## Duke Mathematical Journal

- Duke Math. J.
- Volume 167, Number 5 (2018), 835-921.

### Picard–Lefschetz oscillators for the Drinfeld–Lafforgue–Vinberg degeneration for ${SL}_{2}$

#### 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 We begin the study of the singularities of a canonical compactification of ${Bun}_{G}$ due to Drinfeld (unpublished), which we refer to as the *Drinfeld–Lafforgue–Vinberg compactification* ${\overline{Bun}}_{G}$. For $G={GL}_{2}$ and $G={GL}_{n}$, certain smooth open substacks of this compactification have already appeared in the work of Drinfeld and Lafforgue on the Langlands correspondence for function fields. The stack ${\overline{Bun}}_{G}$ 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 ${\overline{Bun}}_{G}$ for a general reductive group $G$ relies on the Vinberg semigroup of $G$ (we will study this case in a forthcoming article). In the present article we focus on the case $G={SL}_{2}$, where 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 one-parameter degeneration via the weight-monodromy theory of its nearby cycles. We give an explicit description of the nearby cycles sheaf together with its monodromy action in terms of certain novel perverse sheaves which we call *Picard–Lefschetz oscillators* and which govern the singularities of ${\overline{Bun}}_{G}$. We then use this description to determine the intersection cohomology sheaf of ${\overline{Bun}}_{G}$ and other invariants of its singularities. Our proofs rely on the construction of certain local models which themselves form one-parameter families of spaces which are *factorizable* in the sense of Beilinson and Drinfeld. We also briefly 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 on the level of functions: 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.

#### Article information

**Source**

Duke Math. J., Volume 167, Number 5 (2018), 835-921.

**Dates**

Received: 8 January 2017

Revised: 26 August 2017

First available in Project Euclid: 13 March 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.dmj/1520906430

**Digital Object Identifier**

doi:10.1215/00127094-2017-0044

**Mathematical Reviews number (MathSciNet)**

MR3782063

**Zentralblatt MATH identifier**

06870396

**Subjects**

Primary: 14D24: Geometric Langlands program: algebro-geometric aspects [See also 22E57]

Secondary: 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} 14D23: Stacks and moduli problems 11R39: Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E55] 14H60: Vector bundles on curves and their moduli [See also 14D20, 14F05] 55N33: Intersection homology and cohomology 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.) 14D06: Fibrations, degenerations 11F99: None of the above, but in this section

**Keywords**

geometric representation theory geometric Langlands program moduli spaces of G-bundles nearby cycles intersection cohomology Picard–Lefschetz theory weight-monodromy theory Vinberg semigroup wonderful compactification miraculous duality strange invariant bilinear form on automorphic forms Bernstein asymptotics

#### Citation

Schieder, Simon. Picard–Lefschetz oscillators for the Drinfeld–Lafforgue–Vinberg degeneration for $\operatorname{SL}_{2}$. Duke Math. J. 167 (2018), no. 5, 835--921. doi:10.1215/00127094-2017-0044. https://projecteuclid.org/euclid.dmj/1520906430