数理解析研究所講究録
第 1659 巻 2009 年 141-156

141

Spherical functions on $U(n, n)/(U(n)\cross U(n))$ andhermitian Siegel series
Yumiko Hironaka\S 0

Introduction

Let be an unramified quadratic extension over a non-archimedian local filed of characteristic . We fix a prime element of , and the additive value
and the normalized
absolute value
on , where $|\pi|^{-1}=q$ is the cardinality of the residue class field of .
We consider hermitian matrices with respect to the involution onwhich is identity
on , and set
$k’$

$k$

$0$

$\pi$

$k$

$v_{\pi}()$$k^{x}$

$||$

$k$

$k’$

$*$

$k$

$\mathcal{H}_{m}=\{A\in M_{m}(k’)|A^{*}=A\}$,$\mathcal{H}_{m}^{nd}=\mathcal{H}_{m}\cap GL_{m}(k’)$

,

(0.1)

where, for a matrix $A=(a_{ij})\in M_{mn}(k’)$ , we denote by $A$ “ the matrix $(a_{ji^{*}})\in M_{nm}(k’)$ .
For, we define the spaces$T\in \mathcal{H}_{n}^{nd}$,$X_{T}=\{x\in M_{2n_{\{}n}(k’)|x^{*}H_{n}x=T\}$

where$X_{T}=X_{T}/U(T)$ ,and $U(T)=\{g\in GL_{n}(k’)|g^{*}Tg=T\}$ . We consider
spherical functions on, which is isomorphic to $U(n, n)/(U(T)\cross U(T))$ over , where$U(n, n)=U(H_{n})$ (cf. Lemma 1.1). We consider the following integral$H_{n}=(\begin{array}{ll}0 1_{n}1_{n} 0\end{array})\in \mathcal{H}_{2n}$$X_{T}$$k$$\omega_{T}(\overline{x};s)=\int_{K}|f_{T}(kx)|^{s+\epsilon}dk$,$(\overline{x}\in X_{T}, s\in \mathbb{C}^{n})$

.

(0.2)

Here $dk$ is the normalized Haar measure on $K=U(n, n)\cap GL_{2n}(\mathcal{O}_{k’})$ ,$\epsilon=(-1, \ldots, -1, -\frac{1}{2})+(\frac{\pi\sqrt{-1}}{\log q}, \ldots, \frac{\pi\sqrt{-1}}{\log q})\in \mathbb{C}^{n}$$|f_{T}(x)|^{s}= \prod_{i=1}^{n}|d_{i}(x_{2}T^{-1}x_{2}^{*})|^{s_{i}}$

,

,

where
is the lower half byblock of $x\in X_{T}$ andis the determinant of
the upper left by block of . The right hand side of (0.2) is absolutely convergent$n$$x_{2}$

$i$

$i$

$n$

$d_{i}(y)$$y$2000 Mathematics Subject Classification: Primary llF85; secondly llE95, llF70, $22E50$ .
Key words and phrases: spherical functions, unitary groups, hermitian Siegel series.

This research is partially supported by Grant-in-Aid for scientific Research $(C):20540029$ .
A full version of this article will appear elsewhere (cf. arXiv : 0904.4304).

142. continued to a rational function of
if ${\rm Re}(s_{i})\geq 1(1\leq i\leq n-1)$ and Rc, and becomes a common eigen function with respect to the action of Hecke. It iswith $G=U(n, n)$ ; thus we have a spherical function on
algebraconvenient to introduce the new variable which is related to by$(s_{n}) \geq\frac{1}{2}$$q^{s_{1}},$$\ldots,$$q^{9n}$$X_{7^{i}}$$\mathcal{H}(G, K)$

$s$

$z$

$(1\leq i\leq n-1)$ ,$s_{i}=-z_{i}+z_{i+1}$$s_{n}=-z_{n}$

(0.3)

,

. We denote by $W$ the Weyl group of
and we write
the maximal k-split torus in , which is is isomorphic tothe set of positive roots of
permutation of indices. We denote by
and write
the Borel group, and regard it a subset of$G$$\omega_{T}(\overline{x};z)=\omega_{T}(\overline{x};s)$$S_{n}\ltimes(C_{2})^{n},$$G$$\Sigma^{+}$$\mathbb{Z}^{n}$$\langle\alpha,$with respect tobyacts onwith respect tofor$S_{n}$$G$$z \rangle=\sum_{i=1}^{n}\alpha_{i}z_{i}$$z_{i}$$\alpha\in\Sigma^{+}$(for details, see \S 2.2).

Our main results in \S 1 and \S 2 are the following.
Theorem 1 (i) For any$T\in \mathcal{H}_{n}^{nd}$, the function$\prod_{1\leq i<j\leq n}\frac{(1+q^{z_{i}-z_{j}})}{(-q^{z_{i}-z-1})}\cross\omega_{T}(\overline{x};z)$$\iota s$

holomorphic for all

$z$

in

$\mathbb{C}^{n}$and$S_{n}- inva\tau\dot{n}$ant, and the function$|2|^{-z_{1}-z-z_{n}}2 \prod_{1\leq i<j\leq n}\frac{(1+q^{z_{i}-z}j)(1+q^{z_{i}+z_{j}})}{(1-q^{z_{i}-z_{j}-1})(1-q^{z_{i}+z_{j}-1})}\cross\omega_{T}(\overline{x};z)$also holomorphic for all.
in$\iota s$$z$

in

and W-invariant. In particular the latter is an element$\mathbb{C}^{n}$$\mathbb{C}[q^{\pm z_{1}}, \ldots, q^{\pm z_{n}}]^{W}$(ii) For any$T\in \mathcal{H}_{n}^{nd}$and$\sigma\in W$, the following functional equation holds$\omega_{2^{1}}(x;z)=\Gamma_{\sigma}(z)\cdot\omega_{T}(x;\sigma(z))$

(0.4)

,

where
,

$\Gamma_{\sigma}(z)=$$f_{\alpha}(t)=\{\begin{array}{ll}\frac{1-q^{t-1}}{q^{t}-q^{-1}} if \alpha is short|2|^{t} if \alpha is long\end{array}$$\prod_{\alpha\in\Sigma^{+},\sigma(\alpha)<0}f_{\alpha}(\langle\alpha, z\rangle)$

In \S 3, we give an explicit expression for$\omega_{T}(x_{T};s)$.

