Geometric Eisenstein series: twisted setting

TL;DR

This paper advances the theory of geometric Eisenstein series within the quantum geometric Langlands program, focusing on twisted settings, intersection cohomology, and specific cases like G=SL_2, providing new constructions and properties.

Contribution

It develops the theory of geometric Eisenstein series in the twisted quantum geometric Langlands setting, including intersection cohomology calculations and explicit constructions for G=SL_2.

Findings

Calculated intersection cohomology sheaves on twisted Drinfeld compactification.

Derived results on Fourier coefficients of Eisenstein series for G=SL_2.

Constructed theta-sheaves and established their Hecke property for G=SL_2 on P^1.

Abstract

Let G be a simple simply-connected group over an algebraically closed field k, X be a smooth connected projective curve over k. In this paper we develop the theory of geometric Eisenstein series on the moduli stack Bun_G of G-torsors on X in the setting of the quantum geometric Langlands program (for \'etale l-adic sheaves) in analogy with [3]. We calculate the intersection cohomology sheaf on the version of Drinfeld compactification in our twisted setting. In the case G=SL_2 we derive some results about the Fourier coefficients of our Eisenstein series. In the case of G=SL_2 and X=P^1 we also construct the corresponding theta-sheaves and prove their Hecke property.