数理解析研究所講究録924 巻 1995 年 153-163

153

ON FUNCTIONAL EQUATIONS OFPREHOMOGENEOUS ZETA DISTRIBUTIONSOVER A LOCAL FIELD OF CHARACTERISTIC$\mathrm{P}$TATSUO KIMURA, MAKIKO FUJINAGA AND TAKEYOSHI KOGISO

Institute of Mathematics
University of Tsukuba
IBARAKI, 305, JAPANABSTRACT. For a local field of characteristic , the functional equations of zeta
distributions of prehomogeneous vector spaces are obtained by M.Sato, T.Shintani,
J.Igusa and F.Sato (See [17], [9], [13], [15]). In this paper, we shall consider the.
case of local fields of characteristic$0$$\mathrm{p}>0$\S 1.

$\mathrm{K}$-regular P.V.’s$>0$ . Let
be a connected linear
We fix a local field $K$ of characteristic
algebraic group, its rational representation of on a finite-dimensional vector
of $K$ . We call a triplet $(G, \rho, V)$ a
space , all defined over an algebraic closure. ) if has a Zariski-dense -orbit Y.
prehomogeneous vector space (abbrev.
Any point of is called a generic point and the isotropy subgroupis called a generic isotropy$G_{y}=\{g\in G;\rho(g)y=y\}$ of a generic point
if and only if $y\in$ Y.
subgroup. Note that we have $dimG_{y}=dim$ G–dim$f(x)$
invariant of $(G, \rho, V)$relativeaiscalledon
A non-zero rational function
if $f(\rho(g)X)=\chi(g)f(X)$ holds for any $g\in G$ and $x\in Y$ where : $Garrow GL_{1}$ is arational character of .
The complement of is a Zariski-closed set which is called the singular set ofare. $(G, \rho, V)$ . Now we assume that $(G, \rho, V)$ is defined over $K$ , i.e., $G,$theall defined over $K$ . Let $S_{i}=\{x\in V;f_{i}(X)=0\}(i=1, \ldots., l)$ be the K-irreducibleof of codimension one defined by a Kcomponent of the $K$ -rational points
irreducible (not necessarily absolutely irreducible) polynomial $f_{i}(x)(\dot{i}=1, \cdots, l)$ .
are algebraically independent relative invariants and any
Then $f_{1}(x),$relative invariant $f(x)$ in $K(V)$ is of the form $f(x)=c\cdot f_{1}(x)^{m_{1}}\cdots\cdot f\iota(x)^{m\iota}(c\in$the basic $K$ -relative invariants of. We call $f_{1}(x),$.
corresponding to$(G, \rho, V)$ . Let
be the rational character of$G,$$X_{1}(G)_{K}$$K$
its subgroup
Let $X(G)_{K}$ be the group of -rational characters of$X_{1}(G)_{K}$$K$
group ofabelian
free
is
a
Then
corresponding to -relative invariants.
.
rank generated by
$G$

$p$

$G$

$\rho$$\overline{K}$

$V$

$G$

$V$

$\mathrm{P}.\mathrm{V}$

$Y$

$y$

$V$

$V$

$\chi$

$G$

$S$

$Y$

$\rho,$$\mathrm{P}.\mathrm{V}$$S_{K}$$\ldots\ldots.,$$K^{\cross},$$S$$f_{l()}X$$(m_{1}, \ldots.., m_{l})\in \mathrm{Z}^{t})$$\cdots,$$f\iota(x)$$G$$\chi_{i}$

$l$

$V$

$\chi_{1},$$\ldots..,$$\chi_{l}$$f_{i}(\dot{i}=1, \ldots, l)$

154

Let
be a subgroup of generated by the commutator subgroup $[G, G]$ anda generic isotropy subgroup. This does not depend on a choice of a generic point.
For $\chi\in X(G)_{K}$ , it is in $X_{1}(G)_{K}$ if and only if $\chi|_{G_{1}}=1$ . For a relative invariant$f(x)$ of $(G, \rho, V)$ , we can define a rational map
:
by
$G$

$G_{1}$$\mathrm{Y}arrow V^{*}$$\varphi_{f}$$\varphi_{f}(x)=^{t}(\frac{1}{f(x)}\cdot\frac{\partial f}{\partial x_{1}}(_{X}),$$\frac{1}{f(x)}\cdot\frac{\partial f}{\partial x_{n}}(_{X)})$$\ldots\ldots.,$

