数理解析研究所講究録909 巻 1995 年 154-159

154

Shintani Functions and Rankin-Selberg Convolution
I. Local Theory

京都産業大理村瀬篤(Atsushi Murase)

広島大理

菅野

孝史

(Takashi Sugano)In this note, we report a recent progress of the local theory of Shintani
functions on split orthogonal groups (joint work with Shin-ichi Kato).

\S 1. Notationand denote
Let be a non-archimedean local field with charthe integer ring of F. Fix a prime element of and put. For a positive integer , we put$(\mathrm{F})\neq 2$$\mathrm{F}$by$\mathit{0}$$\pi$$\#(\mathit{0}/\pi \mathit{0})$$\mathrm{F}$$\mathrm{q}=$$\mathrm{n}$$\mathrm{S}_{\mathrm{n}}=\{_{\{\begin{array}{l}00\int_{\mathrm{V}}020\mathrm{J}_{\mathrm{v}}00\end{array}\}}^{\{\begin{array}{l}0\mathrm{I}_{\mathrm{v}}\mathrm{I}_{\mathrm{v}}0\end{array}\}}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{f}\mathrm{n}\mathrm{n}\mathrm{i}\mathrm{s}\mathrm{e}_{\mathrm{d}}\mathrm{i}_{\mathrm{S}\mathrm{O}}\mathrm{v}\mathrm{e}_{\mathrm{d}}\mathrm{n}$

where
of
of

$\mathrm{S}_{\mathrm{n}}$$\mathrm{v}=[\frac{\mathrm{n}}{2}]$over$\mathrm{G}_{\mathrm{n}-1}$

and$\mathrm{I}_{\mathrm{v}}=\in \mathrm{G}\mathrm{L}_{\mathrm{V}}(\mathrm{p})$. Let$\mathrm{F}:\mathrm{G}_{\mathrm{n}}=\{\mathrm{g}\in \mathrm{G}\mathrm{L}_{\mathrm{n}}(\mathrm{F})|{}^{\mathrm{t}}\mathrm{g}\mathrm{S}_{\mathrm{n}}\mathrm{g}=\mathrm{S}_{\mathrm{n}}\}$

into$\mathrm{G}_{\mathrm{n}}$as follows (we put$\mathrm{v}’=[\frac{\mathrm{n}-1}{2}]$$\mathrm{G}_{\mathrm{n}}$be the orthogonal group. We define an embedding):$\iota_{\mathrm{n}}$155(a) If$\mathrm{n}$is even,$\iota_{\mathrm{n}}()=$

where $\in \mathrm{G}_{\mathrm{n}-1}$ is the block decomposition according to thepartition(b) If$\mathrm{n}$$\mathrm{n}-1=\mathrm{v}’+1+\mathrm{v}’$.

is odd,$\iota_{\mathrm{n}}()=$

where$\in \mathrm{G}_{\mathrm{n}-1}$

is the block decomposition according to the partition.

$\mathrm{n}-1=\mathrm{v}’+\mathrm{v}’$In what follows, we writeLet$\mathrm{v}=[\mathrm{n}/2]$

identify$\mathrm{G}’$(resp.

$\mathrm{G}’\cap \mathrm{G}\mathrm{L}_{\mathrm{n}}-1(\mathit{0}))$algebra of$\mathrm{v}’=[(\mathrm{n}-1)/2]$

with a subgroup of, and let(resp.

$(\mathrm{G}, \mathrm{K})$and$\mathrm{G}$$\mathrm{G}$$\mathrm{K}’)$

for

and

$\mathrm{G}_{\mathrm{n}}$be the Witt index ofvia . Put$\mathrm{S}_{\mathrm{n}}$respectively.

$\mathrm{G}_{\mathrm{n}-1}$(resp.

$\mathrm{K}=\mathrm{G}\cap \mathrm{G}\mathrm{L}_{\mathrm{n}}(\mathit{0})$$\iota_{\mathrm{n}}$$H=H(\mathrm{G}, \mathrm{K})$$(\mathrm{G}’,$$\mathrm{G}’$(resp.

$ff’=ff(\mathrm{G}’, \mathrm{K}\gamma)$). To parametrize$\mathrm{S}_{\mathrm{n}-1}\rangle$. We(resp.

