Geometric Whittaker models and Eisenstein series for Mp_2

TL;DR

This paper explores the geometric Langlands program for the metaplectic group Mp_2, focusing on Whittaker models, Eisenstein series, and Fourier coefficients within the derived category of sheaves on the moduli stack of metaplectic bundles.

Contribution

It develops the theory of geometric Eisenstein series and characterizes Whittaker sheaves for Mp_2, proposing conjectural relations with quantum Langlands and theta-lifting.

Findings

Describes the Whittaker category for Mp_2

Calculates Fourier coefficients of Eisenstein series

Proposes conjectural relations with quantum Langlands

Abstract

Let X be a smooth projective curve over an algebraically closed field of characteristic >2. Let Bun_{Mp_2} be the stack of metaplectic bundles on X of rank 2. In this paper we study the derived category of genuine l-adic sheaves on Bun_{Mp_2} in the framework of the quantum geometric Langlands. We describe the corresponding Whittaker category, develop the theory of geometric Eisenstein series and calculate the most non-degenerate Fourier coefficients of these Eisenstein series. The existing constructions of automorphic sheaves for GL_n are based on using Whittaker sheaves. Our calculations lead to a conjectural characterization of the Whittaker sheaf for Mp_2, though its existence is not clear. We also formulate a conjectural relation between the quantum Langlands functors and the theta-lifting functors for the dual pair (Mp_2, PGL_2).