(x)\overline{\psi(x)}dx=\int_{\mathfrak{a}^{*}}F(\varphi)(z)\overline{F(\psi)(z)}d\mu(z) , (\varphi, \psi\in S(K\backslash X))$.

In [H5], we have investigated spherical functions on a similar spaceassociated toeach nondegenerate hermitian matrix , and obtained functional equations of hermitian
Siegel series as an application. Both spaces,
and the present $X$ , are isomorphic to$U(2n)/U(n)\cross U(n)$ over the algebraic closure of , and the former realization was usefulfor the application to hermitian Siegel series. But it was not easily understandable, andwe could not obtain its Cartan decomposition, nor complete parametrization of sphericalfunctions. For the present space $X$ , we give an explicit Cartan decomposition in \S 2,
and complete parametrization for spherical functions in \S 4, using explicit formulas ofparticular spherical functions given in \S 4. We discuss the correspondence between both
spaces in Appendix.
$X_{T}$$T$$X_{T}$$k$Throughout of this article, we denote by a non-archilnedian local field of characteristic , fix an unramified quadratic extension
and consider unitary and hermitian$k’/k$
matrices with respect to. We fix a prime elementof , denote by
the additive
value on , and normalize the absolute value
by $|\pi|^{-1}=q=\#(\mathcal{O}_{k}/(\pi))$ . Weonalso fix a unit
for which. We may take such as, then1,
forms an -basis for(cf. [Om], 64.3 and 64.4). From \S 2 to \S 4, we assume
that is odd.
$k$

$k’$

$0$

$\pi$

$k$

$||$

$k’=k(\sqrt{\epsilon})$$\epsilon\in \mathcal{O}_{k}^{\cross}$$\frac{1+\sqrt{\epsilon}}{2}\}$

$\{$

$k$

$\mathcal{O}_{k}$$v_{\pi}()$$k^{\cross}$$\epsilon-1\in 4\mathcal{O}_{k}^{\cross}$$\epsilon$$\mathcal{O}_{k’}$$q$The space $X$ and its $K$ -orbit decomposition and-orbit decomposition

\S 1

$G$

Let
be an unramified quadratic extension of a -adic field and consider hermitianmatrices and unitary matrices with respect to $k’/k$ . For a matrix $A\in M_{mn}(k’)$ , we denote
by $A^{*}\in M_{nm}(k’)$ its conjugate transpose with respect to $k’/k$ , and say is hermitian if
$k’$

$k$

$\mathfrak{p}$$A$$A^{*}=A.$

We consider the unitary group$G=G_{n}=\{g\in GL_{2n}(k’)|g^{*}j_{2n}g=j_{2n}\},$

the space $X$ of unitary hermitian matrices in$j_{2n}=(\begin{array}{ll}0 11 0\end{array})\in M_{2n}$$\tilde{X}$

(1.1)

$G$

$X=X_{n}=\{x\in G|x^{*}=x, \Phi_{xj_{2n}}(t)=(t^{2}-1)^{n}\}$ ,and a supplementary space

,

(1.2)

containing $X$$\tilde{X}=\tilde{X}_{n}=\{x\in G|x=x^{*}\}$

,

(1.3)

114
whcre
by

$\Phi_{y}(t)$is the characteristic polynomial of the matrix . The group$y$acts on $X$ and$G$$\tilde{X}$$g\cdot x=gxg^{*}=x[g^{*}]=gxj_{2n}g^{-1}j_{2n}, g\in G, x\in\tilde{X}.$Over the algebraic closure$\overline{k}$of , we may understand as follows.
$k$$G(\overline{k})=GL_{2n}(\overline{k}) , \tilde{X}(\overline{k})=\{x\in \mathbb{G}(\overline{k})|(xj_{2n})^{2}=1_{2n}\}$

(1.4)

,

with action given by$g\star x=gxj_{2n}g^{-1}j_{2n}, (g\in G(\overline{k}), x\in\tilde{X}(\overline{k}))$Then
we take$\tilde{X}(\overline{k})$.

is decomposed into $2n+1G(\overline{k})$ -orbits according to the shape of$X(\overline{k})=G(\overline{k})\star 1_{2n}=\{x\in\tilde{X}(\overline{k})|\Phi_{xj_{2n}}(t)=(t^{2}-1)^{n}\}$$\Phi_{xj_{2n}}(t)$

, and

.

(1.5)

Then
$X=X(\overline{k})\cap G=X(\overline{k})\cap\tilde{X}.$We fix the maximal compact subgroup $K$ of

$G$

by

$K=K_{n}=G\cap M_{2n}(\mathcal{O}_{k’})$

(1.6)

.

The main purpose of this section is to give the Cartan decomposition of $X$ , i.e., the-orbit decomposition of $X$ for odd (Theorem 1.2), and -orbit decomposition of $X$(Theorem 1.4).
To start with, we recall the case of unramified hermitian matrices. The group $G_{0}=$$GL_{n}(k’)$ acts on the space $\mathcal{H}_{n}(k’)=\{y\in G_{0}|y^{*}=y\}$ by $g\cdot y=gyg^{*}$ , and there are two-orbits in, thedetermined by the parity of $v_{\pi}(\det(y))$ . Setting
Cartan decomposition is known(cf. [Jac]) as follows:
$K$

$G$

$q$

$G_{0}$$K_{0}=GL_{n}(\mathcal{O}_{k’})$$\mathcal{H}_{n}(k’)$$\mathcal{H}_{n}(k’) = \prod_{\lambda\in\Lambda_{n}}K_{0}\cdot\pi^{\lambda}$

,

where
$\pi^{\lambda}=Diag(\pi^{\lambda_{1}}, \ldots, \pi^{\lambda_{n}}) , \Lambda_{n}=\{\lambda\in \mathbb{Z}^{n}|\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}\}.$

Proposition 1.1 Let $n=1$ . Then$x_{1}=u_{0^{K_{1}}}\ell\geq\cdot(\begin{array}{ll}\pi^{\ell} 00 \pi^{-\ell}\end{array})\sqcup_{1\leq r\leq}\sqcup_{v_{\pi}(2)}(^{\pi^{-r}(1-\epsilon)}\sqrt{\epsilon} -\sqrt{\epsilon}\pi^{r)},$where the latter union is empty if$q$is odd.

For general , we have the following.
$n$

(1.7)

115
Theorem 1.2 Assume that has odd residual chamcteristic. Then, the $K$ -orbit decomposition ofis given as follows:$k$$X_{n}$$X_{n}=u_{n}\lambda\in\Lambda^{+}K\cdot x_{\lambda}$

(1.8)

,

where
$\Lambda_{n}^{+}=\{\lambda\in \mathbb{Z}^{n}|\lambda_{1}\geq\cdots\geq\lambda_{n}\geq 0\},$.

