数理解析研究所講究録 1468 巻 2006 年 158-162

158

AUTOMORPHIC GREEN FUNCTIONS FOR SYMMETRIC SPACESMASAO TSUZUKI

1. CLASSICLA CASE
be the Poincare’ upper half-plane,Letand a Fuchsian group of the first kind which acts on 5 by the usual Mobius transassociatedand the Laplacian
formation. Since the volume form-important, they yield the volume fromwith the Poincare metric $y^{-2}(dx^{2}+dy^{2})$ is. The resolvent operatorof the Riemanian surface
and the Laplacian$R_{s}=(\triangle \mathrm{x}+s(s-1))^{-1}$ of the shifted Laplacian $\triangle x+s(s-1)$ is an integral operator,:
is constructed as
whose kernel function$\ovalbox{\tt\small REJECT}$$=\{\tau\in \mathbb{C}|{\rm Im}(\tau)>0\}\cong \mathrm{S}\mathrm{L}_{2}(\mathbb{R})/\mathrm{S}\mathrm{O}(2)$$\Gamma$$-y^{2}( \frac{\partial^{2}}{\partial x^{2}} !

$\frac{dx\wedge d^{l}y}{y^{2}}$\frac{\partial^{2}}{\partial y^{2}})$$\mathrm{S}\mathrm{L}_{2}(\mathbb{R})$$X=\Gamma\backslash 5$$\triangle_{X}$$\omega_{X}$$G_{s}(z,$ $w,1$(1.1)$X\mathrm{x}$$G_{s}(z, w)= \sum_{\gamma\in\Gamma}\phi_{s}^{S\mathrm{j}}(\gamma z, w)$,$X-\triangle Xarrow \mathbb{C}$$({\rm Re}(s)>1, z\not\equiv w (\mathrm{m}\mathrm{o}\mathrm{d} \Gamma)))$$\phi\xi(z, w)=\frac{-1}{4\pi}\frac{\Gamma(s)^{2}}{\Gamma(2s)}(1-|\frac{z-w}{\overline{z}-w}|^{2}/)_{2}^{s}F_{1}(S,$$s_{1}\cdot 2s$;$1-| \frac{z-w}{\overline{z}-w}|^{2})$.

The series (1.1) is absolutely convergent if ${\rm Re}(s)>1$ , and the convergence is locallyuniform for and $(z, w)\in X><X-\triangle X$ . The function
is called the free space Greenfunction of and $G_{s}(z, w)$ the automorphic Green function of $X$ , which has been an
important object of reserch in the analytic theory of automorphic functions.
Among many properties of $G_{s}(z, w)$ , we focus on the following two.
(a) (Poisson equation) For each $w\in X$ ,$s$$\phi_{s}^{\mathfrak{H}}$$ffi$$([2]_{7\lfloor}^{\lceil}3], [4])$$(\triangle_{X,z}+s(s-1))G_{s}(z, u))=\delta_{w}(z)$.

(b) (square-integrability) $G_{s}(z, w)\in L^{2}(X\mathrm{x} X)$ .
These two properties are important because they enable us to have the spectral expansionof $G_{s}(z, w)$ in the space $L^{2}(X)$ in terms of basic wave functions,
Maass wave functions
and Eisenstein series.
The aim of this article is first to provide a proper definition of automorphic Green function for a pair of a higher dimensional locally symmetric space and its modular subvariety
generalising the classical construction, and then to state the basic properties of Green
funciton generalizing (a) and (b) above.
$\mathrm{i}.\mathrm{e}.$)2. GREEN FUNCTIONS
2.1. Notations and assumptions. Let G be a reductive Lie group with compact center.
Let 0 and a be involutions of G such that
(1) and are commutative, i.e,.
(2) 0 is a Cartan involution of G.
$\theta$$\sigma$$\theta\sigma=\sigma\theta$