whereis the dual vector space of . We sometimes denoteby grad $logf(x)$ .
By a direct calculation, we have(1) $\varphi f(\rho(g)_{X})=\rho^{*}(g)\varphi f(X)$ for $g\in G$ andwheredenotes the contragradient representation of ,and(2)
for $x\in Y$ and $\mathrm{A}\in Lie(G)$ where(resp.
) isthe infinitesimal representation of (resp. the infinitesimal character of ) of the
Lie algebra Lie $(G)$ of .
$V^{*}$$V$$\varphi_{f}(x)$$x\in \mathrm{Y}$$\rho^{*}$$\rho$$\langle d\rho(\mathrm{A})x, \varphi f(x)\rangle=\delta\chi(A)$$d\rho$$\rho$$\delta\chi$$\chi$$G$A relative invariant $f(x)$ is called non-degenerate if:
is dominant
and the Hessian
is not identically zero. In this case, a rational$\mathrm{Y}arrow V^{*}$$\varphi_{f}$$H_{f}(x)= \det(\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}})$function $F(x)= \frac{f(x)l\iota}{H_{f}(x)}(n=d_{\dot{i}}mV)$ is a relative invariant corresponding to thecharacter.
If there exists a non-degenerate relative invariant $f(x)$ in $K(V)$ , we say that$(G, \rho, V)$ is a $K$ -regular. Then we have $\det\rho(g)^{2}\in X_{1}(G)_{K}$ . In general, we$K$denote by, etc.
-rational points of, etc. We write(resp.
$X^{*}(G)_{K},$ $Y^{*,s*}$ , etc.) for
which corresponds to $X_{1}(G)_{K}$ (resp. $X(G)_{K},$etc. ) for $(G, \rho, V)$ .
$\chi_{0}(\mathit{9})=\det\rho(g)^{2}$$\mathrm{P}.\mathrm{V}$$\mathrm{Y}_{K},$$S_{K}$$Y,$ $S$$X_{1}^{*}(c)_{K}$$(G, \rho^{*}, V^{*})$$\mathrm{Y},$Proposition 1.1
Assume that $(G, \rho, V)$ andare $K$ -regular P. V. ’s. Then we have thefollowing assertion.
(1) $X_{1}(c)_{K}=X_{1}*(G)_{K}$ .
(2) For a non-degenerate $K$ -relative invariant , the map $\varphi=gradlogf$ :is bijective.
$(G, \rho^{*}, V^{*})$$f$$\mathrm{Y}arrow$$Y^{*}$[Proof]Since
is a Zariski-dense -orbit in, we have, i.e.,
surjective. Since, we havefor $x\in$ Y. Now let
be a non-degenerate relative invariant in $K(V^{*})$ , and put $\varphi^{*}=grad$ $logf^{*}$ :. Similarly we havefor $y=\varphi(x)$ and $x’=\varphi^{*}(y)$ , and hence$x’=\rho(g_{0})x$. Sincefor some $g_{0}\in G$ , we have $G_{x’}=g0G_{x}g_{0}-1\supset G_{x}$ .
Since $d_{\dot{i}}mG_{x’}=dimG_{x}$ , the algebraic groupandhave the same connectedcomponent $H$ of the identity. Since
is isomorphic to, the numbers of theirconnected components coincide, i.e., $[G_{x’} : H]=[G_{x} : H]$ with. Thisimplies $G_{x’}=G_{x}$ , and hence $G_{x}=G_{y}$ with $y=\varphi(x)$ .
$\varphi(\mathrm{Y})$$G$$V^{*}$$\rho^{*}(g)\varphi(x)=\varphi(\rho(g)_{X})$$\varphi(\mathrm{Y})=\mathrm{Y}^{*}$$G_{x}\subset G_{\varphi(x)}$$f^{*}$$\mathrm{Y}^{*}arrow Y$$G_{y}\subset G_{x’}$$G_{x}\subset G_{y}\subset G_{x’}$$G_{x’}$$G_{x’}$$G_{x}$$G_{x}$$G_{x’}\supset G_{x}$$\varphi=$

$S$

,

155

Thus we haveand hence $X_{1}(G)_{K}=x_{1^{*}}(G)_{K}$ . Note that $X_{1}(G)_{K}=$Now assume thatwith $x_{2}=\rho(g)x_{1}$ forsome $g\in G$ . Then we have $\varphi(x_{1})=\varphi(x_{2})=\varphi(\rho(g)x1)=\rho^{*}(g)\varphi(X_{1})$ and hence, i.e., $x_{2}=\rho(g)X_{1}=x_{1}$ . Thus is injective.
$G_{1}=G_{1}^{*}$$\{\chi\in X(G)K;x|c_{1}=1\}$ .

$\varphi(x_{1})=\varphi(x_{2})$$g\in G_{\varphi(x_{1})}=G_{x_{1}}$$\square$$\varphi$

Now assume that $(G, \rho, V)$ is a $K$ -regular. Then, as we have seen above, thedual triplet
is a. For a generic point, a dominant morphism: $Garrow V^{*}$ defined by $\psi(g)=\rho^{*}(g)y$ is called an open orbit morphism.
$\mathrm{P}.\mathrm{V}$$(G, \rho^{*}, V^{*})$$\mathrm{P}.\mathrm{V}$$y\in \mathrm{Y}^{*}$$\psi$