$x_{\lambda}=(\begin{array}{ll}\pi^{\lambda} 00 \pi^{-\lambda}\end{array})=Diag(\pi^{\lambda_{1}}, \ldots, \pi^{\lambda_{n}}, \pi^{-\lambda_{n}}, \cdots, \pi^{-\lambda_{1}})$For dyadic case, we show the following.

Theorem 1.3 Assume that$q$

is even. Then

$X_{n}= \bigcup_{r=0}^{n}\bigcup_{\lambda\in\Lambda_{r}^{+}}\mu\in\Lambda_{n-r}^{(2)}K\cdot x_{\lambda,\mu}, x_{\lambda,\mu}=(\begin{array}{lll}D_{r}(\lambda) 0 00 E_{n-r}(\mu) 00 0 D_{r}(-\lambda)\end{array})$Here$x_{\lambda}=(\begin{array}{ll}D_{r}(\lambda) 00 D_{r}(-\lambda)\end{array})\in X_{r}, (\lambda\in\Lambda_{r}^{+})$,$\Lambda_{m}^{(2)}=\{\mu\in\Lambda_{m}^{+}|v_{\pi}(2)\geq\mu_{1}\geq\cdots\geq\mu_{m}\geq 1\}, (m=n-r)$,$E_{m}(\ell\iota)=(\pi^{-\mu_{m}}(1-\sqrt{\epsilon} \epsilon) \pi^{-\mu_{1}}(1-\sqrt{\epsilon} \epsilon) -\sqrt{\epsilon}\pi^{\mu_{1}} -\sqrt{\epsilon}\pi^{\mu_{m}})\in X_{m},$where any entry of(resp.
stood asexcept in the diagonal or anti-diagonal is , and) if $r=n$ (resp. $r=0$ ). Further,$E_{m}(\mu)$$E_{n}(\mu)$$x_{\lambda}$$0$$x_{\lambda,\mu}$

is under-$\bigcup_{\lambda\in\Lambda_{n}^{+}}K\cdot x_{\lambda}=\sqcup\lambda\in\Lambda_{n}^{+}K\cdot x_{\lambda}\not\supset E_{n}(\mu) , \mu\in\Lambda_{n}^{(2)}.$

As for -orbits, we have the following. For
to be even or odd according to the parity of$G$$\lambda\in\Lambda_{n}^{+}$, we set$| \lambda|=\sum_{i=1}^{n}\lambda_{i}$and call$\lambda$$|\lambda|.$Theorem 1.4 There are precisely two$G$-orbits in$X_{n}$:$X_{n}=G\cdot x_{0}\sqcup G\cdot x_{1}, x_{0}=1_{2n}, x_{1}=Diag(\pi, 1, \ldots,1, \pi^{-1})$

If

$q$

is odd, then

evenK

.

$x_{\lambda},$$G\cdot x_{0}=\sqcup\lambda\in\Lambda_{n}^{+}$

If

$q$

is even,$x_{\lambda,\mu}$

is

$G$

-equivalent to$x_{0}$$G\cdot x_{1}=u\lambda\in\Lambda^{+}odd^{n}K\cdot x_{\lambda}.$if and only if$|\lambda|+|\mu|$is even.

.

(1.9)

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\S 2 Spherical function

on $X$$\omega(x;s)$

For simplicity, we write $j=j_{n}$ , and take a Borel subgroup$B=$$bisuppera+a^{*}=0$$\{(\begin{array}{ll}b 00 jb^{*-l}j\end{array})(\begin{array}{ll}1_{n} aj0 1_{n}\end{array})\in G$

$B$

of

by

$G$

triangular of size $n\},$where consists of all the upper triangular matrices in $G.$
on $X$ by Poisson transform from relative
We introduce a spherical function
the determinant of lower right by-invariants. For a matrix $g\in G$ , denote by$1\leq i\leq n$ are relative-invariants on $X$ associated with rational
block of . Thenof , wherecharacters$B$$\omega(x;s)$$B$$d_{i}(x),$

$g$

$i$

$i$

$d_{i}(g)$

$B$

$B$

$\psi_{i}$$d_{i}(p\cdot x)=\psi_{i}(p)d_{i}(x) , \psi_{i}(p)=N_{k’/k}(d_{i}(p)) , (x\in X,p\in B)$

(2.1)

.

We set(2.2)$X^{op}=\{x\in X|d_{i}(x)\neq 0,1\leq i\leq n\}$ .

For $x\in X$ and$s\in \mathbb{C}^{n}$, we consider the integral(2.3)$\omega(x;s)=\int_{K}|d(k\cdot x)|^{s+\epsilon}dk, |d(y)|^{s}=\prod_{i=1}^{n}|d_{i}(y)|^{s_{i}}$where $dk$ is the normalized Haar measure on $K,$
and

$k$

runs over the set $\{k\in K|k\cdot x\in X^{op}\},$$\epsilon=\epsilon_{0}+(\frac{\pi\sqrt{-1}}{\log q}, \ldots, \frac{\pi\sqrt{-1}}{\log q}) , \epsilon_{0}=(-1, \ldots, -1, -\frac{1}{2})\in \mathbb{C}^{n}.$The right hand side of (2.3) is absolutely convergent if, and we use the notation, and continued to a rational function ofsuch sense. We note here that${\rm Re}(s_{i})\geq-{\rm Re}(\epsilon_{i})=-\epsilon_{0,i},$$q^{s_{n}}$$q^{s_{1}},$$n$$\ldots,$$| \psi(p)|^{\epsilon}(=\prod_{i=1}^{n}|\psi_{i}(p)|^{\epsilon_{i}})=|\psi(p)|^{\epsilon_{0}}=\delta^{\frac{1}{2}}(p)$$1\leq i\leq$$\omega(x;s)$

in

(2.4)

,

$(i.e., d(pp’)=\delta(p’)^{-1}dp$ for the left invariant
where is the modulus character on
measure $dp$ on ).
-common eigen function
becomes an
By a general theory, the function
on $X$ (cf. [H2]-\S 1, or [H4]-\S 1), and we call it a spherical function on $X$ . More precisely, theof with respect to $K$ is the commutative -algebra consisting
Hecke algebraof compactly supported two-sided $K$ -invariant functions on , which acts on the spaceof left $K$ -invariant functions on $X$ by$B$$\delta$$B$$\omega(x;s)$$\mathcal{H}(G, K)$$\mathcal{H}(G, K)$$G$$\mathbb{C}$$G$$C^{\infty}(K\backslash X)$,$(f* \Psi)(y)=\int_{G}f(g)\Psi(g^{-1}\cdot y)dg, (f\in \mathcal{H}(G, K), \Psi\in C^{\infty}(K\backslash X))$

where $dg$ is the Haar measure on

$G$

(2.5)

normalized by $\int_{K}dk=1$ , and we see$(f*\omega(; s))(x)=\lambda_{s}(f)\omega(x;s) , (f\in \mathcal{H}(G, K))$