As an application, we consider the hermitian Siegel series in \S 4. For eachhermitian Siegel series $b_{\pi}(T;s)$ is defined by$b_{\pi}(T;s)= \int_{\mathcal{H}_{n}(k’)}\nu_{\pi}(R)^{-s}\psi(tr(TR))dR$,$T\in \mathcal{H}_{n}$

, the

(0.5)

, tr $()$ is the trace of matrix
of conductor
whereis an additive character on$R$ , which is certain non-negative powers of(cf.
is the “denominator” ofand(4.1) As for Siegel series (for symmetric matrices), F. Sato and the author have given$\psi$$\nu_{\pi}(R)$

$)$

$k$

$\mathcal{O}_{k}$

$q$

143
a new integral expression and related it to a spherical function on the symmetric space$O(2n)/(O(n)\cross O(n))$ (cf. [HS]). In the present paper we develop the similar argument
for hermitian Siegel series. Since we know well about the functional equations of sphericalfunctionswith respect to $W$ as above, we can bring out the functional equationofas an application; thus we obtain an integral expression ofand itsfunctional equation.
$\omega_{T}(\overline{x};s)$$b_{\pi}(T;s)$$b_{\pi}(T;s)$

Theorem 2(i) If ${\rm Re}(s)>2n$ , one has$b_{\pi}(T;s)= \zeta_{n}(k’;\frac{s}{2})^{-1}\cdot\int_{X_{T}(\mathcal{O}_{k},)}|N_{k’/k}(\det x_{2})|^{\frac{s}{2}-n}|\Theta_{T}|(x)$where$X_{T}(\mathcal{O}_{k’})=X_{T}\cap M_{2n,n}(\mathcal{O}_{k’}),$$M_{n}(k’)$, and(ii) For any$|\Theta_{T}|(x)$$T\in \mathcal{H}_{n}^{nd}$$\zeta_{n}(k’; )$

,

(0.6)

is the zeta function of the matrix algebm

is a certain normalized measure on$X_{T}$.

, one has,$\frac{b_{\pi}(T;s)}{\prod_{i=0}^{n-1}(1-(-1)^{i}q^{-s+i})}=\chi_{\pi}(\det T)^{n-1}|\det(T/2)|^{s-n}\cross\frac{b_{\pi}(T;2n-s)}{\prod_{i=0}^{n-1}(1-(-1)^{i}q^{-(2n-s)+i})}$where$\chi_{\pi}$is the chamcter on$k^{\cross}$

determined by$\chi_{\pi}(a)=(-1)^{v_{\pi}(a)}=|a|^{\frac{\pi\sqrt{-1}}{\log q}}$$a\in k$’.

We note here that the above functional equation is related to an element of the Weylgroup of $U(n, n)$ , which was not the case for symmetric case when is odd. The existenceof functional equation of $b_{\pi}(T;s)$ was known in an abstract form as functional equationsof Whitakker functions of a p-adic group by Karel [Kr](cf. also Kudla-Sweet [KS], Ikeda[Ik] .
$n$

$)$

\S 1
We follow the notations in the introduction. For$A\in \mathcal{H}_{m}$and $X\in M_{mn}(k’)$ , we write$A[X]=X^{*}AX=X^{*}\cdot A\in \mathcal{H}_{n}$then our spaces are given for each$T\in \mathcal{H}_{n}^{nd}$by$X_{T}=\{x\in M_{2n,n}(k’)|H_{n}[x]=T\}$ ,$x_{T}=(\begin{array}{l}\frac{1}{2}T1_{n}\end{array})\in x_{T}$,$X_{T}=X_{T}/U(T)$ ,

(1.1)

.

, as well as on
The group $G=U(n, n)$ acts on, through left multiplication, which
is transitive by Witt’s theorem for hermitian matrices (cf. [Sch], Ch.7, \S 9). Our firstobservation is the following.
$X_{T}$$X_{T}$

144
Lemma 1.1 The stabilizer subgroup of $G=U(n, n)$ at $x_{T}U(T)\in X_{T}$ is given as$\{\overline{T}^{-1}(\begin{array}{ll}h_{1}^{*} 00 h_{2}^{*}\end{array})\tilde{T}$In particular, the space$h_{1},$$h_{2}\in U(T)\}$,$- \frac{1}{2}T\frac{1}{2}T)\in GL_{2n}(k’)$$\tilde{T}=(1_{n}1_{n}$.

is $\iota somorphic$ to $G/(U(T)\cross U(T))$ .

$X_{T}$

We flx the Borel subgroup $B$ of

as

$G$

$a+a^{*}=0bisupper$$B=\{(\begin{array}{ll}b 00 b^{*-1}\end{array})(\begin{array}{ll}l_{n} a0 1_{n}\end{array})$

and introduce the B-relative invariants on$\psi_{i}$of$f_{T,i}(bx)=\psi_{i}(b)f_{T,i}(x)$

$\}$

,

(1.2)

$X_{T}$$f_{T,i}(x)=d_{i}(x_{2}T^{-1}x_{2}^{*})$

associated with k-rational characters

triangular of size $n,$,$B$

$1\leq i\leq n$

(1.3)

,

by
$\psi_{i}(b)=N(d_{i}(b))^{-1}$

(1.4)

,

is the determinant of upper left
is the lower half by block of $x\in X_{T},$whereby block of and $N=N_{k’/k}$ . Since $f_{T,i}(xh)=f_{T,i}(x)$ for any $h\in U(T)$ , we understand$1\leq i\leq n$ .
$f_{T,i}(x)$ as B-relative invariants on$i$$d_{i}(y)$

$n$

$n$

$x_{2}$$i$$y$$X_{T},$

Remark 1.2 It is possible to realize above objects as the sets of k-rational points ofalgebraic sets defined over and develop the arguments, but we take down to earth way foris isomorphic to $U(n, n)/(U(n)\cross U(n))$
simplicity of notations. We only note here that$\{x\in
X_{T}|f_{T,i}(x)\neq
0,1\leq i\leq n\}$ is a Zariski open
over the algebraic closure of andB-orbit over , where $U(n)=U(1_{n})$ .
$k$$X_{T}$$\overline{k}$$k$$\overline{k}$Hereafter, we write an element hi $=xU(T)$ in
by its representative infor
simplicity of notations. We set $|0|=0$ for the absolute value onfor convenience.
The modulus character on $B$ (which is characterized by $d_{l}(bb’)=\delta(b’)^{-1}d_{l}(b)$ for theon $B)$ is given byleft invariant measure$X_{T}$$x$$X_{T}$$k^{\cross}$$\delta$$d_{l}(b)$$\delta^{\frac{1}{2}}(b)=\prod_{i=1}^{n-1}|\psi_{i}(b)|^{-1}\cross|\psi_{n}(b)|^{-\frac{1}{2}}$.