$\mathrm{K}’=$be the Hecke$\mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{C}}(P\zeta \mathrm{C})$, let$\mathrm{X}_{\mathrm{u}\mathrm{n}\mathrm{r}}(\mathrm{F}^{\cross \mathrm{v}})$be the group of -tuples of unramified characters of . Let be thesubgroup of upper triangular matrices in (the standard minimal parabolicis regarded as a character of in a
subgroup of G). Then$\mathrm{F}^{\mathrm{x}}$$\mathrm{P}$$\mathrm{v}$$\mathrm{G}$$\chi\in \mathrm{X}_{\mathrm{u}\mathrm{n}\mathrm{r}}(\mathrm{F}\cross)^{\mathrm{v}}$natural manner. Define a function$\mathrm{p}\in \mathrm{P}$and, where$\mathrm{k}\in \mathrm{K}$$\delta_{\mathrm{P}}$$\Phi_{\chi}$$\mathrm{P}$on$\mathrm{G}$to be$\Phi_{\chi}(\mathrm{p}\mathrm{k})=(x6_{\mathrm{P}}^{1/})(\mathrm{p})2$

is the module of P. For$\chi^{\mathrm{A}}(\varphi)=\int_{\mathrm{c}}\varphi(\mathrm{g})\Phi x^{()\mathrm{d}}\mathrm{g}\mathrm{g}$

.

, put

$\varphi\in H$

for

156

Then

$\varphi\vdash i\chi^{\mathrm{A}}(\varphi)$defines an element of$\mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{C}}(P\zeta \mathrm{c})$. The correspondence X

onto
gives rise to a bijection fromis the Weyl group of (cf. [Sal). Similarly we can identify), whereis the Weyl group of .
C) with$\mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{C}}(g\zeta \mathrm{C})$$\mathrm{x}_{\mathrm{u}\mathrm{n}\mathrm{r}^{()^{\mathrm{V}}}\mathrm{c}}\mathrm{p}\mathrm{x}/\mathrm{w}$$|\Rightarrow \mathrm{X}^{\mathrm{A}}$$\mathrm{G}$, where,$\mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{C}}(ff’$$\mathrm{w}_{\mathrm{G}}$$\mathrm{G}’$$\chi_{\mathrm{u}\mathrm{n}\mathrm{r}^{(\mathrm{F}^{\cross}}}$$\mathrm{V}’/\mathrm{w}_{\mathrm{c}}’$$\mathrm{W}_{\mathrm{G}’}$\S 2. Main results
As in [MSI], we define the spaceon$\mathrm{G}$attached toof local Shintani functions$\mathrm{S}\mathrm{h}(\mathrm{X}, \mathrm{X}’)$’by$(l\mathrm{X}^{r})\in \mathrm{X}(\mathrm{F}\mathrm{u}\mathrm{n}\mathrm{r}\mathrm{u})\mathrm{m}\cross \mathrm{v}_{\cross \mathrm{x}}(\mathrm{F})\cross \mathrm{v}$$\mathrm{s}\mathrm{h}(\mathrm{X},\mathrm{X}’)=\mathrm{t}\mathrm{w}:\mathrm{G}arrow \mathrm{C}|(\mathrm{i})$(ii)W(k’gk)$=\mathrm{W}(\mathrm{g})(\mathrm{k}’\in \mathrm{K}’, \mathrm{k}\in \mathrm{K}, \mathrm{g}\in \mathrm{c})$$\varphi’*\mathrm{W}^{*}\varphi=\mathrm{X}’\mathrm{A}(\varphi’)_{\mathrm{X}^{\mathrm{A}}}(\varphi)\mathrm{w}(\varphi’\in H’, \varphi\in ff)\}$.

Here we put$( \varphi’*\mathrm{W}*\varphi)(\mathrm{g})=\mathrm{G}\int,$$\mathrm{d}\mathrm{X}’\int \mathrm{G}$dx.