,

(2.6)

117
where

$\lambda_{s}$is the$\mathbb{C}$-algebra homomorphism defined by$\lambda_{s}:\mathcal{H}(G, K)arrow \mathbb{C}(q^{s_{1}}, \ldots, q^{s_{n}})$,$f \mapsto\int_{B}f(p)|\psi(p)|^{-s+\epsilon}dp.$

We introduce a new variable$z$which is related to

by

$s$

(2.7)

$s_{i}=-z_{i}+z_{i+1} (1\leq i\leq n-1) , s_{n}=-z_{n}$and write $\omega(x;z)=\omega(x;s)$ . Denote by $W$ the Weyl group ofwith respect to the
maximal -split torus in . Then $W$ acts on rational characters ofas usual (i.e.,$W$
by taking a representativeof ), soacts onand onas well. We will determine the functional equations ofwith respect to this
Weyl group action. The group $W$ is isomorphic toacts on by permutationof indices, and $W$ is generated by
and : $(z_{1}, \ldots, z_{n})\mapsto(z_{1}, \ldots, z_{n-1}, -z_{n})$ . Keeping
the relation (2.7), we also write. Since
$G$

$k$

$B$

$B$

$\sigma(\psi)(b)=\psi(n_{\sigma}^{-1}bn_{\sigma})$$z\in \mathbb{C}^{n}$$\sigma$$n_{\sigma}$$s\in \mathbb{C}^{n}$$\omega(x;s)$$S_{n}\ltimes C_{2}^{n},$$S_{n}$$S_{n}$$z$$\tau$$\lambda_{z}(f)=\lambda_{S}(f)$$| \psi(p)|^{-s+\epsilon}=\prod_{i=1}^{n}|N(p_{i})|^{-z_{i}}\cross\delta^{\frac{1}{2}}(p)$,where is the i-th diagonal component of $p\in B$ , the -algebra map(the Satake isomorphism)$\mathbb{C}$$p_{i}$$\lambda_{z}$:$\lambda_{z}$is an isomorphism
(2.8)

,

$\mathcal{H}(G, K)arrow^{\sim}\mathbb{C}[q^{\pm 2z_{1}}, \ldots, q^{\pm 2z_{n}}]^{W}$

where the ring of the right hand side is the invariant subring of the Laurent polynomial
ring
by $W.$$\mathbb{C}[q^{2z1}, q^{-2z}1, \ldots, q^{2z_{n}}, q^{-2z_{n}}]$By using a result on spherical functions on the space of hermitian forms, we obtainthe following results.

Theorem 2.1 The function$G_{1}(z)\cdot\omega(x;s)$

is invariant under the action of$G_{1}(z)= \prod_{1\leq i<j\leq n}\frac{1+q^{z_{i}-z_{j}}}{1-q^{z_{l}’-z_{j}-1}}$$S_{n}$

on , where
$z$

(2.9)

.

Hereafter till the end of \S 4, we assume has odd residual characteristic, i.e.
is nondyadic and is odd.
Next we study the functional equation with respect to . To begin with, based on
Proposition 1.1, we calculate
explicitly and obtain the following.
$k$

$k$

$q$

$\tau$$\omega^{(1)}(x_{\ell};s)$

Proposition 2.2 For $n=1$ , the spherical function
and satisfies the functional equation$\omega^{(1)}(x;s)$$s\in \mathbb{C}$$\omega^{(1)}(x;s)=\omega^{(1)}(x;-s)$.

is holomorphic for any

118
Theorem 2.3 For general size , the spherical function satlsfies the functional equation$n$$\omega(x;z)=\omega(x;\tau(z))$.

We assume $n\geq 2$ and introduce the following standard parabolic subgroup
to :

$P$

attached$\tau$$P=$$\alpha 0101$$\{(\begin{array}{llll} a b q’ c d q\end{array})$$q$$(1_{n-1}$$-\alpha^{*}j1_{n-1})(\begin{array}{llll}1_{n-1} \beta \gamma j 1 0 -\beta^{*}j 0 1 1_{n-1}\end{array})\in G|$

is upper triangular in $GL_{n-1}(k’),$$(\begin{array}{ll}a bc d\end{array})\in U(j_{2}), \alpha, \beta\in M_{n-1,1}(k’)$(2.10), $q’=jq^{*-1}j|$ ,$\gamma\in M_{n-1}(k’), \gamma+\gamma^{*}=0$

where $j=j_{n-1}$ and each empty place in the above expression means zero-entry.
$1\leq i\leq n-1$ are relative-invariants, but
The relative -invariants$\tilde{P}=P\crossGL_{1}(k’),\tilde{X}=X\cross
V$ with$X$
is not. So we enlarged
andas follows: Set$V=M_{21}(k’)$ , and
$B$

$P$

$d_{i}(x),$$d_{n}(x)$$P$$(p, r)\star(x, v)=(p\cdot x, \rho(p)vr^{-1}) , (p, r)\in\tilde{P}, (x, v)\in\tilde{X},$where$\rho(p)=(\begin{array}{ll}a bc d\end{array})$

for the decomposition of $p\in P$ as in (2.10).

Then we have the following relative -invariant $g(x;v)$ on$\tilde{P}$$\tilde{X}$instead of$d_{n}(x)$.

Lemma 2.4 Set$g(x, v)=\det[(\begin{array}{lll}-v_{2} v_{1} 00 1_{n-1}\end{array})\cdot x_{(n+1)}], (x, v)\in\tilde{X}, v=(\begin{array}{l}v_{i}c_{2}\end{array}),$is the lower right $(n+1)$ by $(n+1)$ block of . Then(i) $g(x, v)$ is a relative -invariant on
associated with the -rational chamcter$\tilde{\psi}(p, r)=\psi_{n-1}(p)N(r)^{-1}$$g(x,v_{0})=d_{n}(x)$, andwith $v_{0}=t(10)$ .
(ii) $g(x, v)$ is expressed as $g(x, v)=D(x)[v]$ by some hermitian matrix $D(x)$ of size 2.
For $x\in X^{op},$ $D_{1}(x)=d_{n-1}(x)^{-1}D(x)$ belongs to

where

$x$

$x_{(n+1)}$$\tilde{P}$$\tilde{X}$$\tilde{P}$$X_{1}.$

By the embedding$K_{1}=U(j_{2})\hookrightarrow K=K_{n}, h\mapsto\tilde{h}=(1_{n-1} h 1_{n-1})$(2.11),

we have$\omega(x;s) = \int_{K_{1}}dh\int_{K}|d(k\cdot x)|^{s+\epsilon}dk$$= \int_{K}\prod_{i<n}|d_{i}(k\cdot x)|^{s_{i}+\epsilon_{i}}\int_{K_{1}}|d_{n}(\tilde{h}k\cdot x)|^{s_{n}+\epsilon_{n}}dhdk$.

