Relative trace formula for compact quotient and pseudocoefficients for relative discrete series

TL;DR

This paper introduces the concept of relative pseudocoefficients for relative discrete series in real spherical spaces, constructs examples for hyperbolic spaces, and connects these to the spectral decomposition via a relative trace formula.

Contribution

It defines relative pseudocoefficients, proves their non-existence in certain symmetric spaces, constructs them for hyperbolic spaces, and links them to spectral decomposition using a relative trace formula.

Findings

Relative pseudocoefficients do not exist for certain symmetric spaces.

Strong relative pseudocoefficients are constructed for hyperbolic spaces.

Existence of strong pseudocoefficients implies presence in spectral decomposition.

Abstract

We introduce the notion of relative pseudocoefficient for relative discrete series of real spherical homogeneous spaces of reductive groups. We prove that such relative pseudocoefficient does not exist for semisimple symmetric spaces of type G(C)/G(R) and construct strong relative pseudocoefficients for some hyperbolic spaces. We establish a toy model for the relative trace formula of H.Jacquet for compact discrete quotient {\Gamma}\G. This allows us to prove that a relative discrete series which admits strong pseudocoefficient with sufficiently small support occurs in the spectral decomposition of L^2({\Gamma}\G) with a nonzero period.