$\varphi’(\mathrm{x}’)\mathrm{w}(_{\mathrm{X}}-1\mathrm{g}\mathrm{x})\varphi(\mathrm{x})$Note that Shintani functions can be regarded as spherical functions on a, wherespherical homogeneous spacein the sense of [Brl. The following
is a spherical subgroup ofhas been conjectured in [MSI].
$X=\mathrm{G}’\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\backslash \mathrm{G}’\cross \mathrm{G}$$\mathrm{G}^{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}’=\{(\mathrm{g}’, \mathrm{g}’)|\mathrm{g}’$$\mathrm{G}’\cross \mathrm{G}$$\in \mathrm{G}’\}$

Theorem 1 Let$=1$ . Moreover, there exists a

Then we have.
with$(l\mathrm{X}’)\in\chi_{\mathrm{u}\mathrm{n}\mathrm{r}}(\mathrm{F}\cross)\mathrm{v}\cross \mathrm{x}_{\mathrm{u}\mathrm{n}\mathrm{r}}(\mathrm{p})^{\mathrm{v}’}\cross$$\in \mathrm{S}\mathrm{h}(\chi, \chi’)$$\mathrm{W}_{l\mathrm{X}},$

To state an explicit formula forLet$\mathrm{W}_{\chi,\chi},$, where$\Pi_{\mathrm{m}}=\mathrm{d}_{\mathrm{n}}()\in \mathrm{G}$$\mathrm{w}_{l\mathrm{X}’}(1\rangle=1$, we need several preparations.

$\Lambda_{\mathrm{V}}=\{(\mathrm{m}_{1}, \ldots, \mathrm{m}_{\mathrm{V}})\in \mathrm{Z}^{\mathrm{V}}|\mathrm{m}_{1}\geq\ldots\geq \mathrm{m}_{\mathrm{V}}\geq 0\}$

we put$\mathrm{d}\mathrm{i}\mathrm{n}_{\mathrm{k}}\mathrm{S}\mathrm{h}(\chi, \chi’)$. For$\mathrm{m}=(\mathrm{m}_{1}, \ldots, \mathrm{m}_{\mathrm{V}})\in\Lambda_{\mathrm{v}}$

,

157

for A

$\mathrm{d}_{\mathrm{n}}(\mathrm{A})=\{_{\{\begin{array}{lll}\mathrm{A} 0 1 0 \mathrm{J}_{\mathrm{v}}^{\mathrm{t}_{\mathrm{A}^{-1}}}\mathrm{L}\end{array}\}}^{\{\begin{array}{ll}\mathrm{A} 00 \mathrm{J}_{\mathrm{v}\mathrm{h}\prime}^{\mathrm{t}-}\mathrm{A}^{1}\end{array}\}}\mathrm{i}\mathrm{f}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{n}\mathrm{i}_{\mathrm{S}\mathrm{O}}\mathrm{i}_{S}\mathrm{e}_{\mathrm{d}\mathrm{d}}\mathrm{v}\mathrm{e}\mathrm{n}$Similarly we define
given by$\Pi_{\mathrm{m}}’,$$\in \mathrm{G}’$for$\mathrm{m}’\in\Lambda_{V}$. Let$\mathrm{g}_{0}$$\in \mathrm{G}\mathrm{L}_{\mathrm{V}}(\mathrm{F})$.

be an element of$\mathrm{G}$$\mathrm{g}_{0}=\{_{\{\begin{array}{ll} 1_{\mathrm{V}}-2\eta-\eta\eta \mathrm{J}i\mathrm{t}0 \mathrm{t}_{\eta \mathrm{J}_{\mathrm{v}}}10 01_{\mathrm{V}}\end{array}\}}^{\mathrm{d}}\mathrm{n}(\mathrm{A}_{\mathrm{o}})\mathrm{i}\mathrm{f}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{n}\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}_{\mathrm{d}\mathrm{d}}\mathrm{V}\mathrm{e}\mathrm{n}$

where

and

$\mathrm{A}_{\mathrm{O}}=\in \mathrm{G}\mathrm{L}_{\mathrm{V}}(\mathrm{F})$the ”Cartan decomposition” for

Proposition2$\eta=\in \mathrm{F}^{\mathrm{V}}$.

$X$

Wehave(disjoint union)$\mathrm{G}=\prod_{\mathrm{m}\in\Lambda_{\mathrm{v}},\mathrm{m}\in\Lambda_{\mathrm{V}}\prime}\mathrm{K}\prime\prime(\mathrm{g}\mathrm{m}, \mathrm{m})\prime \mathrm{K}$where. The following result is$\mathrm{g}(\mathrm{m}, \mathrm{m}’)=\Pi_{\mathrm{m}}’,$$\mathrm{g}\mathrm{o}\Pi_{\mathrm{m}^{\in \mathrm{c}}}$

.