(2.12)

119
By Lemma 2.4, we see$d_{n}(\overline{h}\cdot y) = g(\tilde{h}\cdot y_{)}v_{0})=g((\tilde{h}, 1)\star(y, h^{-1}v_{0}))$$= d_{n-1}(y)d_{1}((h^{*-1}\cdot D_{1}(y))^{-1})=d_{n-1}(y)d_{1}(h\cdot D_{1}(y)^{-1})$,

and we obtain$\omega(x;s) = \int_{K}\prod_{i<n}|d_{i}(k\cdot x)|^{s_{i}+\epsilon_{i}}|d_{n-1}(k\cdot x)|^{s_{n}+\epsilon_{n}}\omega^{(1)}(D_{1}(k\cdot x)^{-1};s_{n})dk.$

Then, by Proposition 2.2, we obtain$\omega(x;s) = \omega(x;s_{1}, \ldots, s_{n-2}, s_{n-1}+2s_{n}, -s_{n})$,which shows in -variable$z$$\omega(x;z)=\omega(x;\tau(z)) , \tau(z)=(z_{1}, \ldots, z_{n-1}, -z_{n})$,and we conclude the proof of Theorem 2.3.

1

In order to describe functional equations of, we prepare some notations. We
denote by the set of roots of with respect to the maximal -split torus of containedinand bythe set of positive roots with respect to . We may understand as a
, and set
subset in$\omega(x;z)$$G$$\Sigma$

$B$

$G$

$k$

$\Sigma^{+}$$B$$\Sigma$$\mathbb{Z}^{n}$$\Sigma^{+}=\Sigma_{s}^{+}\cup\Sigma_{p}^{+}$(2.13),$\Sigma_{S}^{+}=\{e_{i}-e_{j}, e_{i}+e_{j}|1\leq i<j\leq n\}, \Sigma_{\ell}^{+}=\{2e_{i}|1\leq i\leq n\},$$1\leq i\leq n$ . Then
where is the i-th unit vector in
dominant weights. We define a pairing onby$\mathbb{Z}^{n},$$e_{i}$$\Lambda_{n}^{+}$

can be regarded as the set of$\mathbb{Z}^{n}\cross \mathbb{C}^{n}$$\langle t, z\rangle=\sum_{i=1}^{n}t_{i}z_{i}, (t\in \mathbb{Z}^{n}, z\in \mathbb{C}^{n})$,which satisfies$\langle\alpha, z\rangle=\langle\sigma(\alpha), \sigma(z)\rangle, (\alpha\in\Sigma, z\in \mathbb{C}^{n}, \sigma\in W)$.

By Theorem 2.1 and Theorem 2.3 and cocycle relations of Gamma factors, we havethe following.

Theorem 2.5 The spherical function
for each$\omega(x;z)$satisfies the following functional equation$\sigma\in W$$\omega(x;z)=\Gamma_{\sigma}(z)\cdot\omega(x;\sigma(z))$

,

where
$\Gamma_{\sigma}(z)=\prod_{\alpha\in\Sigma_{S}^{+}(\sigma)}\frac{1-q^{\langle\alpha,z)-1}}{q^{\langle\alpha,z\rangle}-q^{-1}}, \Sigma_{s}^{+}(\sigma)=\{\alpha\in\Sigma_{s}^{+}|-\sigma(\alpha)\in\Sigma^{+}\}.$

and we understand$\Gamma_{\sigma}(z)=1$

if$\Sigma_{s}^{+}(\sigma)=\emptyset.$(2.14)

120
Further we obtain the following in the similar line to the proof of Theorem 2.9 in [H5].
Theorem 2.6 The function$\prod_{\alpha\in\Sigma_{s}^{+}}\frac{1+q^{\langle\alpha,z\rangle}}{1-q^{\langle\alpha,z)-1}}\cdot\omega(x;z)$

is holomorphic for all

$z$

in

$\mathbb{C}^{n}$

and $W$ -invariant, in particular it is an element in$\mathbb{C}[q^{\pm z_{1}})\ldots, q^{\pm z_{n}}]^{W}.$\S 3 The explicit formula for$\omega(x;z)$. Since $\omega(x;z)$ is stable on each $K$ -orbit, it is
We give the explicit formula ofby Theorem 1.2.
enough to show the explicit formula for each$\omega(x;z)$$x_{\lambda},$Theorem 3.1 For$\lambda\in\Lambda_{n}^{+}$$\lambda\in\Lambda_{n}^{+}$, one has the explicit formula:$\omega(x_{\lambda};z) = \frac{(1-q^{-2})^{n}}{w_{2n}(-q^{-1})}\cdot\prod_{\alpha\in\Sigma_{s}^{+}}\frac{1-q^{\langle\alpha,z\rangle-1}}{1+q^{\langle\alpha,z\rangle}}\cdot c_{\lambda}\cdot Q_{\lambda}(z)$

,

(3.1)

where
$\prime U)_{m}(t)=\prod_{i=1}^{m}(1-t^{i}) , c_{\lambda}=(-1)^{\Sigma_{i}\lambda_{i}(n-i+1)}q^{-\Sigma_{i}\lambda_{i}(n-i+\frac{1}{2})},$,$Q_{\lambda}(z)= \sum_{\sigma\in W}\sigma(q^{-\langle\lambda,z\rangle}c(z))$$c(z)= \prod_{\alpha\in\Sigma_{s}^{+}}\frac{1+q^{\langle\alpha,z\rangle-1}}{1-q^{\langle\alpha,z\rangle}}\prod_{\alpha\in\Sigma_{\ell}^{+}}\frac{1-q^{\langle\alpha,z\rangle-1}}{1-q^{\langle\alpha,z\rangle}}$

.

(3.2)

belongs toofRemark 3.2 We see that the main partby Theorem 2.6. On the other handis a specialization of Hall-Littlewood polynoup to constant multiple, which is introduced in a general context ofmialof typeorthogonal polynomials associated with root systems ([M2], \S 10), and $Q_{0}(z)$ is a specialization of Poincar\’e polynomial ([Ml],Th.2.8.). More precisely,$Q_{\lambda}(z)$$\omega(x_{\lambda};z)$$\mathcal{R}=\mathbb{C}[q^{\pm z_{1}}, \ldots, q^{\pm z_{n}}]^{W}$$Q_{\lambda}(z)$$P_{\lambda}$$C_{n}$$Q_{\lambda}(z)= \frac{\tilde{w_{\lambda}}(-q^{-1})}{(1+q^{-1})^{n}}\cdot P_{\lambda}(z)$

(3.3)

,

$\overline{w_{\lambda}}(t)=w_{mo(\lambda)}(t)^{2}\cdot\prod_{\ell\geq 1}w_{m\ell(\lambda)}(t) , m_{\ell}(\lambda)=\#\{i|\lambda_{i}=\ell\},$and it is known that the set$Q_{0}(z)$ is a constant independent of$\{Q_{\lambda}(z)|\lambda\in\Lambda_{n}^{+}\}$$z.$forms a -basis for$\mathbb{C}$$\mathcal{R}$, and in particular,