Now we introduce the spherical function $\omega(x;s)$ on $X_{T}=X_{T}/U(T)$$\omega_{T}(x;s)=\omega_{T}^{(n)}(x;s)=\int_{K}|f_{T}(kx)|^{s+\epsilon}dk$

(1.5)

,

where $dk$ is the normalized Haar measure on $K=G\cap GL_{2n}(\mathcal{O}_{k}/),$$s\in \mathbb{C}^{n}$$\epsilon=(-1, \ldots, -1, -\frac{1}{2})+(\frac{\pi\sqrt{-1}}{\log q}, \ldots, \frac{\pi\sqrt{-1}}{\log q})\in \mathbb{C}^{n}$$f_{T}(x)= \prod_{i=1}^{n}f_{T,i}(x)$,,$|f_{T}(x)|^{s}= \prod_{i=1}^{n}|f_{T,i}(x)|^{s_{i}}$145
The right hand side of (1.5) is absolutely convergent if ${\rm Re}(s_{i})\geq 1(1\leq i\leq n-1)$
and, continued to a rational function of, and becomes a common
eigenfunction with respect to the action of the Hecke algebra(cf. [H2], \S 1).
Since we see${\rm Re}(s_{n}) \geq\frac{1}{2}$$q^{s1},$$\ldots,$$q^{s_{n}}$$\mathcal{H}(G, K)$$\omega_{T[h]}(x;s)=\omega_{T}(xh^{-1};s)$,$h\in GL_{n}(k’),$ $x\in X_{T[h]}$

,

(1.6)

it suffices to consider only for diagonal ’s for the study of functional properties of(e.g., Theorem 1 in the introduction),
We write $\omega_{T}(x;z)=\omega_{T}(x;s)$ for the new variable introduced by (1.7). The Weylgroup $W$ of $G$ relative to the maximal k-split torus in $B$ acts on rational characters of$B$
as usual (i.e.,
by taking a representativeof ), so $W$ acts on
and onas well. We will determine the functional equations ofwith respect to this Weyl group action. The group $W$ is isomorphic toacts on by permutation of indices and $W$ is generated by
and : $(z_{1}, \ldots, z_{n})\mapsto$$(z_{1}, \ldots, z_{n-1}, -z_{n})$ .
$T$$\omega_{T}(x;s)$$z$$\sigma(\psi)(b)=\psi(n_{\sigma}^{-1}bn_{\sigma})$$z\in \mathbb{C}^{n}$$\sigma$$n_{\sigma}$$s\in \mathbb{C}^{n}$$\omega_{T}(x;s)$$S_{n}\ltimes C_{2}^{n},$$z$$S_{n}$$S_{n}$$\tau$

By using a result on spherical functions on the space of hermitian forms (cf. [Hl]-\S 2
or [H3]-\S 4.2) , we obtain the following.
$($

$)$

Theorem 1.3 For any$T\in \mathcal{H}_{n}^{nd}$, the function$\prod_{1\leq i<j\leq n}\frac{q^{z_{j}}+q^{z_{i}}}{q^{z_{j}}-q^{z_{i}-1}}\cross\omega_{T}(x;z)$

is holomorphic for any$\mathbb{C}[q^{\pm z_{1}}, \ldots, q^{\pm z_{n}}]^{S_{n}}$

.

$z$

in

$\mathbb{C}^{n}$and$S_{n}$-invariant. In particular it is an element inRemark 1.4 For the transposition $\tau_{i}=(ii+1)\in W$ ,functional equation holds by Theorem 1.3$\omega_{T}(x;z)=\frac{1-q^{z-zi+1}i-1}{q^{z-z_{i+1}}i-q^{-1}}\cross\omega_{T}(x;\tau_{i}(z))$,$1\leq i\leq n-1$ ,$1\leq i\leq n-1$

the following

.

(1.7)

On the other hand, one can obtain (1.8) directly in the similar way to the case of\S 3, then Theorem 1.3 follows from (1.8).

$\tau$

in\S 2
2.1. We fix a unit$[Om]- 63.3$ and 63.4).

$\epsilon\in \mathcal{O}_{k}^{x}$

Theorem 2.1 For any
equation:

for which$T\in \mathcal{H}_{n}^{nd}$$k’=k(\sqrt{\epsilon})$and$\epsilon\in 1+4\mathcal{O}_{k}^{x}$

if

$k$

is dyadic (cf.

, the spherical function satisfies the following functional$\omega_{T}(x;z)=|2|^{2z_{n}}\omega_{T}(x;\tau(z))$.

146
The case $n=1$ is easy; we calculate spherical functions explicitly for representatives, where $K_{1}=U(H_{1})\cap GL_{2}(\mathcal{O}_{k’})$ , and obtain the functional equation.
-orbits inofof $\tau\in W$ by
For $n\geq 2$ we take a representative$X_{T}$$K_{1}$$w_{\tau}$$w_{\tau}=(\begin{array}{llll}1_{n-1} 0 l 1 1_{n-1} 0\end{array})\in G$

and take the parabolic subgroup $P=P_{\tau}$ attached to$\tau$,(cf. [Bo], 21.11)$P=B\cup Bw_{\tau}B$$\{(qac+q^{*-}bd)(\begin{array}{llll}1_{n-1} \alpha 1 1_{n-1} -\alpha^{*} 1\end{array})(\begin{array}{lll}1_{n} B \beta -\beta^{*} 0 l_{n} \end{array})\in G|$

$=$

$q$

is upper triangular in $GL_{n-1}(k’)$ ,$(\begin{array}{ll}a bc d\end{array})\in U(1.1),$$\alpha,$$\beta\in M_{n-1,1}(k’),$

$\}$

,

(2.1)

$B+B^{*}=0$$B\in M_{n-1}(k’),$where each empty place in the above expression means zero-entry. Hereafter we fix a, and write $f_{i}(x)=f_{T,i}(x)$ by abbreviating the suffix $T$ . The B-relative
diagonal
invariants $f_{i}(x)$ become P-relative invariants associated with except $i=n$ . We consider
the following action of $\tilde{P}=P\cross GL_{1}$ on $\tilde{X}_{T}=X_{T}\cross V$ with $V=M_{21}(k’)$ :$T\in \mathcal{H}n^{d}$$\psi_{i}$$(p, r)\cdot(x, v)=(px, \rho(p)vr^{arrow 1})$ ,where$\rho(p)=(\begin{array}{ll}a bc d\end{array})$

for the decomposition of $p\in P$ as in (2.1). For$g(x, v)$is the lower half
where
Then we obtain$x_{2}$$\det[(1_{n-l} t_{v})(\begin{array}{l}x_{2}-y\end{array})\cdot T^{-1}]$

