Geometric Eisenstein series: twisted setting

Abstract

Let $G$ be a simple simply-connected group over an algebraically closed field $k$, and $X$ 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 étale ${\mathbb{Q}}_{\ell }$-sheaves) in analogy with [3]. We calculate the intersection cohomology sheaf on the version of Drinfeld compactification in our twisted setting. In the case of $G$ = SL${}_2$ we derive some results about the Fourier coefficients of our Eisenstein series. For $G$ = SL${}_2$ and $X={\mathbb{P}}^1$ we also construct the corresponding theta-sheaves and prove their Hecke property.