SINGULARITIES OF HOLOMORPHICALLY EXTENDED SPHERICAL FUNCTIONS

Abstract

1. Motivation and background This is a preliminary account of joint work in progress which serves as an (very) extended abstract for a presentation of one of us ( $E.M$ .O.) in the Awajishima conference on Representation Theory, November 16-19, 2004, Japan. To motivate the questions which we will address we will describe in some detail the beautiful ideas that were initiated by Sarnak [18] and by Bernstein and Reznikov [2], and then further explored by Kr\otz and Stanton [9]. Inspired by Sarnak [18], Bernstein and Reznikov [2] proposed a new method for estimating the coefficients in the expansion of the square of a Maass form on a compact locally symmetric space $Z=\Gamma\backslash X$ (where $X=G/K$ denotes a noncompact Riemannian symmetric space, and $\Gamma\subset G$ is a $co$-compact discrete subgroup of $G$ ) with respect to an orthonormal basis of $L^{2}(Z)$ consisting of Maass forms. The method is based on holomorphic extension of irreducible representations of $G$ to a certain $G$-invariant domain in $X_{\mathbb{C}}:=G_{\mathbb{C}}/If_{\mathbb{C}}$ (we assume that $G\subset G_{\mathbb{C}})$ . In [2] the method was applied in the case of $G=SL_{2}(\mathbb{R})$ . The method was carried further by Kr\otz and Stanton in [9], where the results of [2] were slightly improved for $G=SL_{2}(\mathbb{R})$ , and similar results for other rank 1 Riemannian symmetric spaces $G/K$ were obtained. In addition some higher rank cases were considered in [9]. These considerations gave rise to various interesting issues concerning holomorphic extensions of representations and their matrix coefficients.