$=$

by

$n$

$n$

Lemma 2.2 $g(x, v)$ is a relative

block of$\tilde{P}$$x$(the same before) and-invariant on$\overline{X_{T}}$$(x, v)\in\tilde{X}_{T}$

, set

,
$y$

is the n-th row of .
$x$

associated with character$\tilde{\psi}(p, r)=N(d_{n-1}(p))^{-1}N(r)^{-1}=\psi_{n-1}(p)N(r)^{-1}$,$(p, r)\in\tilde{P}=P\cross GL_{1}$,satisfies$g(x, v_{0})=f_{n}(x)$ ,$v_{0}=(\begin{array}{l}l0\end{array})$,

and is expressed as$g(x, v)=D(x)[v]$ ,with some hermitian matrix$D(x)=(\begin{array}{ll}a b+c\sqrt{\epsilon}b-c\sqrt{\epsilon} d\end{array})$,$a,$ $c,$ $d\ink,$$b=- \frac{1}{2}f_{n-1}(x),$$ad=b^{2}-c^{2}\epsilon$

.

(2.2)

147
In order to prove Theorem 2.2, we need the functional equation of the following function$\zeta_{K_{1}}(A;s)=\int_{K_{1}}|d_{1}(h\cdot A)|^{s-\frac{1}{2}}dh$where $dh$ is the normalized Haar measure on,$K_{1}$$(A\in \mathcal{H}_{2}, s\in \mathbb{C})$,.

Lemma 2.3 Let $x\in X_{T}$ such that $f_{T}(x)\neq 0$ and $D(x)$ be given by (2.4). Then one has$\zeta_{K_{1}}(D(x), s)=|2|^{-2s}|f_{n-1}(x)|^{2s}\zeta_{K_{1}}(D(x), -s)$.

Now Theorem 2.2 is proved as follows. By the embedding$KJarrow K=K_{n}$ ,$h=(\begin{array}{ll}a bc d\end{array})\mapsto\tilde{h}=(\begin{array}{llll}l_{n-1} a b c 1_{n-1} d\end{array})$,

we have$\omega_{T}(x;s)$$=$$\int_{K_{1}}dh\int_{K}|f(\tilde{h}kx)|^{s+\epsilon}dk$$=$$\int_{K}\chi_{\pi}(\prod_{i<n}f_{i}(kx))\prod_{i<n}|f_{i}(kx)|^{s_{i}-1}(\int_{K_{1}}\chi_{\pi}(f_{n}(\tilde{h}kx))|f_{n}(\tilde{h}kx)|^{s_{n}-\frac{1}{2}}dh)dk$.

By definition of $f_{n}(x)$ and $g(x, v)$ and Lemma 2.3, we see$f_{n}(\tilde{h}x)$$=$$g(x, (\begin{array}{l}d-c\end{array}))=D(x)[(\begin{array}{l}d-c\end{array})]=d_{1}(h^{*-1}\cdot D(x))$,$(h\in K_{1})$,hence we have$\omega_{T}(x;s)=\int_{K}\chi_{\pi}(\prod_{i<n}f_{i}(kx))\prod_{i<n}|f_{i}(kx)|^{s_{i}-1}\zeta_{K_{1}}(D(kx);s_{n}+\frac{\pi\sqrt{-1}}{\log q})dk$

Then the functional equation of.

follows from Lemma 2.4.

$\omega_{T}(x;s)$12.2. We denote by the set of roots of with respect to the k-split torus of containedin and by
the set of positive roots with respect to . We may understand$G$$\Sigma$$B$$G$$\Sigma^{+}$$B$$\Sigma^{+}=\{e_{i}-e_{j}, e_{i}+e_{j}|1\leq i<j\leq n\}\cup\{2e_{i}|1\leq i\leq n\}$where$e_{i}\in \mathbb{Z}^{n}$whose j-th component is given by the Kronecker delta,$\delta_{ij}$, and the set$\Sigma_{0}=\{e_{i}-e_{i+1}|1\leq i\leq n-1\}\cup\{2e_{n}\}$forms the set of simple roots. We denote by the subset of $W$ consisting of the reflections
associated to elements in . Then $\triangle=\{\tau_{i}|1\leq i\leq n-1\}\cup\{\tau\}$ generates $W$ . We write$\alpha<0$ if
is negative. We see the pairingongiven by$\triangle$$\Sigma_{0}$$\alpha\in\Sigma$$\langle,$$\langle\alpha,$$z \rangle=\sum_{i=1}^{n}\alpha_{i}z_{i}$is W-invariant. Then we obtain,$\rangle$$\Sigma\cross \mathbb{C}^{n}$$(\alpha\in\Sigma, z\in \mathbb{C}^{n})$.

148
and $\sigma\in W$ , the sphencal function
Theorem 2.4 For
following functional equation$T\in \mathcal{H}_{n}^{nd}$$\omega_{T}(x;z)=\Gamma_{\sigma}(z)\cdot\omega_{T}(x;\sigma(z))$

satisfies the$\omega_{T}(x;z)$

(2.3)

,

where
,

$\Gamma_{\sigma}(z)=$$\prod_{\alpha\in\Sigma^{+},\sigma(\alpha)<0}f_{\alpha}(\langle\alpha, z\rangle)$$f_{\alpha}(t)=\{\begin{array}{ll}|2|^{t} if \alpha=2e_{i} for some i\frac{1-q^{t-1}}{q^{t}-q^{-1}} otherwise,\end{array}$in particular, the Gamma factor

Pmof. For an element ofIn general, assume that$\sigma\in W$$\Gamma_{\sigma}(z)$does not depend on, we know the Gamma factor by (1.8) and Theorem 2.2.
has the shortest expression

associted by someW-invariant,
we haveis$\alpha_{i}\in\Sigma_{0}$$\sigma_{i}\in\Delta$$(,$$\rangle$$\Gamma_{\sigma}(z)$Hence$\Gamma_{\sigma}(z)$$x$$\Delta$$\sigma=\sigma\ell\cdots\sigma_{1}$

with
and

nor .