Proposition 1.2
Assume that $(G, \rho, V)$ is a $K$ -regular P. V. and an open orbit morphism : $Garrow$is a separable morphism. Then there exists If-relative invariant
such thatgrad $logf^{*}$ : $Y^{*}arrow V$ is dominant.
$\psi$$V^{*}$$a$$f^{*}$[Proof]
Let be a non-degenerate relative invariant in $K(V)$ and put $\varphi=gradlogf$ :. First we show thatis injective. Assume that $\varphi(x)=\varphi(x’)$ . Since, we have $\langle x-xd’,(\rho^{*}A)\varphi(x)\rangle=0$$GarrowV^{*}$
for all $A\in Lie(G)$ . Since:with $\psi(g)=\rho^{*}(g)\varphi(x)$ is separable, wehave $\{d\rho^{*}(A)\varphi(x);A\in Lie(G)\}=V^{*}$ , and hence $x-x’=0$ , i.e., $x=x’$ . For any, we have $\varphi(\rho(g)_{X})=\rho^{*}(g)\varphi(x)=\varphi(x)$ . As
is injective, wehave $\rho(g)x=x$ , i.e., $g\in G_{x}$ . This implies thatand hence $X_{1}(G)_{K}=$. A rational characterto is in $X_{1}(G)_{K}$ and hence. This implies that there exists a relative invariant
in $K(V^{*})$ satisfying$f^{*}(\rho^{*}(g)y)=\chi(\mathit{9})^{-}1f*(y)$ for $g\in G$
and $y\in Y^{*}$ .
Put $\varphi^{*}=gradlogf^{*}$ . Then we have
. Since $\delta\chi(A)=$, we have$\langle x-\varphi^{*}(y), d\rho(*A)y\rangle=0$
for $y=\varphi(x)$ and all $A\in Lie(G)$ .
Since the open orbit morphism is separable , we have$f$$\mathrm{Y}arrow \mathrm{Y}^{*}$$\varphi$$\delta\chi(A)=\langle d\rho(A)_{X}, \varphi(x)\rangle=-\langle x, d\rho^{*}(A)\varphi(x)\rangle$$\psi$$g\in G_{\varphi(x)}(\supset G_{x})$$\varphi$$G_{x}=G_{\varphi(x)}$$X_{1}^{*}(G)_{K}$$\chi \mathrm{C}\mathrm{O}\Gamma \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}_{\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}}1$$X_{1}^{*}(G)_{K}$$f$$\chi^{-1}\in$$f^{*}$$\langle\varphi^{*}(y), d\rho^{*}(A)y\rangle=-\delta\chi(A)$$\langle d\rho(A)x, \varphi(x)\rangle=-\langle_{X,d\rho^{*}}(A)\varphi(x)\rangle$$\psi$,, i.e., $\varphi^{*}(Y^{*})=Y$.

$\{d\rho^{*}(A)y;A\in L_{\dot{i}e(G)\}}=V^{*}$and hence$\varphi^{*}(y)=x\in \mathrm{Y}$$\square$Note that in the case of $ch(K)=0$ , the proof of Proposition 1.2 gives theequivalence between If-regularity of $(G, \rho, V)$ and that of $(G, \rho^{*}, V^{*})$ .
Proposition 1.3
Assume that $(G, \rho, V)$ and $(G, \rho^{*}, V^{*})$ are $K$ -regular P. V. ’s. Then we have$\#\rho(G)_{K}\backslash \mathrm{Y}_{K}=\#\rho^{*}(G)_{K}\backslash Y^{*}K$$[\mathrm{P}_{\Gamma 0\mathrm{O}}\mathrm{f}]$

.

156

$\varphi=gradlogf$ .
Let be a non-degenerate relative invariant in $K(V)$ and put
Then for any $x\in Y_{K}$ , we haveto an orbit in, i.e., maps an orbit in$f$$Y_{K}$$\varphi(\rho(c)_{K}\cdot x)=\rho^{*}(G)_{K}\cdot\varphi(X)\subset Y_{K}^{*}$$Y_{K}^{*}$.

By Propositionl.1, this map
Similarly we have$\varphi$$\varphi$

is injective, and hence$\#\rho(G)_{K}\backslash Y_{K}\leqq\#\rho^{*}(G)K\backslash Y^{*}K$$\#\rho^{*}(G)K\backslash Y^{*}K\leqq\#\rho(G)_{K}\backslash Y_{K}\square$is finite.
Now we shall consider a sufficient condition that
.Serre kindly let us know the following theorem with the proof which
Professor
was explained by Tits to him.
$\#\rho(G)K\backslash Y_{K}$$\mathrm{J}.\mathrm{P}$Theorem 1.4$K$ be afield
Let $K$ be a local field of characteristic $p>0$ (or more generally let
type
residue
the
with
of
and
valuation,
fieldcomplete with respect to a discrete
over
group
reductive. Let be a connected smooth$(F)$ in the sense of Serre [K. Then $H^{1}(K, G)$ is finite.
( after Serre’s letter on 9th. September 1992. )$K’$ is known to$K$
Let $K’$ be the maximal unraInified extension of . The field\S 3). By a theorem of Steinberg (for(in the sense of
be of $dim$ .
$K’$ imperfectsee Borel Col. Papers II,$K’$ perfect) and of Borel-Springer (for
$K$ is killed
p.761) we have $H^{1}(K’, G)=0$ . Hence the Galois cohomology of overof Bruhatby $K’$ , i.e., it is equal to $H^{1}(K’/K, c)$ . We may now apply a theorem$H^{1}(K^{J}/K,
c)$
that
says
Tits (J.Fac.Sci.Tokyo, 34 (1987), p.693, th.3.12 ); thisare$H^{1}(k, G_{i})$ , where theis contained in a finite union of cohomology setsconnected)
over . Since is type (F),algebraic linear groups (non neccessarily
Coh. Gal.
each $H^{1}(k, G_{i})$ is finite (see e.g. Borel, Col.Papers II, p.404 , th.6.2, or
III-30, th.4). Hence $H^{1}(K, G)$ is finite.
$k$$G^{t}$$\mathit{1}\mathit{8}J$$[\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{f}]$$\mathrm{C}\mathrm{G},$$\leq 1$$\mathrm{I}\mathrm{I},$-$G$$G_{i}’ \mathrm{s}$