121
By Theorem 3.1 and Remark 3.2, we have the following corollary.
Corollary 3.3 For $x_{0}=1_{2n}$ , one has$\omega(1_{2n};z)=\frac{(1-q^{-1})^{n}w_{n}(-q^{-1})^{2}}{w_{2n}(-q^{-1})}\cross\prod_{\alpha\in\Sigma_{s}^{+}}\frac{1-q^{\langle\alpha,z\rangle-1}}{1+q^{\langle\alpha,z\rangle}}.$Theorem 3.1 is proved by using a general expression formula given in [H4] (or in [H2])of spherical functions on homogeneous spaces, which is based on functional equations offiner spherical functions and some data depending only on the group . We need tocheck the assumptions there. Let be a connected reductive linear algebraic group andbe an affine algebraic variety which is -homogeneous, where everything is assumed
field . For an algebraic set, we use the same ordinary
to be defined over aletter to indicate the set of -rational points. Let $K$ be a special good maximal compact
open subgroup of , and a minimal parabolic subgroup of defined over satisfying
the group of rational character of defined over
$G=KB=BK$ . We denote by
and bythe subgroup consisting of those characters associated with some relative-invariant on defined over . In these situation, the assumptions are the following:$G$$\mathbb{G}$$\mathbb{X}$$\mathbb{G}$$k$$1\succ adic$

$k$

$G$

$k$

$\mathbb{G}$$\mathbb{B}$$k$$\mathbb{B}$$\mathfrak{X}(\mathbb{B})$$\mathfrak{X}_{0}(\mathbb{B})$$k$$\mathbb{X}$$\mathbb{B}$$(A1)\mathbb{X}$

has only a finite number of -orbits$\mathbb{B}$$(A2)A$ basic set of relative

functions on$\mathbb{B}$-invariants onhence there $is only one open$ orbit$(,$

defined over$\mathbb{X}$$k$$\mathbb{X}^{op})$.

can be taken by regular$\mathbb{X}.$, there exists some inFor
whose restriction to the identity
component of the stabilizer
of at is not trivial.
$(A3)$$y\in \mathbb{X}\backslash \mathbb{X}^{op}$$\psi$$\mathfrak{X}_{0}(\mathbb{B})$$\mathbb{B}$$\mathbb{H}_{y}$$(A4)$The rank of$\mathfrak{X}_{0}(\mathbb{B})$$y$coincides with that of$\mathfrak{X}(\mathbb{B})$.

In the present situation, our space $X$ is isomorphic to $U(j_{2n})/U(1_{2n})$ over (cf. (1.5),
which is a symmetric space and $(A1)$ is satisfied. $(A2)$ and $(A4)$ are satisfied by ourandrelative -invariants $\{d_{i}(x)|1\leq i\leq n\}$ , where is the rank of$\mathbb{X}^{op}=\{x\in \mathbb{X}|d_{i}(x)\neq 0,1\leq i\leq n\}$ . To check (A3) is crucial and rather complicated.
It is proved by showing the existence of as above for each$\overline{k}$

$B$

$n$

$y\in \mathbb{X}\backslash \mathbb{X}^{op}.$$\psi$

According to thefollows$B$-orbit decomposition of$\omega_{u}(x;s)=\int_{K}|d(k\cdot x)|_{u}^{s+\epsilon}dk,$$\mathfrak{X}_{0}(\mathbb{B})=\mathfrak{X}_{0}(\mathbb{B})$$X^{op}$, we define finer spherical functions as$|d(y)|_{u}^{s}=\{\begin{array}{ll}\prod_{i=1}^{n}|d_{i}(y)|^{si} if y\in X_{u},0 otherwise.\end{array}$where$X^{op}=\lfloor\rfloor X_{u}u\in \mathcal{U}$’$\mathcal{U}=(\mathbb{Z}/2\mathbb{Z})^{n}$$X_{u}=\{x\in X^{op}|v_{\pi}(d_{i}(x))\equiv u_{1}+\cdots+u_{i} (mod 2), 1\leq i\leq n\}.$122
For each characterof , we may represent as follows$\mathcal{U}$$\chi$$\sum_{u\in u}\chi(u)\omega_{u}(x;s)=\omega(x;z_{\chi})$where$z_{\chi}$is obtained by adding$\frac{\pi\sqrt{-1}}{\log q}$to$z_{i}$(3.4),

for suitableaccording to$i$linearly independent (for generic z) as varying charactersfor each character of and$\chi$$\chi$, and they are. By Theorem 2.5, we have,$\sigma\in W,$$\mathcal{U}$$\chi$$\omega(x;z_{\chi}) = \Gamma_{\sigma}(z_{\chi})\omega(x;\sigma(z_{\chi}))$$= \Gamma_{\sigma}(z_{\chi})\omega(x;\sigma(z)_{\sigma(\chi)})$

(3.5)

,

by taking a suitable characterof . If is trivial character 1, then (3.5) coincides
with the original one. Further we obtain vector-wise functional equations as follows$\sigma(\chi)$$\mathcal{U}$$\chi$$(\omega_{u}(x;z))_{u\in \mathcal{U}}=A^{-1}\cdot G(\sigma, z)\cdot\sigma A\cdot(\omega_{u}(x;\sigma(z)))_{u\in u}, \sigma\in W$

(3.6)

,

where
,

$A=(\chi(u))_{\chi,u}, \sigma A=(\sigma(\chi)(u))_{\chi,u}\in GL_{2^{n}}(\mathbb{Z})$runs over characters of, andis the diagonal matrix of size
whose-component is. Here we fix the first entry of to be 1. Applying Theorem 2.6
in [H4] to our present case, we obtain for generic , by virtue of (3.6),$\chi$$(\chi, \chi)$$\mathcal{U},$$u\in \mathcal{U}$$G(\sigma, z)$$2^{n}$$\Gamma_{\sigma}(z_{\chi})$$\chi$$z$$( \omega_{u}(x_{\lambda};z))_{u\in u}=\frac{1}{Q}\sum_{\sigma\in W}\gamma(\sigma(z))(A^{-1}\cdot G(\sigma, z)\cdot\sigma A)(\delta_{u}(x_{\lambda},\sigma(z)))_{u\in u}$