$T$

,

. Since the Gamma factors satisfy cocycle relations$=$$\Gamma_{\sigma_{\ell}}(\sigma_{\ell-1}\cdots\sigma_{1}(z))\cdots\Gamma_{\sigma 2}(\sigma_{1}(z))\cdot\Gamma_{\sigma 1}(z)$$=$$f_{\alpha_{\ell}}(\langle\alpha_{\ell}, \sigma_{\ell-1}\cdots\sigma_{1}(z)\rangle)\cdots f_{\alpha}2(\langle\alpha_{2}, \sigma_{1}(z)\rangle)\cdot f_{\alpha_{1}}(\langle\alpha_{1}, z\rangle)$$=$$f_{\alpha\ell}(\langle\sigma_{1}\cdots\sigma_{\ell-1}(\alpha_{\ell}), z\rangle)\cdots f_{\alpha 2}(\langle\sigma_{1}(\alpha_{2}), z\rangle)f_{\alpha_{1}}(\langle\alpha_{1}, z\rangle)$.

has the required form, since we have$\{\alpha\in\Sigma^{+}|\sigma(\alpha)<0\}=\{\sigma_{1}\cdots\sigma_{k-1}(\alpha_{k})|1\leq k\leq\ell\}$.
1Corollary 2.5 Set $\rho\in W$ by$\rho(z_{1}, \ldots, z_{n})=(-z_{n}, -z_{n-1}, \ldots, -z_{1})$

.

(2.4)

Then
$\Gamma_{\rho}(z)=|2|^{2(z_{1}+\cdots+z_{n})}\prod_{1\leq i<j\leq n}\frac{1-q^{z_{l}+z_{j}-1}}{q^{z+z_{j}}i-q^{-1}}$

.

(2.5)

Remark 2.6 The above gives the functional equation of the hermitian Siegel series(cf. \S 4), and it is interesting that such corresponds to the unique automorphism of the, which was pointed out byextended Dynkin diagram of the root system of type
Y. Komori.
$\rho$$\rho$$(C_{n})$By Theorem 1.3 and Theorem 2.5, we obtain the following.

149
Theorem 2.7 Set$F(z)= \prod_{\alpha\in\Sigma^{+}}g_{\alpha}(z)$where, for$\alpha\in\Sigma$,,$g_{\alpha}(z)=\{\begin{array}{ll}|2|^{-\frac{\langle\alpha,z\rangle}{2}} if \alpha=\pm 2e_{i} for some i\frac{1+q^{\langle\alpha,z\rangle}}{1-q^{\langle\alpha,z\rangle-1}} otherwise\end{array}$, the function $F(z)\omega_{T}(x;z)$ is holomorphic for all
Then, for any.
W-invariant. In particular it is an element in$T\in \mathcal{H}_{n}^{nd}$

in

$z$

$\mathbb{C}^{n}$and$\mathbb{C}[q^{\pm z}1, \ldots , q^{\pm z_{n}}]^{W}$

since

Pmof.

Take any

associated byand$W$
generates . Set$\sigma\in\Delta$$\triangle$. Then $F(z)\omega_{T}(x;z)$ is -invariant,. Thus, $F(z)\omega_{T}(x, z)$ is W$\sigma$$\Gamma_{\sigma}(z)=g_{-\alpha}(z)/g_{\alpha}(z)$$g_{\alpha}(\sigma z)=g_{\sigma\alpha}(z)=g_{-\alpha}(z)$invariant, since$\alpha\in\Sigma_{0}$$F_{1}(z)= \prod_{1\leq i<j\leq n}\frac{1+q^{z_{i}-z_{j}}}{1-q^{z-z_{j}-1}i}$,$F_{2}(z)=|2|^{-z-z_{n}}1 \prod_{1\leq i<j\leq n}\frac{1+q^{z+z}ij}{1-q^{z_{i}+z_{j}-1}}$.

and -invariant by
Then $F(z)=F_{1}(z)F_{2}(z)$ and $F_{1}(z)\omega_{T}(x;z)$ is holomorphic in, since it is W-invariant andTheorem 1.3. Hence $F(z)\omega_{T}(x;z)$ is holomorphic in.
holomorphic for certain region e.g.,1$z\in \mathbb{C}^{n}$$S_{n}$$z\in \mathbb{C}^{n}$$\{z\in \mathbb{C}^{n} {\rm Re}(z_{i})\leq 0\}$\S 3
by using the generalat3.1. In this section we give an explicit formula offormula of Proposition 1.9 in [H2] (or Theorem 2.6 in [H4]). In order to apply it, we haveto check several conditions $((A1)-(A4)$ in [H4]-\S 1 , and it is obvious our $(B, X_{T})$ satisfies
them except (A3), which is the same as $(C)$ below.
$\omega_{T}(x;s)$$x_{T}$$)$

Proposition 3.1 The following condition $(C)$ is satisfied.
$(C)$ : For $y\in X_{T}$ such that $f_{T}(y)=0$ , there exists a chamcter $\psi\in\langle\psi_{i}|1\leq i\leq n\rangle$
whose restriction to the identity component of the stabilizer of $B$ at is not trivial.
$y$

Theorem 3.2 Let$T=Diag(\pi^{\lambda_{1}}, \ldots, \pi^{\lambda_{n}})$with$\lambda_{1}\geq\lambda_{2}\cdots\geq\lambda_{n}\geq v_{\pi}(2)$. Then$\omega_{T}(x_{T};z)=\frac{(-1)^{\Sigma_{i}\lambda_{i}(n-i+1)}q^{\Sigma_{i}\lambda_{i}(n-i+\frac{1}{2})}(1-q^{-2})^{n}}{\prod_{i=1}^{2n}(1-(-1)q-i)}\sum_{\sigma\in W}\gamma(\sigma(z))\Gamma_{\sigma}(z)q^{<\lambda,\sigma(z)>}$where$<\lambda,$ $z>= \sum_{i=1}^{n}\lambda_{i}z_{i},$$\Gamma_{\sigma}(z)$is defined in Theorem 2.5, and$\gamma(z)=\prod_{1\leq i<j\leq n}\frac{(1-q^{2z_{i}-2z_{j}-2})(1-q^{2z_{i}+2z_{j}-2})}{(1-q^{2z-2z_{j}}i)(-q^{2z_{0}+2z})}\cdot\prod_{i=1}^{n}\frac{1-q^{2z-1}i}{1-q^{2z_{i}}}$

.

