Duke Mathematical Journal

Picard–Lefschetz oscillators for the Drinfeld–Lafforgue–Vinberg degeneration for SL2

Simon Schieder

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Let G be a reductive group, and let BunG denote the moduli stack of G-bundles on a smooth projective curve. We begin the study of the singularities of a canonical compactification of BunG due to Drinfeld (unpublished), which we refer to as the Drinfeld–Lafforgue–Vinberg compactification Bun¯G. For G=GL2 and G=GLn, certain smooth open substacks of this compactification have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 Bun¯G is, however, already singular for G=SL2; questions about its singularities arise naturally in the geometric Langlands program, and form the topic of the present article. Drinfeld’s definition of 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=SL2, where the compactification can alternatively be viewed as a canonical one-parameter degeneration of the moduli space of SL2-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 Bun¯G. We then use this description to determine the intersection cohomology sheaf of 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=SL2 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

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

Received: 8 January 2017
Revised: 26 August 2017
First available in Project Euclid: 13 March 2018

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Zentralblatt MATH identifier

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

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


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

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