where, taking$U$

as the Iwahori subgroup of$K$

compatible with $B,$

$Q= \sum_{\sigma\in W}[U\sigma U:U]^{-1}=\frac{w_{2n}(-q^{-1})}{(1-q^{-2})^{n}},$$\gamma(z)=\prod_{\alpha\in\Sigma_{s}^{+}}\frac{1-q^{2\langle\alpha,z\rangle-2}}{1-q^{2\langle\alpha,z\rangle}}\cdot\prod_{\alpha\in\Sigma_{\ell}^{+}}\frac{1-q^{\langle\alpha.z\rangle-1}}{1-q^{\langle\alpha,z\rangle}},$$\delta_{u}(x_{\lambda};z)=\{\begin{array}{ll}c_{\lambda}q^{-\langle\lambda,z\rangle} if x_{\lambda}\in X_{u}0 otherwise.\end{array}$Then, we obtain$\omega(x_{\lambda};z) = \sum_{u\in u}1(u)\omega_{u}(x_{\lambda};z)$$=$

the first entry of$A(\omega_{u}(x_{\lambda};z))_{u\in u}$$= \frac{(1-q^{-2})^{n}}{w_{2n}(-q^{-1})}\cross\sum_{\sigma\in W}\gamma(\sigma(z))\Gamma_{\sigma}(z)\sum_{u}\delta_{u}(x_{\lambda}, \sigma(z))$$= \frac{c_{\lambda}(1-q^{-2})^{n}}{w_{2n}(-q^{-1})}\cross\sum_{\sigma\in W}\gamma(\sigma(z))\Gamma_{\sigma}(z)q^{-\langle\lambda.\sigma(z)\rangle}.$

,

(3.7)

123
Setting$G(z)= \prod_{\alpha\in\Sigma_{s}^{+}}\frac{1+q^{\langle\alpha,z\rangle}}{1-q^{\langle\alpha,z\rangle-1}},$

we have$\Gamma_{\sigma}(z)=\frac{G(\sigma(z))}{G(z)}$(by Theorem 2.6),$\gamma(z)\cdot G(z)=c(z)$ .

Thus we obtain the required explicit formula ofevery, sinceis a polynomial in$\omega(x_{\lambda};z)$$z\in \mathbb{C}^{n}$\S 4$G(z)\cdot\omega(x_{\lambda};z)$

for generic , and it is valid for$z$$q^{\pm z_{1}},$$\ldots$,I$q^{\pm z_{n}}.$Spherical Fourier transform and Plancherel formula on$\mathcal{S}(K\backslash X)$

We consider the Schwartz space$S(K\backslash X)${$=$$\varphi$:$Xarrow \mathbb{C}|$which is an-submodule ofby the characteristic function of$\mathcal{H}(G, K)$

left $K$ -invariant, compactly supported},by the convolution product and spanned
We introduce a modified spherical function$C^{\infty}(K\backslash X)$$K\cdot x,$$x\in X$ .

(4.1)

,

$\Psi(x;z)=\omega(x;z)/\omega(1_{2n};z)\in \mathbb{C}[q^{\pm z_{1}}, \ldots, q^{\pm z_{n}}]^{W}(=\mathcal{R}, say)$and define the spherical Fourier transform$F$:$\mathcal{S}(K\backslash X)$$arrow$$\mathcal{R}$(4.2)$\varphi \mapsto F(\varphi)(z)=\int_{X}\varphi(x)\Psi(x;z)dx,$

where $dx$ is a -invariant measure on $X$ . There is a -invariant measure on $X$ , since $X$ is adisjoint union of two -orbits, and is reductive. We don’t need to fix the normalizationof $dx$ at this moment, we will determine suitably afterward(cf. Theorem 4.5). We denoteby $v(K\cdot y)$ the volume of-module through theby $dx$ . We regard
as anSatake isomorphism
$G$

$G$

$G$

$G$

$K\cdot y$$\mathcal{R}$$\mathcal{H}(G, K)$$\lambda_{z}:\mathcal{H}(G, K)arrow^{\sim}\mathbb{C}[q^{\pm 2z_{1}}, \ldots, q^{\pm 2z_{n}}]^{W}(=\mathcal{R}_{0}, say)$

By Theorem 3.1, Corollary 3.3 and (3.3), we see, for any$\Psi(x_{\lambda};z)=c_{\lambda}w_{\lambda}P_{\lambda}(z)$.

$\lambda\in\Lambda_{n}^{+},$,$F(ch_{\lambda})(z)=v(K\cdot x)\Psi(x_{\lambda};z)=c_{\lambda}w_{\lambda}v(K\cdot x_{\lambda})P_{\lambda}(z)$where$c_{\lambda}$

is the same as in Theorem 3.1,$ch_{\lambda}$

is the characteristic function of$w_{\lambda}= \frac{\overline{w_{\lambda}}(-q^{-1})}{w_{n}(-q^{-1})^{2}}.$

(4.3)

,
$K\cdot x_{\lambda}$

, and

124
is an
Theorem 4.1 The spherical Fourier tmnsform$mnk2^{n}.$-module ofin particular
is a free$F$

Proof. Since$\{ch_{\lambda}|\lambda\in\Lambda_{n}^{+}\}$$\mathcal{H}(G, K)$-module isomorphism,$\mathcal{H}(G, K)$$\mathcal{S}(K\backslash X)$$\{P_{\lambda}(z)|\lambda\in\Lambda_{n}^{+}\}$

forms a -basis for$\mathbb{C}$forms a -basis for( $cf$ . Remark 3.2) and( . Theorem 1.2), $F$ is bijective by (4.3). It$\mathbb{C}$$S(K\backslash X)$$\mathcal{R}$$cf$

is easy to check,$F(f* \varphi)=\lambda_{z}(f)F(\varphi) , (\int\in \mathcal{H}(G, K). \varphi\in\mathcal{S}(K\backslash X))$and we seerank$S(K\backslash X)$

is a free$\mathcal{H}(G, K)$-module of rank

$2^{n}$, since$\mathcal{R}$is a free$\mathcal{R}_{0}$-module ofI$2^{n}.$As a corollary we have the following.
Theorem 4.2 All the spherical functions on $X$ are pammetnzed by eigenvalues $z\in$through. The set $\{\Psi(x;z+u)|u\in\{0, \pi\sqrt{-1}/\log q\}^{n}\}$ forms abasis of the space of spherical functions oncorresponding to$( \mathbb{C}/\frac{2\pi\sqrt{-1}}{\log q}\mathbb{Z})^{n}/W$$\lambda_{z}(f)$$X$

In order to give the Plancherel formula on
on by

$z.$

$S(K\backslash X)$, we introduce an inner product$\mathcal{R}$$\langle P, Q\rangle_{\mathcal{R}}=\int_{\mathfrak{a}^{*}}P(z)\overline{Q(z)}d\mu(z) , (P, Q\in \mathcal{R})$.

