On a tamely ramified local relative Langlands conjecture via categorical representations

TL;DR

This paper advances the understanding of the local relative Langlands conjecture for complex reductive groups by providing a spectral description of certain D-modules on loop spaces, confirming a variant of the conjecture under specific conditions.

Contribution

It offers a spectral characterization of Iwahori-equivariant D-modules on loop spaces of spherical varieties, linking them to the relative Langlands dual, thus supporting a tamely ramified local relative Langlands conjecture.

Findings

Spectral description of Iwahori-equivariant D-modules on loop spaces.

Confirmation of a variant of the tamely ramified local relative Langlands conjecture.

Connection between D-modules and the relative Langlands dual of the variety.

Abstract

Let $G$ be a complex reductive group. For a smooth affine spherical $G$-variety $X$, assume that the unramified relative local Langlands conjecture of Ben-Zvi-Sakellaridis-Venkatesh for $X$ holds, the loop space $LX$ is an $L^+G$--placid ind--scheme, and there exists a dimension theory for $LX$, we give a spectral description of a full subcategory of Iwahori equivariant D-modules on $LX$ in terms of the relative Langlands dual of $X$, confirming a slight variant of the tamely ramified local relative Langlands conjecture proposed by Devalapurkar.