$k$

$k$

$\square$Proposition 1.5
subgroup.
Let $(G, \rho, V)$ be a P. V. defined over $K$ with a reductive generic isotropy

is finite.
Then[Proof]. Then there exists a
Let $H$ be a generic isotropy subgroup of a point inand $Ker(H^{1}(K, H)arrow H^{1}(K, G))$ (see Serre [18]).
bijection between
is a finite set.
By Theorem 1.4, $H^{1}(K, H)$ is finite , and hence$\#\rho(G)_{K}\backslash YK$$Y_{K}$$\rho(G)_{K}\backslash Y_{K}$$\rho(G)_{K}\backslash Y_{K}$$\square$Example 1.6
Let
Let be the subgroup of $GL_{n}$ consisting of all lower triangular matrices.
matrices and define by
be the totality of symmetricfor all $g\in G$$G$$n\cross n$$\rho(g)_{X}=gx^{\mathrm{t}}g$$\rho$$V$

157

and $x\in V$. Since $dimG=d\dot{i}mV$, a generic isotropy subgroup is a finite subgroupand hence we haveby Proposition 1.5.
Moreover $\det x$ is a non-degenerate $K$ -relative invariant. By $tr(xy)(x, y\in V)$ ,we identify
with its dual.
Then $(G, \rho, V)$ andare $K$ -regular P. V. ’s. Hence, by Proposition
1.3, we have.
$\#\rho(G)_{K}\backslash \mathrm{Y}_{K}=\nu<+\infty$$V$$V^{*}$$(G, \rho^{*}, V^{*})$$\#\rho^{*}(G)_{K}\backslash \mathrm{Y}^{*}K=\nu<+\infty$

Proposition 1.7
Let $(G, \rho, V)$ be an irreducible regular P. V. defined over K. Then we have$+\infty$$\#\rho(G)_{K}\backslash \mathrm{Y}K<$.

[Proof]By a classification of irreducible P.V.’s (see Z.Chen [4]), we know that a genericisotropy subgroup is reductive.
$\square$\S 2. Zeta distributions
Let $K$ be a local field of characteristic $p>0$ . Assume that $(G, \rho, V)$ and its dualare -regular P.V.’s. Moreover we shall assume that
decomposes into a finite union of-orbits $Y_{i}(1\leqq i\leqq\nu)$ , i.e., $\#\rho(G)_{K}\backslash YK=$$\nu<+\infty$ . Then by Proposition 1.3 , we have.
( resp.
) be basic -relative invariants of
Let $f_{1}(x),$$(G, \rho, V)$ (resp. $(G,$). Let ( resp.
) be the corresponding character of(resp.
). Then we have,)
By
that there exists uniquely amatrix$(G, \rho^{*}, V^{*})$$IC$$Y_{K}=Y_{1}\cup\cdots\cup Y_{\nu}$$\rho(G)_{K}$$Y_{K}^{*}=Y_{1}^{*}\cup\cdots\cup Y_{\nu}^{*}$$\cdot\cdot.,$$\rho^{*},$$f_{i}$$f_{1}^{*}(y),$$f_{\ell}(X)$$V^{*})$$\chi_{i}$$\cdot\cdot.,$$f_{\ell}^{*}(y)$$IC$$\chi_{i}^{*}$$f_{i}^{*}$$X_{1}(G)_{K}=\langle\chi_{1}$$\mathrm{p}\mathrm{r}\mathrm{o}_{\mathrm{P}}\mathrm{o}\mathrm{S}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}1.1’,\mathrm{w}\mathrm{e}xl$$\mathrm{a}\mathrm{n}\mathrm{d}_{d}\mathrm{Y}*(1)_{K}c_{G}=\mathrm{h}\mathrm{a}\mathrm{V}\mathrm{e}X1()_{K}=x_{1}^{1}*’(\langle x^{**}\ldots,\chi l\rangle c)_{K}\mathrm{S}\mathrm{o}$$U=(u_{ij})\in GL_{l}(\mathrm{Z})$satisfying$\chi_{i}=\prod_{j=1}^{l*u}x_{j}‘ j$. Since $\det\rho(g)^{2}\in X_{1}(G)_{K}$ , we$\mathrm{h}\mathrm{a}.\mathrm{v}\mathrm{e}*2\lambda_{1}*.\mathrm{d}.\mathrm{e}.\mathrm{t}\rho\chi l*2(g)^{2}=\lambda^{*}l\mathrm{f}\mathrm{o}\mathrm{r}$$\chi_{1}^{2\lambda_{1}\ldots 2\lambda\iota}\chi l$somefor some