Here$\mathfrak{a}^{*}=\{\sqrt{-1}(\mathbb{R}/\frac{2\pi}{\log q}\mathbb{Z})\}^{n}$and the measure $d\mu=d\mu(z)$ on$\mathfrak{a}^{*}$

is given by$d \mu=\frac{1}{n!2^{n}}\cdot\frac{w_{n}(-q^{-1})^{2}}{(1+q^{-1})^{n}}\cdot\frac{1}{|c(z)|^{2}}dz,$

where $c(z)$ is defined in (3.2) and and $dz$ is the Haar measure on. Then,withthe following lemma is essentially reduced to a result of Macdonald([M2], \S 10).
$\mathfrak{a}^{*}$

Lemma 4.3 For$\lambda,$$\mu\in\Lambda_{n}^{+}$, one has$\langle P_{\lambda}, P_{\mu}\rangle_{\mathcal{R}}=\langle P_{\mu}, P_{\lambda}\rangle_{\mathcal{R}}=\delta_{\lambda,\mu}w_{\lambda}^{-1}$

On the other hand, one may obtain
Lemma 4.4 For$\lambda,$$\mu\in\Lambda_{n}^{+}$

such that $|\lambda|\equiv|\mu|(mod 2)$ ,$\underline{v(K\cdot x_{\lambda})}=\frac{c_{\mu}^{2}w_{\mu}}{2}.$$v(K\cdot x_{\mu}) c_{\lambda}w_{\lambda}$$\int_{\mathfrak{a}^{*}}=1$

125
Since we may normalize $dx$ on $X$ according to$G$-orbits, we normalize as$v(K \cdot x_{0})=1, v(K\cdot x_{1})=q^{2n-1}\frac{(1-(-q^{-1})^{n})^{2}}{1+q^{-1}},$

where $x_{0}=1_{2n}$ and $x_{1}=Diag(\pi, 1, \ldots, 1, \pi^{-1})$ (cf. Theorem 1.4). Then we obtain
Theorem 4.5 (Plancherel formula onmeasure $dx$ such that

$S(K\backslash X)$) By the normalization of$G$-invariant$v(K\cdot x_{\lambda})=c_{\lambda}^{-2}w_{\lambda}^{-1}, \lambda\in\Lambda_{n}^{+},$

one has, for any$\varphi,\psi\in S(K\backslash X)$,$\int_{X}\varphi(x)\overline{\psi(x)}dx=\int_{a^{*}}F(\varphi)(z)\overline{F(\psi)(z)}d\mu(z)$Corollary 4.6 (Inversion formula) For any$\varphi\in S(K\backslash X)$.

,$\varphi(x)=\int_{a^{*}}F(\varphi)(z)\Psi(x;z)d\mu(z) , x\in X.$

Appendix
In [H5], we have considered, for each
$X_{T}=\mathfrak{X}_{T}/U(T)$,$T\in \mathcal{H}_{n}(k’)$$\mathfrak{X}_{T}=\{x\in M_{2n,n}(k’)|H_{n}[x]=T\},$$H_{n}=(\begin{array}{ll}0 1_{n}1_{n} 0\end{array}),$by the left multiplication, and the stabilizer atwhere $U(H_{n})$ acts homogeneously ona point in
is isomorphic to $U(T)\cross U(T)$ (cf. [H5] Lemma 1,1).
in Theorem 3.1 is the same as the explicit formula of
The explicit formula ofin Theorem 3.3 in [H5]. We explain
parametrized by
on
at
$X$
and
the relation between the spaces
as a set of -rational points in an algebraic
We assume is diagonal and realize
set defined over . We consider the space$X_{T}$$X_{T}$$\omega(x_{\lambda};z)$$\omega_{T}(y_{\lambda};z)$$X_{T}$$y_{\lambda}\in X_{T}$$\lambda\in\Lambda_{n}^{+}$$X_{T}’ s.$$T$$X_{T}$$k$$k$$\mathfrak{X}_{T}(\overline{k}) = \{(x, y)\in M_{2n,n}(\overline{k})\oplus M_{2n,n}(\overline{k})|tyH_{n}x=T\},$with the action of$\Gamma=Gal(\overline{k}/k)$given by$\sigma(x, y)=\{\begin{array}{l}(x^{\sigma}, y^{\sigma}) if \sigma|_{k’}=id(y^{\sigma}, x^{\sigma}) if \sigma|_{k’}=\tau,\end{array}$

126
where

$x^{\sigma}=(x_{ij^{\sigma}})$for $x=(x_{ij})$ and$\sigma\in\Gamma$, and $\langle\tau\rangle=Gal(k’/k)$ . We set$\mathbb{U}(H_{n})=U(H_{n})(\overline{k})=\{(g_{1}, g_{2})\in GL_{2n}(\overline{k})\cross GL_{2n}(\overline{k})|^{t}g_{2}H_{n}g_{1}=H_{n}\},$$\mathbb{U}(T)=U(T)(\overline{k})=\{(h_{1}, h_{2})\in GL_{n}(\overline{k})\crossGL_{n}(\overline{k})|th_{2}Th_{1}=T\},$$\mathbb{X}_{T}(\overline{k})=\mathfrak{X}_{T}(\overline{k})/\mathbb{U}(T)\supset \mathbb{X}_{T}(k)=\mathbb{X}_{\tau^{\Gamma}},$where we may consider the similar-invariant. We identify$\Gamma$-action on$\mathbb{U}(H_{n})$and$\mathbb{U}(T)$, since$H_{n}$

and

$T$

are

$\Gamma$$\mathbb{U}(H_{n})^{\Gamma}=\{(g, \overline{g})\in GL_{2n}(k’)\cross GL_{2n}(k’)|t\overline{g}H_{n}g=H_{n}\}$with $U(H_{n})$ ,$\mathbb{U}(T)^{\Gamma}=\{(h,\overline{h})\in GL_{n}(k’)\cross GL_{n}(k’)|t\overline{h}Th=T\}$with $U(T)$ ,where and henceforth we write
injective map$\overline{g}$instead of$g^{\tau}$

for a matrix$g$with entries in$\varphi\tau:X_{T}arrow X_{T}(k), xU(T)\mapsto(x,\overline{x})\mathbb{U}(T)$we understandas a subspace of$X_{T}$$\mathbb{X}_{T}(k)$

,

. Set

$T_{1}=(\begin{array}{ll}\pi 00 1_{n-1}\end{array}), \tilde{\eta_{\pi}}=(\eta_{\pi}, \eta_{\pi}) , \eta_{\pi}=(^{\sqrt{\pi}^{-1}}0 1_{n-1}0)$.