Put
1

if

$(1-\mathrm{q}^{-})\mathrm{V}$if$\mathrm{n}$is even$\mathrm{Q}_{\mathrm{n}}(\mathrm{q})=\prod^{1}(1-\mathrm{q}\mathrm{i}\mathrm{v}^{-}\mathrm{i}=1-2)\cross\{$Let$\chi=(\mathrm{X}_{1}, \ldots, \mathrm{X}_{})\in \mathrm{x}_{\mathrm{u}\mathrm{n}\mathrm{r}}(\mathrm{F}\cross \mathrm{v})$notation, we often write

respectively. Define$\chi_{\mathrm{i}}’$

and

and

$\chi_{\mathrm{j}}$$\mathrm{n}$is odd.

$\chi’=(\chi_{1’}’’\ldots, \chi_{\mathrm{V}},)\in \mathrm{X}_{\mathrm{u}\mathrm{m}^{()}}\mathrm{F}^{\cross \mathrm{V}’}$for the values) and. To simplify$\chi_{\mathrm{i}^{(}}’\pi$$\chi_{\mathrm{i}}(\pi)$158$\prod$$(1-\mathrm{q}^{-1}/2_{(_{\mathrm{X}\chi \mathrm{q}}\mathrm{i}}\prime 1\epsilon_{\mathrm{i}\dot{|}}(1--1/2_{(\mathrm{i}^{)}}-1_{)}\mathrm{i}^{\rangle)}\mathrm{X}_{\mathrm{i}^{\chi}}’$$\alpha_{\mathrm{X},\mathrm{X}}\gamma=.\frac{1<\lrcorner\leq \mathrm{v}’,1\leq_{i^{\leq \mathrm{v}}}}{\Delta_{\mathrm{G}}(\mathrm{X}^{)\cdot\Delta}\mathrm{c}\prime(\mathrm{x}\gamma}$

where

1
$\epsilon_{\mathrm{i}\mathrm{i}}=\{$

$-1$

if
if

$\mathrm{i}<\mathrm{j}$$\mathrm{i}\geq \mathrm{j}$and$\Delta_{\mathrm{G}^{(x^{)}=}\ell}.\prod_{\leq\ell\leq \mathrm{v}}(1-\chi_{\mathrm{k}}^{-1}\chi)(1-\mathrm{X}_{\mathrm{k}}x^{-1}\ell)-1\cross$($k^{\prime(\chi}\gamma$

is similarly defined).

Theorem 3Let$(\mathrm{m}, \mathrm{m}’)\in\Lambda_{\mathrm{V}}\cross\Lambda_{\sqrt}$$\mathrm{W}_{\chi,\chi},$$\in \mathrm{S}\mathrm{h}(\mathrm{X}, \mathrm{X}’)$$\mathrm{W}_{\chi,\chi},$be as in Theorem 1. Then, for, we have$(\mathrm{g}(\mathrm{m}, \mathrm{m}\gamma)$$\mathrm{W}\chi \mathrm{w}\chi\cdot\S^{1.1}/2_{)(}\Pi)(_{\mathrm{W}}\prime x’\mathrm{m})6/2(\Pi’,)’\gamma(\mathrm{P}\mathrm{p}\prime \mathrm{m}$’$=^{\frac{1}{\mathrm{Q}_{\mathrm{n}}(\mathrm{q})}} \mathrm{w}’\in \mathrm{w}_{\mathrm{c}}^{\mathrm{G}}’\sum_{\mathrm{w}\in \mathrm{W}}\alpha_{\mathrm{W}}x,$where$\mathrm{W}_{\mathrm{G}}$(resp.

) is the Weyl group$\mathrm{W}_{\mathrm{G}’}$of$\mathrm{G}$(resp.

module of the standard $\min$ imal parabolic subgroup$\mathrm{P}$) and$\mathrm{G}’$(resp.

$\delta_{\mathrm{P}}$)$\mathrm{P}’$of(resp. $*$ ) is the$\mathrm{G}$(resp.

$\mathrm{G}’$).