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Then $K=G^{\theta}$ is a maximal compact subgroup of and $H=G^{\sigma}$ is a reductive closed
such that $H\cap K$ is maximally compact in $H$ . We further make two
subgroup ofassumptions. The first is that(3) the symmetric pair $(G, H)$ has -rank one,is a maximal abelian subspace ofsuch thatwhich means there exists a vector
belongsothatthe eigenvalues ofis supposed to be taken; the vector
andto $\{0, \pm 1, \pm 2\}$ . For $j\in$ { , il,}, let be the corresponding eigenspace ofassumption
is.
Thesecondandset(4) $m:=2^{-1}(rr\iota_{1}^{+}+m_{2}^{+}+1)\in \mathbb{Z}$ .
Note $m$ is the half of the -codimension of $H/H\cap K$ in $G/K$ .
$G$

$G$

$\mathbb{R}$$\mathbb{R}Y_{0}$$Y_{0}\in \mathfrak{g}$$\mathrm{a}\mathrm{d}(Y_{0})$$Y_{0}$$\mathfrak{g}^{-\sigma}\cap \mathfrak{g}^{-\theta}$$0$$\pm 2$$\mathrm{a}\mathrm{d}(Y_{0})$$\mathfrak{g}_{j}$$m_{j}^{\pm}=\dim_{\mathbb{R}}(\mathfrak{g}_{j}\cap \mathfrak{g}^{\pm\sigma\theta})$$m_{j}=\dim_{\mathbb{R}}(\mathfrak{g}_{j})$$\mathbb{R}$2.2. Free space Green function. Set$(m_{2}^{-}=0)$$Y_{\mathrm{O}}$$\tilde{Y}_{0}$

$=$

,

$\{$

$2$

$-$ $1$$(m_{2}^{-}>0)$$Y_{0}$

By a general theory, the set $\{a_{t}=\exp(tY_{0})|t\geq 0\}$ comprises a complete set of repre$\{a_{t}|t>$and the natural smooth map
sen.tatives for the double coset space$\phi_{s}(g)\inC^{\infty}(H\backslash(G-HK)/K)$$Karrow G-HK$ is a submersion. Let us define a function
depending on a complex one parameter by$H\mathrm{x}$$H\backslash G/K$

$0\}\mathrm{x}$$s$$\phi_{s}(a_{t})=C_{m}\frac{\Gamma(\frac{s+\rho 0}{2})\Gamma(\frac{s-\rho_{0}}{2}+m)}{\Gamma(s+1)}(\cosh t)_{2}^{-(s+\rho 0)}F_{1}(\frac{s+\rho_{0}}{2},$ $\frac{s-\rho_{0}}{2}+m;s[perp] 1;\frac{1}{\cosh^{2}\#})$

with$\rho_{0}=2^{-1}\mathrm{t}\mathrm{r}(\mathrm{a}\mathrm{d}(\tilde{Y}_{0})|\mathfrak{g}_{1}+\mathfrak{g}_{2})$,$(t \neq 0)$,$-2^{-1}$$(m=1)$ ,$\overline{[perp]}(m-1)^{-1}$$(m>1)$ .

$C_{m}=\{$

has the following three properties, which
Proposition 1. Let ${\rm Re}(s)>0$ . The function
characterize .
be a -invariant symmetric -bilinear forrn on such that:
(1) Let. Let$-B(X,$
is -invariant and positive definite and such thatbe the Casimir element corresponding to B. Then$\phi_{s}$$\phi_{s}$$B$$\mathfrak{g}\cross$$\theta Y\grave{)}$$\mathfrak{g}arrow \mathbb{R}$$G$$\mathbb{R}$$\mathfrak{g}$$B(\tilde{1}_{0}^{\Gamma},\tilde{Y}_{0})=1$$\theta$$C_{\mathfrak{g}}$$C_{9}\phi_{s}(g)=(s^{2}-\rho_{0}^{2})\phi_{s}(g)$,$g\in G-HK$ .