Categorification of local relative Langlands duality

TL;DR

This paper formulates a categorical version of the local relative Langlands duality conjecture, verifies it for specific periods under certain assumptions, and explores its implications for distinction problems in representation theory.

Contribution

It introduces a categorical framework for the local relative Langlands duality conjecture and verifies it for key periods assuming the existence of a categorical local Langlands correspondence for GL(2).

Findings

Verification of the conjecture for Iwasawa-Tate periods

Verification of the conjecture for Hecke periods

Connections established between the conjecture and distinction problems

Abstract

We formulate the normalized period conjecture proposed by Ben-Zvi, Sakellaridis and Venkatesh in the framework of the categorical local Langlands correspondence and study its relation to distinction problems. Motivated by the work of Feng and Wang in the geometric setting, we verify the conjecture for the Iwasawa-Tate and Hecke periods, assuming the existence of the categorical local Langlands correspondence for $\mathrm{GL}_2$ with the Eisenstein compatibility.