\S 3. Sketch of proof
The existence part of Theorem 1 is proved by using an integral
expression of Shintani functions similar to that of [MS21. We can prove
Theorem 3 following the method of [KM1, where an explicit formula for local
Shintani functions on GL(n) is shown.
We now give an outline of the proof of the uniqueness part of Theoremby, we define an element { , ml of
1. For$\mathrm{z}^{\mathrm{v}+}\mathrm{V}’$$(\mathrm{m}, \mathrm{m}’)\in\Lambda=\Lambda_{\mathrm{V}}\cross\Lambda_{\mathrm{V}’}$$\mathrm{m}$159$(\mathrm{m}_{1}, \mathrm{m}_{1’ 22\mathrm{V}}’’\mathrm{m}, \mathrm{m}, \ldots, \mathrm{m}, , \mathrm{n}_{’}, \mathrm{m})\mathrm{V}$ifis even (in this case$\mathrm{n}$$\mathrm{v}=\mathrm{v}’+1$$\{\mathrm{m}, \mathrm{m}’\}=\{$

if’$(\mathrm{m}_{1}, \mathrm{m}_{1’ 2}’’\mathrm{m}, \mathrm{m}_{2}, \ldots, \mathrm{m}_{\mathrm{v}}, , \mathrm{m}_{\mathrm{v}},)$We define a total ordering of A as follows:$l’\}<\{\mathrm{m}, \mathrm{m}’\}$

is odd (in this case$\mathrm{n}$$(\ell, l’)\prec(\mathrm{m}, \mathrm{m}’)$(in the usual lexicographic ordering of).

$\mathrm{v}=\mathrm{v}’$

if and only if

$\{l$

,

). The proof of the$\mathrm{Z}^{\mathrm{V}+\mathrm{V}’}$uniqueness of Shintani functions is reduced to the following:
Proposition 4Let$\mathrm{W}\in \mathrm{S}\mathrm{h}(\chi, \chi’)$and$(\mathrm{m}, \mathrm{m}’)\in\Lambda$. Then we have,$\mathrm{w}(\mathrm{g}(\mathrm{m}, \mathrm{m})’)=\sum_{\mathrm{c}}\ell l^{()(}’\chi,\chi’\mathrm{W}\mathrm{g}(\ell, \ell 0)$

where the summation is overis an element of$(\ell,l’)\in\Lambda$

with$(\ell,x’)\prec(\mathrm{m},\mathrm{m}’)$$\mathrm{C}[\chi_{1\pi}^{\pm 1}, \ldots,+1,(\mathrm{X}_{1}’)\pm 1, \ldots, (\mathrm{X}_{\mathrm{V}}’,)^{\pm 1}]$$\mathrm{c}_{p\ell},’(x, x’)$$(\ell, l’)$and$(\chi, \chi)’$, anddepending only on

and not on W.

The proposition follows from the next result, which is an analogue of
Proposition (4.4.4) in$[\mathrm{B}\mathrm{T}1\cdot$Let

Key lemma$\in \mathrm{K}’\mathrm{g}(\ell, x’)$$(\mathrm{m}, \mathrm{m}’),$

K. Then we have$(l, l’)\in\Lambda$

and$\mathrm{k}\in \mathrm{K}$$(\ell, l’)\prec=(\mathrm{m}, \mathrm{m}’)$, and$su_{W^{\mathrm{o}se}}$that$\Pi_{\mathrm{m}’}’\mathrm{k}\Pi_{\mathrm{m}}$.

References
Brion, M.: Classification des espaces homog\‘enes sph\’eriques,Compositio Math. 63, 189-208 (1987)[BT]
Bruhat, F. and Tits, J.: Groupes r\’eductifs sur un corps local: I.
Donn\’ees radicielles valu\’ees, I.H.E.S. Publ. Math. 41, 5-252 (1972)
[Br]

[KM]

Kato, S. and Murase, A.: in preparation[MS1]

Murase, A. and Sugano, T.: Shintani function and its application toautomorphic -functions for dassical groups: I. The orthogonal group case,
Mathematische Annalen 299, 17-56 (1994)[MS2] Murase, A. and Sugano, T.: Shintani functions and automorphic Lfunctions for GL(m), preprint[Sa]
Satake, I.: Theory of spherical functions on reductive algebraic groups
over -adic fields, I.H.E.S. Publ. Math. 18, 5-69 (1963)$\mathrm{L}$$\mathrm{P}$)