Deformations of local systems and Eisenstein series

Abstract

Let $X$ be a (smooth and complete) curve and $G$ a reductive group. In [BG] we introduced the object that we called "geometric Eisenstein series". This is a perverse sheaf $\bar{Eis}_E$ (or rather a complex of such) on the moduli stack $Bun_G(X)$ of principal $G$-bundles on $X$, which is attached to a local system $E$ on $X$ with respect to the torus $\check{T}$, Langlands dual to the Cartan subgroup $T\subset G$. In loc. cit. we showed that$\bar{Eis}_E$ corresponds to the $\check{G}$-local system induced from $E$, in the sense of the geometric Langlands correspondence. In the present paper we address the following question, suggested by V. Drinfeld: what is the perverse sheaf on $Bun_G(X)$ that corresponds to the universal deformation of $E$ as a local system with respect to the Borel subgroup $\check{B}\subset \check{G}$? We prove, following a conjecture of Drinfeld, that the resulting perverse sheaf if the classical, i.e., non-compactified Eisenstein series.