,

(3.1)

150
We admit Proposition 3.1 for the moment and prove Theorem 3.2.
The set $X_{T}^{op}=\{x\in X_{T}|f_{T}(x)\neq 0\}$ becomes a disjoint union of B-orbits as follows.
”$\mathcal{U}=(\mathbb{Z}/2\mathbb{Z})^{n-1}$$X_{T}^{op}=u\urcorner u\in \mathcal{U}^{x_{1,u}}$,$X_{T,u}=\{x\in X_{T}|v_{\pi}(f_{T,i}(x))\equiv u_{1}+\cdots+u_{i} (mod 2), 1\leq i\leq n-1\}$ .

We set$\omega_{T,u}(x;s)=\int_{K}|f_{T}(kx)|_{u}^{s+\epsilon}dk$

,

where
$|f_{T}(y)|_{u}^{s+\epsilon}=\{\begin{array}{ll}|f_{T}(y)|^{s+\epsilon} if y\in X_{T,u},0 otherwise.\end{array}$

For a character $\chi=(\chi_{1}, \ldots, \chi_{n-1})$ of$\mathcal{U}$, we set$L_{T}(x; \chi;z)=\int_{K}\chi(f_{T}(kx))|f_{T}(kx)|^{s+\epsilon}dk=\sum_{u\in \mathcal{U}}\chi(u)\omega_{T,u}(x;z)$

where $\chi(u)=\prod_{i=1}^{n-1}\chi_{i}(u_{1}+\cdots+u_{i})$ . Adjusting
if necessary, we may write$z$according to$L_{T}(x;\chi;z)=\omega_{T}(x)z_{\chi})$$\chi$,, by adding$\frac{\pi\sqrt{-1}}{\log q}$to$z_{i}$.

Then, by the functional equations of $\omega_{T}(x;z)$ (Theorem 2.5), we have
$L_{T}(x;\chi;z)=\Gamma_{\sigma}(z_{\chi})L_{T}(x;\sigma(\chi);\sigma(z))$

by taking suitable characterof . If
with the original functional equation of$\sigma(\chi)$$\mathcal{U}$$\chi$,(3.2)$\sigma\in W$

is the trivial character 1, then (3.2) coincides. We obtain$\omega_{T}(x;z)$$(\omega_{T,u}(x_{T)}\cdot z))_{u}=(A^{-1}\cdot G(\sigma, z)\cdot\sigma A)(\omega_{T,u}(x_{T};\sigma(z)))_{u}$

,

where
$A=(\chi(u))_{\chi,u}$,$\sigma A=(\sigma(\chi)(u))_{\chi,u}\in GL_{2^{n}}(\mathbb{Z})$and $G(\sigma, z)$ is the diagonal matrix of size
given as in Theorem 3.2, we obtain$\int_{U}|f_{T}(ux_{T})|^{s+\epsilon}du$$2^{n}$

whose$(\chi, \chi)$,-component is$=$$|f_{T}(x_{T})|^{s+\epsilon}$$=$$(-1)^{\Sigma_{i}\lambda_{i}(n-i+1)}q^{\Sigma_{i}\lambda_{i}(n-i+\frac{1}{2})}q^{<\lambda,z>}$$\Gamma_{\sigma}(z_{\chi})$. For $T$,is the Iwahori subgroup of $K$ compatible with $B$ and $du$ is the normalized Haar
measure on . Setting

where

$U$

$U$

$\delta_{u}(x_{T}, z)=\{\begin{array}{ll}(-1)^{\Sigma_{i}\lambda_{i}(n-i+1)}q^{\Sigma_{i}\lambda_{i}(n-i+\frac{1}{2})}q^{<\lambda_{t}z>} if x_{T}U(T)\in X_{T,u}0 otherwise,\end{array}$151
we have, by Proposition 1.9 in [H2] (or its generalization Theorem 2.6 in [H4]),$( \omega_{T,u}(x_{T};z))_{u}=\frac{1}{Q}\sum_{\sigma\in W}\gamma(\sigma(z))(A^{-1}\cdot G(\sigma, z)\cdot\sigma A)(\delta_{u}(x_{T}, \sigma(z)))_{u}$

,

where
$Q= \sum_{\sigma\in W}[U\sigma U:U]^{-1}=\prod_{i=1}^{2n}(1-(-1)^{i}q^{-i})/(1-q^{-2})^{n}$.

Hence we obtain$\omega_{T}(x_{T};z)$$=$$\sum_{u\in \mathcal{U}}1(u)\omega_{u}(x_{T};z)$$=$.

$\frac{(-1)^{\Sigma_{i}\lambda_{i}(n-i+1)}q^{\Sigma_{i}\lambda_{i}(n-i+\frac{1}{2})}}{Q}\sum_{\sigma\in W}\gamma(\sigma(z))\Gamma_{\sigma}(z)q^{<\lambda,\sigma(z)>}$1

3.2. In order to prove Proposition 3.1, we consider the action of $G\cross U(T)$ onby$(g, h)\circ x=gxh^{-1}$ . Then, the stabilizer$yU(T)\in
X_{T}$
of atcoincides with the imageof the projection toof the stabilizer $(B\cross U(T))_{y}$ at $y\in X_{T}$ to . Hence the$X_{T}$$B_{y}$$B$$B$$B_{(y)}$$B$

condition $(C)$ is equivalent to the following:: For $y\in X_{T}$ such that $f_{T}(y)=0$ there exists
to the identity component of
is not trivial.
$(C’)$$\psi\in\langle\psi_{i}|1\leq i\leq n\rangle$

whose restriction$B_{(y)}$It is sufficient to prove the condition $(C)$ (equivalently, $(C’)$ ) over the algebraic closure ,since, for a connected linear algebraic groupis dense in. In the rest of thissection, we consider algebraic sets over , extend the involution on
to and denote
it by $-$ , and write
for any matrix $x=(x_{ij})$ . Since
is isomorphic toby$x\mapsto xh$ and $B_{(x)}=B_{(xh)}$ for $h\in GL_{n}$ ,we may assume that $T=1_{n}$ . Then, our situationis the following:$\overline{k}$$\mathbb{H},$$\mathbb{H}(k)$$\mathbb{H}(\overline{k})$$k’$$\overline{k}$$*$$\overline{x}=(\overline{x_{ij}})$$\overline{k}$$X_{T}$$X_{T[h]}$$X=X_{1_{n}}=\{x\in M_{2n,n}|H_{n}[x]=1_{n}\}$ ,$(U(n, n)\cross U(n))\cross Xarrow\chi$,$((g, h), x)\mapsto(g, h)\circ x=gxh^{-1}$ .

