Relative Trace Formula for Compact Quotient and Pseudo-Coefficients for Relative Discrete Series

Abstract

Abstract We introduce the notion of relative pseudo-coefficient for relative discrete series representations of real spherical homogeneous spaces of reductive groups. We prove that $K$-finite relative pseudo-coefficient does not exist for semisimple symmetric spaces of type $G_{\mathbb{C}}/G_{\mathbb{R}}$, where $K$ is a maximal compact subgroup of $G_{\mathbb{C}}$, and construct strong relative pseudo-coefficients for some hyperbolic spaces. We establish a toy model for the relative trace formula of H. Jacquet for compact discrete quotient $\Gamma \backslash G$. This allows us to prove that a relative discrete series representation, which admits strong pseudo-coefficients with sufficiently small support, occurs in the spectral decomposition of $L^2(\Gamma \backslash G)$ with a nonzero period.