Geometric Weil representation: local field case

TL;DR

This paper constructs a geometric analog of the Weil representation over local fields using perverse sheaves, aiming to advance the geometric Langlands program for dual reductive pairs.

Contribution

It introduces a new geometric framework for the Weil representation using perverse sheaves on stacks, linking representation theory and algebraic geometry.

Findings

Defines a category of perverse sheaves representing the Weil representation

Establishes functorial actions of the metaplectic extension on this category

Lays groundwork for applications in geometric Langlands correspondence

Abstract

Let k be an algebraically closed field of characteristic >2, F=k((t)) and Mp(F) denote the metaplectic extension of Sp_{2d}(F). In this paper we propose a geometric analog of the Weil representation of Mp(F). This is a category of certain perverse sheaves on some stack, on which Mp(F) acts by functors. This construction will be used in math.RT/0701170 (and subsequent publications) for a proof of the geometric Langlands functoriality for some dual reductive pairs.