A relative trace formula identity for non-tempered spherical varieties

TL;DR

This paper establishes a new relative trace formula identity linking period integrals of non-tempered and tempered spherical varieties, advancing the understanding of their duality and setting the stage for broader conjectural comparisons.

Contribution

It introduces a novel relative trace formula identity connecting non-tempered and tempered spherical varieties within the framework of the relative Langlands duality.

Findings

Proves a trace formula identity for non-tempered spherical varieties.

Links period integrals of different types of spherical varieties.

Proposes a conjecture for general non-tempered Hamiltonian spaces.

Abstract

In this paper, motivated by some previous works in residue method and the recent theory of the relative Langlands duality, we prove a relative trace formula identity that compares the period integral of non-tempered spherical varieties with the period integral of a tempered spherical varieties associated to a Levi subgroup. This allows us to incorporate numerous relative trace formula comparisons studied during the last four decades under the relative Langlands duality framework. We will also propose a conjectural comparison for general non-tempered Hamiltonian spaces.