and $\det\rho^{*}(g)^{2}=\chi_{1}$. Since $\det\rho^{*}(g)=\det\rho(\mathit{9})^{-}1$ , we have$\lambda=(\lambda_{1}, \cdots, \lambda_{l})\in(\frac{1}{2}\mathrm{Z})^{l}$$\lambda^{*}=(\lambda_{1}^{*}, \cdots, \lambda_{l}*)\in(\frac{1}{2}\mathrm{Z})^{l}$

Example 2.1
For simplicity, we deal with the case $n=2$ in Example 1.6. Then we have

$G=\{g= ; ab\neq 0\}$and.

$\lambda^{*}=-\lambda U$

158$V=\{X=\}$ .
) are $f_{1}(X)=x$
The basic $K$ -relative invariants of $(G, \rho, V)$ (resp. $(G,$and $f_{2}(X)=\det X$ (resp. $f_{1}^{*}(X)=z,$ $f_{2}^{*}(X)=\det X$ ) corresponding to $\chi_{1}(g)=$$\chi_{2}(g)=a^{2}b^{2}$ (resp. $\chi_{1}^{*}(g)=b^{-2},$ $\chi_{2}^{*}(g)=a^{-2}b^{-2}$ )for$\rho^{*},$$V^{*})$$a^{2},$

$g=$
in .
Hence
$G$

$\chi_{1}=\chi_{1}^{*}\chi_{2}^{*}-1$

and$\chi_{2}=\chi_{2^{-1}}^{*}$

so that we have

$U=$ .
Since$\det\rho()=a^{3}b^{3}$ ,we haveLet
have$\lambda=\lambda^{*}=(0,.\frac{3}{2})$$\{\epsilon_{1}, \cdot\cdot., \epsilon_{\nu}\}$.

be the complete representatives ofwithmod} $(i=1, \cdots, \nu)$ .

in$K^{\cross}/IC^{\mathrm{x}2}$$K^{\cross}$. Then we$\mathrm{Y}_{K}=\mathrm{Y}_{1}\cup\cdots\cup Y_{\nu}$$\mathrm{Y}_{i}=${$K^{\mathrm{x}2}$$y\in \mathrm{Y}_{K;}f_{2}(y)\equiv\epsilon_{i}$be a quasicharacter, i.e., a continuous homo:Letmorphism.
and the basic $K$ -relative invariants $f(x)=(f_{1}(x),$For)$i(fi(x))$$f_{l}(x))$ , we write $\omega(f(x))$ instead offor simplicity of notations.
be the absolute value of If normalized by $|\pi|=q^{-1}$ for a prime elementLet
sowhere is the module of $K$ . For $s=(s_{1}, \cdots, s_{l})$ , we writethat $\omega_{s}(f(x))=\Pi_{i=1}^{l}|f_{i}(x)|^{S}‘$ .
where $R$ is
Let $dx$ be the Haar measure on $V_{K}=K^{n}$ normalized by
the maximal compact subring of $K$ . Since $d(\rho(g)_{X})=|\det\rho(g)|dX$ andis a -invariant measure on Y.
, the measure, wedenotes the Schwartz-Bruhat space of
where
For
define an integral$\omega^{(i)}$$K^{\cross}arrow \mathrm{C}^{\cross}(i=1, \cdots, l)$$\omega=(\omega^{(1)}, \cdots, \omega)(p)$$\cdot\cdot$$\Pi_{i=1}^{l(}\omega$

$.,$

$||$

$\pi$

$\omega_{s}--(||^{s_{1}}, \cdots, ||^{s_{\iota}})$$q$$\int_{R^{\tau\iota}}d_{X}=1$$\omega_{\lambda}(f(\rho(g)x))=$$|\det\rho(g)|\omega_{\lambda}(f(x))$$\Phi\in \mathfrak{S}(V_{K})$$d_{Y}(x)= \frac{dx}{\omega_{\lambda}(f(x))}$$G$$V_{K}$$\mathfrak{S}(V_{K})$$Z_{i}( \omega, \Phi)=\int_{Y_{1}}\omega(f(_{X))\Phi}(X)dY(x)(i=1, \cdots, \nu)$.

can be written:
Now any quasi-character:andfor some
uniquely asis the units of $R$ . Put $Re\omega^{(i)}=Res_{i}(i=1, \cdots, l)$ . The following lemma
whereis easy to prove and we omit the proof (cf. F.Sato [15]).
$\omega^{(i)}$$\omega^{(i)}=||^{s_{i}}\cdot\phi_{i}$$R^{\cross}$

Lemma 2.2

$K^{\cross}arrow \mathrm{C}^{\cross}=\{z\in \mathrm{C};z\neq 0\}$$s_{i}\in \mathrm{C}$$\phi_{i}$$R^{\cross}arrow \mathrm{C}_{1}^{\cross}=\{z\in \mathrm{C};|z|=1\}$

159

If

$Re\omega^{(i)}>\lambda_{i}(i=1, \cdot\cdot, l)$and holomorphic with respect to.