Lemma A.l The map$f$:$\mathbb{X}_{T_{1}}(\overline{k})arrow \mathbb{X}_{1_{n}}(\overline{k}),$is well defined, it sends$\mathbb{X}_{T_{1}}(k)$into$(x, y)\mathbb{U}(T_{1})\mapsto(x\eta_{\pi}, y\eta_{\pi})\mathbb{U}(1_{n})$$\mathbb{X}_{1_{n}}(k)$and$f(X_{T_{1}})\neq X_{1_{n}}.$We set$\mathbb{N}=\{((\begin{array}{ll}h_{1} 00 k_{1}\end{array}), (\begin{array}{ll}h_{2} 00 k_{2}\end{array})) (h_{1}, h_{2}), (k_{1}, k_{2})\in U(1_{n})\}$

with the similar -action on$\Gamma$$\mathbb{N}$as before, and identify$\mathbb{N}^{\Gamma}$with$N=\{(\begin{array}{ll}h 00 k\end{array}) h, k\in U(1_{n})\}.$The stabilizer at $y_{0}U(1_{n})\in X_{1n}$ in $U(H_{n})$ , where$y_{0}= (\begin{array}{l}\xi 1_{n}\end{array}), \xi=\frac{1+\sqrt{c\prime}}{2},$is given by$vNv^{-1}, v=(\begin{array}{ll}\xi 1_{n} \overline{\xi}1_{n}1_{n} -1_{n}\end{array})\in GL_{2n}(\mathcal{O}_{k’})$On the other hand, the stabilizer at$1_{2n}$.

in $G=U(j_{2n})$ is given by$\mu N\mu^{-1}, \mu=(\begin{array}{ll}1_{n} 1_{n}j_{n} -j_{n}\end{array})\in GL_{2n}(k)$

.

$k’$

. By the127
Further we see$\mu\nu^{-1}U(H_{n})\nu\mu^{-1}=G,$$\mu\nu^{-1}\mathbb{U}(H_{n})v\mu^{-1}=G(\overline{k})$( . 1).

$A$

Thus we have a commutative diagram$\varphi$$\cong$$\mathbb{X}_{1_{n}}(\overline{k})$$\mathbb{U}(H_{n})/v\mathbb{N}\nu^{-1}$$arrow^{\sim}$$\cong$$G(\overline{k})/\mu \mathbb{N}\mu^{-1}$$X_{n}(\overline{k})$$\cup$$\cup$( .2)$A$$\varphi$$\mathbb{X}_{1_{n}}(k)$$\supset X_{1_{n}}$$U(H_{n})/\nu Nv^{-1}$$\cong$$arrow^{\sim}$$G/\mu N\mu^{-1}\cong$$G\cdot 1_{2n}$$X_{n},$$\subset$

where is the conjugation determined by ( .1). Then we have the following by ( .2),
Lemma A. 1, and Theorem 1.4.
$A$$\varphi$

Proposition A.2 The above$\varphi$$A$gives an isomorphism between the sets of -mtional points$k$$U(H_{n})\backslash \mathbb{X}_{1_{n}}(k)\cong G\backslash X_{n},$and $U(H_{n})$ -orbit decomposition$\mathbb{X}_{1_{n}}’(k)=X_{1_{n}}uf(X_{T_{1}})$

where$x_{0}$

and$x_{1}$;

are the representatives of -orbits in$G$

Under the assumption$q$$f(X_{T_{1}})\cong G\cdot x_{1},$$X_{1_{n}}\cong G\cdot x_{0},$is odd, we have$X_{n}$

given in Theorem 1.4.

$\mu\in GL_{2n}(\mathcal{O}_{k’})$. Hence we see$K’ :=U(H_{n})\cap GL_{2n}(\mathcal{O}_{k’})=v\mu^{-1}K\mu v^{-1},$

and the space$X_{T}$

inherits the Cartan decomposition ofTheorem A.3 Assume$k$$X_{n}$, and we have

has odd residual chamcteristic and take any$T\in \mathcal{H}_{n}(k’)$. Then’$\mathfrak{X}_{T}=\sqcup K’y_{\lambda}h_{\lambda}U(T)\lambda\in\Lambda^{+}\lambda\sim T^{n}$where$y_{\lambda}=(\begin{array}{l}\xi\pi^{\lambda}1_{n}\end{array})\in \mathfrak{X}_{\pi^{\lambda}},$$\lambda\sim T$

means that $|\lambda|\equiv v_{\pi}(\det(T))(mod 2)$ and guamntees the existence ofsatisfying$h_{\lambda}\in GL_{n}(k’)$$T=\pi^{\lambda}[h_{\lambda}].$

The above decomposition has been expected in [H5] Remark 4.2. In [H5], we have
known the disjointness of orbits in the right hand side by explicit formulas of sphericalfunctions, but we didn’t know they are enough. By Theorem A.3, we see thespherical Fourier transform
is isomorphic in [H5, Theorem 4.1.].
$\omega_{T}(y;z)$$F_{T}$

128

References[Bo] A. Borel: Linear Algebmic Groups, Second enlarged edition, Graduate Texts in
Mathematics 126, Springer, 1991.
[Hl] Y. Hironaka: Spherical functions of hermitian and symmetric forms III, T\^ohoku
Math. J. 40(1988), 651-671.

[H2] Y. Hironaka: Spherical functions and local densities on hermitian forms, J. Math.
Soc. Japan 51 (1999), 553–581.
[H3] Y. Hironaka: Functional equations of spherical functions on p–adic homogeneous
spaces, Abh. Math. Sem. Univ. Hamburg 75(2005), 285–311.

[H4] Y. Hironaka: Spherical functions on p–adic homogeneous spaces, “Algebraic and Analytic Aspects of Zeta Functions and -functions”-Lectures at the French-Japanese
Winter School (Miura, 2008)-, MSJ Memoirs 21(2010), 50–72.
$L$[H5] Y. Hironaka: Spherical functions on $U(2n)/(U(n)\cross U(n))$ and hermitian Siegel
series, “Geometry and Analysis of Automorphic Forms of Seveml Variables”, Series
on Number Theory and Its Applications 7, World Scientific, 2011, 120–159.
[HK] Y. Hironaka and Y. Komori: Spherical functions on the space of p–adic unitary
hermitian matrices, in preparation.
[HS] Y. Hironaka and F. Sato: The Siegel series and spherical functions on $O(2n)/(O(n)\cross$$O(n))$ , “Automorphic forms and zeta functions” –Proceedings of the conference inmemory of Tsuneo Arakawa-, World Scientific, 2006, 150–169.
[Jac] R. Jacobowitz: Hermitian forms over local fields, Amer. J. Math.84(1962), 441

$-$

465.
[K]

S. Kato: On eigenspace of the Hecke algebra with respect to a good maximal compact
subgroup of a -adic reductive group, Math. Ann. 257(1981), 1–7.
$p$[Ml] I. G. Macdonald: The Poincar\’e series of a Coxester group, Math. Ann. 199(1972),

161–174.
[M2] I. G. Macdonald: Orthogonal polynomials associated with root systems, S\’eminaire
Lotharingien de Combinatoire 45(2000), Article B45a.

[Om] O. T. O’Meara: Introduction to quadmtic forms, Grund. math. Wiss. 117, SpringerVerlag, 1973.
[Se] J. P. Serre: Galois cohomology, Springer-Verlag, 1997. (English translation of “ Cohomologie Galoisienne”, 1964.)