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Picard-Lefschetz oscillators for the Drinfeld-Lafforgue-Vinberg degeneration for \(\mathrm{SL}_2\). (English) Zbl 1398.14022

The author studies singularities of \(\overline{\text{Bun}}_G\) for \(G=\mathrm{SL}_2\), where \(\overline{\text{Bun}}_G\) is a canonical compactification of the moduli stack of \(G\)-bundles on a smooth projective curve, discovered by Drinfeld but still unpublished. The compactification is relative to the diagonal morphism of \(\text{Bun}_G\), and relies on the Vinberg semigroup of \(G\). For technical reasons it is more convenient to work with \(\text{VinBun}_G\), the total space of the canonical \(\mathbb{G}_m\) bundle on \(\text{Bun}_G\), that has the same singularities and is a one-parameter degeneration of \(\text{Bun}_G\).
From the author’s summary: “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 \(G=\mathrm{SL}_2\) with the miraculous duality of V. Drinfeld and D. Gaitsgory [Camb. J. Math. 3, No. 1–2, 19–125 (2015; Zbl 1342.14041); Sel. Math., New Ser. 22, No. 4, 1881–1951 (2016; Zbl 1360.14060); Ann. Sci. Éc. Norm. Supér. (4) 50, No. 5, 1123–1162 (2017; Zbl 1423.11118)] in the geometric Langlands program, as well as two applications of our results to the classical theory: to V. Drinfeld and J. Wang’s [Sel. Math., New Ser. 22, No. 4, 1825–1880 (2016; Zbl 1393.11044)] “strange” invariant bilinear form on the space of automorphic forms; and to the categorification of the Bernstein asymptotics map studied by R. Bezrukavnikov and D. Kazhdan [Represent. Theory 19, 299–332 (2015; Zbl 1344.20064)] as well as by Y. Sakellaridis and A. Venkatesh [Periods and harmonic analysis on spherical varieties. Paris: Société Mathématique de France (SMF) (2017; Zbl 1479.22016)].”

MSC:

14D24 Geometric Langlands program (algebro-geometric aspects)
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References:

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