We consider the set$\tilde{X}=\{(x, y)\in M_{2n,n}\oplus M_{2n,n}|{}^{t}yH_{n}x=1_{n}\}$together with $GL_{2n}\cross GL_{n}$ -action defined by$(g, h)\star(x, y)=(gxh^{-1},\dot{g}y^{t}h)$and take the Borel subgroup$P$$B_{n}$$\dot{g}=H_{n}^{t}g^{-1}H_{n}$

,

(3.3)

of $GL_{2n}$ by$P=\{(\begin{array}{ll}p_{1} r0 p_{2}\end{array})\in GL_{2n}$

where

,

is the Borel subgroup of$GL_{n}$$p_{1},{}^{t}p_{2}\in B_{n},$$r\in M_{n}$,consisting of upper triangular matrices.

152
Then, the embedding : Gl;
have the commutative diagram$\mapsto\tilde{X},$$\iota$is compatible with the actions, i.e., we$x\mapsto(x, \overline{x})$$(U(n,n_{II^{\iota}})\cross U(n))\cross x(GL_{2n}\cross GL_{n})\cross\tilde{x}idarrow 0arrow\tilde{x}\star x\downarrow\iota$

For $(x, y)\in$

se and$p\in P$, set$\tilde{f_{i}}(x, y)=d_{i}(x_{2^{t}}y_{2})$,$\tilde{\psi}_{i}(p)=\prod_{1\leq j\leq i}p_{j}^{-1}p_{n+j}$(resp. ) is the lower half
whereentry of . Then for each , we see$x_{2}$$y_{2}$

$n$

by

$n$

block of$x$,$(1\leq i\leq n)$(resp.

$y$), and$p_{j}$,

is the j-th diagonal

$i$

$p$

$\tilde{f_{i}}((p, r)\star(x, y))=\tilde{\psi}_{i}(p)\tilde{f_{i}}(x, y)$$(x\in X)$ ,,$\overline{f_{i}}(x,\overline{x})=f_{i}(x)$,$(p, r)\in P\cross GL_{n}$$\tilde{\psi}_{i}|_{B}=\psi_{i}$,.

We set$S=\{(x, y)\in\tilde{X}$$\prod_{i=1}^{n}\overline{f_{i}}(x, y)=0$,$(P\cross GL_{n})\star(x, y)\cap X\neq\emptyset\}$.

its imagethe stabilizer of $P\cross GL_{n}$ at , and by
For $\alpha=(x, y)\in\tilde{X}$ , we denote by
$P$
$(C)$
, it is sufficient to show theof the projection to . In order to prove the condition
following:whose restriction to the, there exists some
: For eachis not trivial.
identity component ofby taking suitable representatives by $P\cross GL_{n}$ -action.
We show the condition-orbit containing(i) Assume $\alpha=(x, y)\in S$ satisfies $\det(x_{2})\neq 0$ . Then, in the$H_{\alpha}$$\psi\in\langle\tilde{\psi}_{i}|1\leq i\leq n\rangle$$\alpha\in S$$(\tilde{C})$$P_{\alpha}$$\alpha$$P_{\alpha}$$(\tilde{C})$$P\cross GL_{n}$$\alpha$, there is$\beta=((\begin{array}{l}01_{n}\end{array}), (\begin{array}{l}l_{n}h\end{array}))$with some hermitian matrix , further we may assume$h$$h=1_{r}\perp\langle 0\rangle\perp h_{1}$or$h=1_{r}\perp h_{2}$,$(1<i\leq n-r)$ such that each entry inwhere $0\leq r\leq n-1$ , and for , there is somethe first row and column or in the i-th row and column is except at (1, i) or $(i, 1)$ whichare 1.
contains the following elements, according to the above type of
Then
$i,$

$h_{2}$$0$$h$$H_{\beta}$$((\delta_{r+1}(a) 1_{n}), 1_{n})$or$((\delta_{r+1}(a) \delta_{r+i}(a)), \delta_{r+i}(a))$.

,

is the diagonal matrix in $GL_{n}$ whose diagonal entries are 1 except the j-th
where
on the identity component of .
which is $a\in GL_{1}$ . Hence we see$\det(y_{2})\neq
0$ is reduced to the case $\det(x_{2})\neq 0$ , since$\alpha=(x,
y)\in
S$
with
(ii) The case.
$\beta=(y, x)\in S$ and $H_{\beta}=\{(\dot{p},{}^{t}r^{-1})|(p, r)\in H_{\alpha}\}$ and$\delta_{j}(a)$$\tilde{\psi}_{r+1}\not\equiv 1$$P_{\beta}$$\tilde{\psi}_{i}(\dot{p})=\tilde{\psi}_{i}(p)^{-1}$153(iii) Assume $\alpha=(x^{l}, y’)\in S$ satisfies $\det x_{2}’=\det y_{2}’=0$ . Then, in the $P\cross GL_{n^{-}}$
orbit containing , there is some $\beta=(x, y)$ of the following type: for some integers$\alpha$$r_{i},$$e_{j}$

satisfying$1\leq r_{1}<r_{2}<\cdots r_{\ell}\leq n$$(1\leq\ell<n)$ ,$1\leq e_{1}<e_{2}<\cdots<e_{k}\leq n$$x=(\begin{array}{l}x_{1}x_{2}\end{array})$and$y=(\begin{array}{l}y_{1}y_{2}\end{array})$with$x_{i},$$y_{i}\in M_{n}$$(k=n-\ell)$ ,is given by: 1 at $(r_{i}, k+i)$ -entry for $1\leq i\leq\ell$ and at any other entry;$x_{2}:1$ at-entry for $1\leq i\leq k$ and at any other entry;: therow is the same as infor $1\leq i\leq k$ , and the j-the column is if$j>k$ ;
: therow is the same as infor $1\leq i\leq\ell$ , and for each , any $(i, j)$ -entry
is for $j>k$ if some $(i,j’)$ -entry is non-zero entry with $j’\leq k$ .
$0$$x_{1}$$(e_{i}, i)$$0$$e_{i^{-}}th$$y_{1}$$r_{i^{-}}th$$y_{2}$$0$$x_{2}$$i$$x_{1}$$0$

Let $D(a)$ be the diagonal matrix inwhose i-th diagonal entry is $a\in GL_{1}$ (resp. 1)
if every $(i,j)$ -entry ofis for $j\leq k$ (resp. otherwise), where thediagonal entryof $D(a)$ is by this choice. Thencontains$GL_{n}$$y_{2}$$a$$0$$r_{i^{-}}th$$H_{\beta}$$((D(a) 1_{n}), (1_{k} a1_{\ell}))$ ,and$\tilde{\psi}_{r_{i}}\not\equiv 1$on the identity component of$P_{\beta},$$1\leq i\leq\ell$

.