, the integral$Z_{i}(\omega, \Phi)$

is absolutely convergent$s=(s_{1}, \cdot\cdot., s_{\ell})\in(\mathrm{C}/(\frac{2\pi i}{logq}\mathrm{Z}))\iota\cong \mathrm{C}^{\mathrm{x}l}$for$\omega=$$(||^{s_{1}}\cdot\phi_{1}, \cdots, ||^{s_{1}}\cdot\phi_{l})$Let–linear mapping} be the space of distribu{. By Lemma 2.2, the mapping
tions on. defines a distribution on$Re\omega^{(i)}>\lambda_{i}(i=1,\ldots,l)$when.
, we can define similar distribution $Z_{j}^{*}(\omega)(j=1, \cdots, \nu)$ given byFor$\mathfrak{S}’(V_{K})=$$z:\mathfrak{S}(V_{K})arrow \mathrm{C},$$\mathrm{C}$$V_{K}$$\Phi\mapsto Z_{i}(\omega, \Phi)$$V_{K}$$(G, \rho^{*}, V^{*})$$Z_{j}^{*}( \omega, \Phi*)=\int_{Y_{j}^{*}}\omega(f^{*}(y))\Phi^{*}(y)d_{Y^{*()}}y$

Now we fix a non-trivial additive character :transformationby$\psi$$Karrow \mathrm{C}_{1}^{\cross}$.
and define the Fourier$\mathfrak{S}(V_{K}^{*})\ni\Phi^{*}\mapsto\tilde{\Phi}^{*}\in \mathfrak{S}(V_{K})$$\hat{\Phi}^{*}(x)=\int_{V^{*}}\Phi*(Ky)\psi(\langle_{X}, y\rangle)dy$

where $dy$ is a Haar measure on
dual to a fixed Haar measure on
.
, put
For
.
purposeOur
is to show thatandare continued analytically to all
and satisfy the functional equation:$V_{K}$$V_{K}^{*}$$\omega=(\omega^{(1)}, \mathrm{r}\cdot\cdot, \omega)(l)$$\omega^{*}=\omega^{U}=(\Pi_{i=1}^{l}\omega(i)u_{i1}, \cdots, \Pi_{i=1}l)u_{i\iota})\omega^{(i}$$Z_{i}(\omega)$$\omega$(2.1)$Z_{j}^{*}(\omega)$$\hat{Z}_{i}(\omega)=\sum_{j=1}^{\nu}\Gamma_{i}j(\omega)z_{j(\omega^{*}}*\omega\lambda^{*})(i=1, \cdots, \nu)$under some additional conditions where. Recall that$\hat{Z}_{i}(\omega)(\Phi*)=Z_{i}(\omega,\hat{\Phi}^{*})$$\omega_{\lambda^{*}}=(||_{\lambda_{1}^{*}}, \ldots., ||_{\lambda_{l}^{*}})$

with $\det\rho^{*}(g)^{2}=$$\chi_{1}^{2\lambda_{1}^{*}}\cdots\chi_{l}^{2\lambda_{l}^{*}}$

Actually when $K$ is a local field of $ch(K)=0$ , then (2.1) is obtained under someconditions and it is called “ the fundamental theorem of. over $K$ ”.
$\mathrm{P}.\mathrm{V}$\S 3. Rationality for almost all$\mathrm{p}$.
denotes the local field with the constant field
For a rational prime , let, we denote modby . Then we have theForfollowing theorem which is suggested by Professor M.Kashiwara.
$p$$f\in \mathrm{Z}[x_{1n}, \cdots, x]$$IC_{p}$$\mathrm{F}_{p}$$f$$p\in \mathrm{F}_{p}[x_{1,n}\ldots, X]$$f_{p}$

160

Theorem 3.1
For almost all , the integral$p$$Z_{p}(s, \Phi)p=\int_{K_{\mathit{1}}^{\prime\iota}},|fp(x)|^{S}K_{l’}\Phi(px)d_{p}X$on

is a rational function of $t=p^{-s}$ where$K_{p}^{n}$$\Phi_{p}\in \mathfrak{S}(K_{p}^{n})$and$d_{p}x$is a Haar measure.

