A relative trace formula identity for non-tempered spherical varieties
Chen Wan
TL;DR
The paper establishes a relative trace formula identity that links the period integral of non-tempered spherical varieties to the period integral of a tempered Levi-associated model, via an explicit transfer F_chi. Framed within the relative Langlands duality of Ben-Zvi–Sakellaridis–Venkatesh, it provides a conceptual bridge between non-tempered and tempered cases and extends earlier Jacquet–Rallis/Jiang–Mao–Rallis-type comparisons to all split reductive spherical varieties. The main result is proved by unfolding and a detailed case analysis (Table 1), showing that only a distinguished orbit contributes while others vanish due to non-temperedness or parabolic-induced structure, with removals of hypotheses in several models. The work also outlines a broader conjectural framework connecting period integrals to dual L-values and points toward generalizations to non-split and more general Hamiltonian spaces, guided by the BSV program.
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.