Picard-Lefschetz oscillators for the Drinfeld-Lafforgue-Vinberg degeneration for SL_2

TL;DR

This paper investigates the singularities of a canonical compactification of SL_2-bundles on a curve, introducing Picard-Lefschetz oscillators to describe nearby cycles and linking the findings to the geometric Langlands program and classical automorphic theory.

Contribution

It introduces Picard-Lefschetz oscillators to explicitly describe the singularities of the Drinfeld-Lafforgue-Vinberg degeneration for SL_2 and connects these results to broader aspects of the Langlands program.

Findings

Explicit description of nearby cycles sheaf using Picard-Lefschetz oscillators

Determination of intersection cohomology sheaf of the degeneration

Connections to geometric Langlands duality and automorphic forms

Abstract

Let G be a reductive group and let Bun_G denote the moduli stack of G-bundles on a smooth projective curve. We begin the study of the singularities of a canonical compactification of Bun_G due to V. Drinfeld (unpublished), which we refer to as the Drinfeld-Lafforgue-Vinberg compactification. For G=GL_n certain smooth open substacks of this compactification have already appeared in the work of Drinfeld and of L. Lafforgue on the Langlands correspondence for function fields. The Drinfeld-Lafforgue-Vinberg compactification is however already singular for G=SL_2; questions about its singularities arise naturally in the geometric Langlands program, and form the topic of the present article. Drinfeld's definition of the compactification for a general reductive group G relies on the Vinberg semigroup of G, and will be given in [Sch]. In the present paper we focus on the case G=SL_2. In this case the compactification can alternatively be viewed as a canonical one-parameter degeneration of the moduli space of SL_2-bundles. We study the singularities of this degeneration via the weight-monodromy theory of its nearby cycles: We give an explicit description of the nearby cycles sheaf in terms of certain novel perverse sheaves which we call "Picard-Lefschetz oscillators" and which govern the singularities of the degeneration. We then use this description to determine its intersection cohomology sheaf and other invariants of its singularities. We also discuss the relationship of our results for G=SL_2 with the miraculous duality of Drinfeld and Gaitsgory in the geometric Langlands program, as well as two applications of our results to the classical theory: To Drinfeld's and Wang's "strange" invariant bilinear form on the space of automorphic forms; and to the categorification of the Bernstein asymptotics map studied by Bezrukavnikov and Kazhdan as well as by Sakellaridis and Venkatesh.