I

\S 4
We recall the hermitian Siegel series, and give its integral representation and functional
equation. Let
be an additive character of of conductor
. For
, the
hermitian Siegel series $b_{\pi}(T;s)$ is defined by$k$$\psi$$T\in \mathcal{H}_{n}(k’)$$\mathcal{O}_{k}$$b_{\pi}(T;s)= \int_{\mathcal{H}_{n}(k’)}\nu_{\pi}(R)^{-s}\psi(tr(TR))dR$

,

(4.1)

where tr $()$ is the trace of matrix andis defined as follows: if the elementary divisorsof with negative -powers are, then, and $\nu_{\pi}(R)=1$
otherwise (cf. [Sh]-\S 13).
In the following we assume thatis nondegenerate, since the properties of $b_{\pi}(T;s)$can be reduced to the nondegenerate case. We recall the set
for$\nu_{\pi}(R)$

$R$

$\pi$

$\pi^{-e}1,$$\pi^{-e_{r}}$$\ldots,$$\nu_{\pi}(R)=q^{e_{1}+\cdots+e_{r}}$$T$$X_{T}$$T\in \mathcal{H}_{n}^{nd}(k’)$$X_{T}=X_{T}(k’)=\{x\in M_{2n,n}(k’)|H_{n}[x]=T\}$ ,which is the fibre space $g^{-1}(T)$ for the polynomial map $g:M_{2n_{r}n}(k’)arrow \mathcal{H}_{n}(k’),$ $g(x)=$defined over . We may take the measureon
induced by a k-rational
differential form on $M_{2n,n}(k’)$ satisfying $\omega\wedge g^{*}(dT)=dx$ where $dT$ is the canonical$H_{n}[x]$$k$$\omega$$|\Theta_{T}|$$\chi_{T}$154$dx$ is the canonical gauge form on $M_{2n,n}(k‘)$ . Then the following
gauge form on
identity holds (cf. [Ym], [HS]-\S 2):$\mathcal{H}_{n}(k’),$$\int_{X_{T}(k’)}\phi(x)|\Theta_{T}|(x)$,$= \lim_{earrow\infty}\int_{\mathcal{H}_{n}(\pi^{-e})}\psi(-tr(Ty))\int_{M_{2n,n}(k’)}\phi(x)\psi(tr(H_{n}[x]y))dxdy$

where $\phi\in S(M_{2n,n}(k’)))$ a locally constant compactly supported function on $M_{2n,n}(k’)$.
and$\mathcal{H}_{n}(\pi^{-e})=\mathcal{H}_{n}(k’)\cap M_{n}(\pi^{-e}\mathcal{O}_{k’})$

The following lemma can be proved in the similar line to the case of symmetric matrices(cf. [HS]-\S 2).

Lemma 4.1 If ${\rm Re}(s)>n$ , one has(4.2)$\int_{X_{T}(\mathcal{O}_{k};)}|N_{k’/k}(\det x_{2})|^{s-n}|\Theta_{T}|(x)$$=$$\lim_{earrow\infty}\int_{\mathcal{H}_{n}(\pi^{-e}\mathcal{O}_{k’})}\psi(-tr(Ty))dy\int_{M_{2n,n}(\mathcal{O}_{k’})}|N_{k’/k}(\det x_{2})|^{s-n}\psi(tr(H_{n}[x]y))dx$.

Let us recall the zeta function of the matrix algebra $M_{n}(k’)$ and its explicit formula:$\zeta(k’;s)$$=$$\int_{M_{n}(\mathcal{O}_{k’})}|\det x|_{k’}^{s-n}dx=\int_{M_{n}(O_{k’})}|N_{k’/k}(\det x)|^{s-n}dx$$=$$\prod_{i\vec{-}1}^{n}\frac{1-q^{-2i}}{1-q^{-2(s-i+1)}}$.

Then we obtain the following integral expression of hermitian Siegel series, which can beproved in a similar line to the case of Siegel series.

Theorem 4.2 If ${\rm Re}(s)>2n$ , we have$b_{\pi}(T;s)= \zeta_{n}(k’;\frac{s}{2})^{-1}\cross\int_{X_{T}(\mathcal{O}_{k’})}|N_{k’/k}(\det x_{2})|^{\frac{s}{2}-n}|\Theta_{T}|(x)$We introduce the spherical function on

group$P=\{(\begin{array}{ll}a b0 d\end{array})\in G$$a,$ $b,$$X_{T}$$d\in M_{n}(k’)\}$.

with respect to the Siegel parabolic subby$\tilde{\omega}_{T}(x;s)=\int_{K}|N_{k’/k}(\det(kx)_{2})|^{s-n}dk$.

Then we have$\tilde{\omega}_{T}(x;s)=|\det T|^{s-n}\omega_{T}(x;1-\frac{\pi\sqrt{-1}}{\log q}, \ldots, 1-\frac{\pi\sqrt{-1}}{\log q}, s-n+\frac{1}{2}-\frac{\pi\sqrt{-1}}{\log q})$

,

(4.3)

by Theorem 1.3. Next proposition shows the relation
which is holomorphic for
between hermitian Siegel series and spherical functions.
$s\in \mathbb{C}$155
Proposition 4.3 Denote the K-orbit decomposition of$X_{T}(\mathcal{O}_{k’})=u_{i=1}^{r}Kx_{i}$$X_{T}(\mathcal{O}_{k’})$as.

Then one has$b_{\pi}(T;s)= \zeta_{n}(k’;\frac{s}{2})^{-1}\cdot\sum_{i=1}^{r}c_{i}\tilde{\omega}_{T}(x_{i};\frac{s}{2})$,$c_{i}= \int_{Kx_{i}}|\Theta_{T}|(y)$.

By Proposition 4.3 and Corollary 2.6, we obtain the following functional equation ofhermitian Siegel series.

Theorem 4.4$\frac{b_{\pi}(T;s)}{\prod_{i=0}^{n-1}(1-(-1)^{i}q^{-s+i})}=\chi_{\pi}(\det T)^{n-1}|\det(T/2)|^{s-n}\cross\frac{b_{\pi}(T;2n-s)}{\prod_{i=0}^{n-1}(1-(-1)^{i-(2n-s)+i}q)}$.

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Yumiko Hironaka
Department of Mathematics, Faculty of Education and Integrated Sciences,
Waseda University,
Nishi-Waseda, Tokyo 169-8050, JAPAN.