Derived Weil Representation and Relative Langlands Duality

TL;DR

This paper introduces a derived category version of the Weil representation for local fields, providing a new perspective on theta correspondence and relative Langlands duality for dual pairs.

Contribution

It presents a novel derived category formulation of the Weil representation and describes it explicitly for the dual pair (GL_n, GL_m) within the framework of relative Langlands duality.

Findings

Derived category version of Weil representation introduced

Explicit description for the dual pair (GL_n, GL_m)

Advances understanding of theta correspondence and Langlands duality

Abstract

The Weil representation is a particularly significant linear representation of the metaplectic group, used in the study of theta correspondence. In this paper, I introduce a derived category version of the Weil representation in the local field case. For the dual pair $ (\mathrm{GL}_n,\mathrm{GL}_m) $, I give a coherent description of this category, in the philosophy of relative Langlands duality.