[Proof]$X=\Omega^{n}$the affine space
Let $K=\mathrm{Q}((t))$ be a field of formal power series over
$Xarrow\Omega$$f(x)$; then there
defined by
and $X_{K}=K^{n}$ . Let denote the morphism$Y$: $Yarrow X$
morphismprojectiveavarietyandalgebraic
nonsingularaexists$K$
both defined over with the following property: let denote an arbitrary point of
the local ring of at relative to $K$ (consisting of “ functions ” definedthen there exists an ideal basis
the ideal of non-units ofover $K$ ), andfor, and integers $N_{i}\geqq 0,$, elements $u,$ ofof$1\leqq i\leqq n$ such that$\mathrm{Q},$

$f$

$h$

$b$

$\mathrm{Y}_{K},$$Y$$\mathrm{O}_{K}$$b$$\mathit{1}\supset_{K;}$$\mathfrak{M}_{K}$$(y_{1}, \cdots, y_{n})$$\nu_{i}\geqq 1$$\mathrm{O}_{K}-\mathfrak{M}_{K}$$v$$\mathfrak{M}_{K}$$f \circ h=u\cdot\prod_{i=1}^{n}y_{i}^{N}i$,$h^{*}(dx)=v \cdot\prod_{i1}^{n}=y_{i}^{\nu-1}diy$.

The existence of such a pair $(Y, h)$ is guaranteed by Hironaka’s theorem [5] p.109-p.326]. Then for almost all , the reduction modulo is well-defined and we have. etc. Then by just similar argument as in Appendix ofsimilar results for
Igusa [11], we obtain our result.
$p$

$p$

$K_{p},$ $f_{p},$$\cdot$$\square$Remark 3.2
Let $K$ be a number field. For
Theorem 3.1 for almost all prime ideals$f\in 1\supset_{K}[x_{1}, \cdots, x_{n}]$$\mathfrak{P}$of$4\supset_{K}$, we have a similar result as.

\S 4. Functional equations
Lemma 4.1
denote a locally compact totally disconnected group, $H$ a closed subgroupLeta quasicharacter. Putof $G,$ $X=H\backslash G$ , and :$T\in \mathfrak{S}(X)’;gT=\omega(g)^{-1}\tau${for all $g\in G$ }.
. Moreover $dim_{\mathrm{C}}\xi_{X}(\omega)=1$ if and only if
Then we havedenotes the module of $G,$ $H$ respectively.
where
$G$

$\omega$$Garrow \mathrm{C}^{\cross}$$\xi_{X}(\omega)=$$dim_{\mathrm{C}}\xi_{X}(\omega)\leqq 1$$\Delta_{G}.\omega|_{H}=\Delta_{H}$[Proof]See Igusa [9]$\triangle_{G},$$\mathrm{p}.1015.\square$$\Delta_{H}$161

Let $(G, \rho, V)$ and its dual$(G, \rho^{*}, V^{*})$

be $K$ -regular P.V.’s with$\#\rho(G)K\backslash YK=\nu<+\infty$

where $K$ is a local field of characteristic . Then, by Proposition 1.3, we haveand
. i.e.,.
As in \S 2, we can define the zeta distribution(resp.
) which(resp.
is convergent when
) $(1\leq i\leq\nu , 1 \leq j\leq l)$ .
We denote by
the distribution defined by
etc.
$p$

$Y_{K}=\mathrm{Y}_{1}\cup\cdot\cdot\cup Y_{\nu}$$\#\rho^{*}(c)_{K}\backslash Y_{K}*=\nu$$Y_{K}^{*}=\mathrm{Y}_{1}^{*}\cup\cdots\cup \mathrm{Y}_{\nu}^{*}$$Z_{i}(\omega, \Phi)$$Re\omega^{(j)}>\lambda_{j}$$\Phi*)$$Z_{i}^{*}(\omega,$$Re\omega^{(j)}>\lambda_{j}^{*}$$Z_{i}(\omega)$$\Phi\mapsto Z_{i}(\omega, \Phi)$

Proposition 4.2
We have
(1)
and
(2)

$z_{j}^{*}(\omega^{*}\omega\lambda*)\in\xi_{Y^{*(*}}j\omega*\omega_{\lambda})$$\hat{z}_{i}(\omega)\in\xi_{Y_{\mathrm{j}}^{*}}(\omega^{*}\omega\lambda^{*)}\cdot$$(i,j=1, \ldots., \nu)$[Proof]By a direct calculation, we obtain our results.

$\square$Proposition 4.3
Let $K$ be a local field of characteristic $p>0$ with the module . Forwith( $\phi_{i}(\pi)=1$ for a prime element ), assumethatandare rational functions of. Then for all, we have$\omega=$$q$$(\omega^{(1)}, \cdots, \omega^{(})\iota)$$\omega^{(i)}=\omega_{s_{i}}\cdot\phi_{i}$$Z_{i}(\omega, \Phi)$$\pi$$q^{-s_{1}},$$Z_{j}^{*}(\omega, \Phi*)$$\cdots,$$q-\mathit{8}_{l}$$\Phi^{*}\in \mathfrak{S}(Y_{K}^{*})$$Z_{i}( \omega,\hat{\Phi}^{*})=\sum_{j}\Gamma_{i})z^{*}(j\lambda*\omega\omega\Phi*,*)j(\omega$for $i,j=1,$$\cdots,$

$\nu$

.

[Proof]Since
andare rational functions, it is defined for allpoles and hence by Lemma 4.1 and Proposition 4.2, we have our result.
$Z_{i}(\omega, \Phi)$$Z_{j}(\omega, \Phi^{*})$$\omega$

except$\square$

Theorem 4.4
Let $(G, \rho, V)$ be a $K$ -regular P. V. satisfying the following conditions:$(Cl)$ its dualis If-regular P. V. such that,, there exists $\chi\in X_{1}(G)_{K}$ satisfying
where
for
is the units of the maximal compact subring $R$ of $K$
and
is a rational function ofwhere$\omega^{(i)}=\omega_{s}:(1\leqq i\leqq l)$ .
with
Then we have the functional equation$(G, \rho^{*}, V^{*})$$a$$\#\rho^{*}(G)_{K}\backslash V^{*}K<+\infty$$(C\mathit{2})$$\chi(c_{x,K})\not\leqq R^{\cross}$$x\in S_{K}^{*}$$(C\mathit{3})Z_{j}(\omega, \Phi)$$\omega=(\omega^{(1)..(l)},\cdot, \omega)$$q^{-s_{1}},$$\cdots,$$q-s_{l}$$R^{\cross}$162$Z_{i}( \omega,\hat{\Phi}^{*})=\sum_{j}\Gamma_{ij}(\omega)Z_{j}^{*}(\omega*\omega_{\lambda^{*}}, \Phi^{*})$

for all$\Phi^{*}\in \mathfrak{S}(V_{K}^{*})$

for $i,j=1,$$\cdots,$

$\nu$

where

$\nu=\#\rho^{*}(G)K\backslash \mathrm{Y}_{K}*$.

[Proof]corresponds to Lemma 2.2 in F.Sato [15] p474 for the case
The conditionof $ch(K)=0$ . Then the proof is just similar as the case of $ch(K)=0$ (using
Proposition 4.3) (See Igusa [9] and F.Sato [15] p.477).
$(C\mathit{2})$$\square$Now let $(G, \rho, V)$ be a reductive
reduction modulo and we obtain.
field with the constant field$p$$\mathrm{Q}$-regular-regular$\mathrm{P}.\mathrm{V}$$IC_{p}$. Then for almost all , we have a
is a local
where.

$\mathrm{P}.\mathrm{V}$$p$$(G_{p}, \rho_{p}, V_{p})$$K_{p}$$\mathrm{F}_{p}$(Assumption A )Assume that $\#\rho_{p}(G)K,,\backslash SK_{1)}<+\infty$ and forsatisfying
for almost all .
$\chi(G_{p,x,K})l^{l}\not\leqq R_{p}^{\cross}$.,$x\in S_{K_{l}},$, there exists. with (Assumption A). Let
Let $(G, \rho, V)$ be a reductive -regularmod
Denote-coefficients.
withbe basic -relative invariants
mod
, and
, by
$\mathrm{P}.\mathrm{V}$$|f_{1}$$\mathrm{Z}$$\mathrm{Q}$$f\iota$$p|_{K_{1}}^{s_{l}}$.

$p$$\mathrm{Q}$$|f_{l}$$\chi\in X_{1}(G_{p})_{K_{1}}$$f_{1},$$p|_{K_{1}}^{s_{1}},$$\cdot$.

$\cdots$$|f^{(p)}(x)|^{S}K_{\iota}$$Z_{i}^{p}(s, \Phi_{p})=\int_{(}Y_{K_{1^{l}}})‘|f^{(p)}(X)|_{K}s,\Phi(_{X}p)d_{Y(}J\mathrm{J}^{J}x)$for$\Phi_{p}\in \mathfrak{S}(V_{K_{p}})$.

Theorem 4.5(Assumption A ). Then for
Let $(G, \rho, V)$ be a reductive -regular P. V.
almost all rational prime , the integral
is a rational function and satisfies the functional equation:$u\mathit{1}\dot{i}th$$\mathrm{Q}$$p$$Z_{i}^{p}(s, \Phi_{\mathrm{P}})(i=1, \cdots, \nu_{p}, Y_{K_{\mathrm{J}}}, =Y_{1}\cup\cdots\cup \mathrm{Y}_{\nu \mathrm{p}})$$Z_{i}^{p}(S, \hat{\Phi}_{p})=\sum_{1j=}\Gamma_{ij}(S)Z_{j}p(_{S}*, \Phi_{p})\nu \mathrm{z}\mathrm{J}$$(i=1, \ldots., \nu_{p})$.

When $l=1$ , we have $s^{*}= \frac{n}{d}-sw\dot{i}thn=dimV$ and $d=degf$ . In general, for.
, we have$\omega=\omega_{s}=\omega_{s_{1}}\cdots\omega_{s_{\iota}}$$\omega_{s}*=\omega^{*}\omega_{\lambda}*$[Proof]
By Theorem 4.4 and using the results of \S 1 and \S 3, we obtain our result.

$\square$163

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