## 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compactification of the moduli stack of $$G$$-bundles on a smooth projective curve, discovered by Drinfeld but still unpublished.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:

 [1] A. A. Beilinson, “How to glue perverse sheaves” in K-Theory, Arithmetic and Geometry (Moscow, 1984-1986), Lecture Notes in Math. 1289, Springer, Berlin, 1987, 42-51. [2] A. A. Beilinson and J. Bernstein, “A proof of Jantzen conjectures” in I. M. Gelfand Seminar, Part 1(Moscow, 1993), Adv. Sov. Math. 16, Amer. Math. Soc., Providence, 1993, 1-50. · Zbl 0790.22007 [3] A. A. Beilinson, J. Bernstein, and P. Deligne, “Faisceaux pervers” in Analysis and Topology on Singular Spaces, I (Luminy, 1981), Astérisque 100, Soc. Math. France, Paris, 1982, 5-171. [4] A. A. Beilinson and V. Drinfeld, Chiral Algebras, Amer. Math. Soc. Colloq. Publ. 51, Amer. Math. Soc., Providence, 2004. · Zbl 1138.17300 [5] A. A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves, preprint, http://www.math.uchicago.edu/ mitya/langlands/hitchin/BD-hitchin.pdf (accessed 14 October 2017). [6] R. Bezrukavnikov, M. Finkelberg, and V. Ostrik, Character $$D$$-modules via Drinfeld center of Harish-Chandra bimodules, Invent. Math. 188 (2012), 589-620. · Zbl 1267.20058 [7] R. Bezrukavnikov and D. Kazhdan, Geometry of second adjointness for $$p$$-adic groups, with an appendix by Y. Varshavsky, R. Bezrukavnikov, and D. Kazhdan, Represent. Theory 19 (2015), 299-332. · Zbl 1344.20064 [8] T. Braden, Hyperbolic localization of intersection cohomology, Transform. Groups 8 (2003), 209-216. · Zbl 1026.14005 [9] A. Braverman, M. Finkelberg, D. Gaitsgory, and I. Mirković, Intersection cohomology of Drinfeld’s compactifications, Selecta Math. (N.S.) 8 (2002), 381-418. · Zbl 1031.14019 [10] A. Braverman and D. Gaitsgory, Geometric Eisenstein series, Invent. Math. 150 (2002), 287-384. · Zbl 1046.11048 [11] A. Braverman and D. Gaitsgory, Deformations of local systems and Eisenstein series, Geom. Funct. Anal. 17 (2008), 1788-1850. · Zbl 1234.11155 [12] T.-H. Chen and A. Yom Din, A formula for the geometric Jacquet functor and its character sheaf analogue, Geom. Funct. Anal. 27 (2017), 772-797. · Zbl 1401.14087 [13] P. Deligne, La conjecture de Weil, II, Publ. Math. Inst. Hautes Études Sci. 52 (1980), 138-252. [14] V. Drinfeld, Moduli varieties of $$F$$-sheaves (in Russian), Funktsional. Anal. i Prilozhen. 21, no. 2 (1987), 23-41; English translation in Funct. Anal. Appl. 21 (1987), 107-122. [15] V. Drinfeld, Cohomology of compactified moduli varieties $$F$$-sheaves of rank $$2$$ (in Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 162 (1987), 107-158; English translation in J. Sov. Math. 46, no. 2 (1989), 1789-1821. [16] V. Drinfeld and D. Gaitsgory, Compact generation of the category of D-modules on the stack of $$G$$-bundles on a curve, Camb. J. Math. 3 (2015), 19-125. · Zbl 1342.14041 [17] V. Drinfeld and D. Gaitsgory, Geometric constant term functor(s), Selecta Math. (N.S.) 22 (2016), 1881-1951. · Zbl 1360.14060 [18] V. Drinfeld and J. Wang, On a strange invariant bilinear form on the space of automorphic forms, Selecta Math. (N.S.) 22 (2016), 1825-1880. · Zbl 1393.11044 [19] M. Emerton, D. Nadler, and K. Vilonen, A geometric Jacquet functor, Duke Math. J. 125 (2004), 267-278. [20] B. Feigin, M. Finkelberg, A. Kuznetsov, and I. Mirković, “Semi-infinite flags, II: Local and global intersection cohomology of quasimaps’ spaces” in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Amer. Math. Soc. Transl. Ser. 2 194, Amer. Math. Soc., Providence, 1999, 113-148. · Zbl 1076.14511 [21] M. Finkelberg and I. Mirković, “Semi-infinite flags, I: Case of global curve $$\mathbf{P}^{1}$$” in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Amer. Math. Soc. Transl. Ser. 2 194, Amer. Math. Soc., Providence, 1999, 81-112. · Zbl 1076.14512 [22] D. Gaitsgory, On de Jong’s conjecture, Israel J. Math. 157 (2007), 155-191. · Zbl 1123.11020 [23] D. Gaitsgory, A “strange” functional equation for Eisenstein series and miraculous duality on the moduli stack of bundles, preprint, arXiv:1404.6780v3 [math.AG]. · Zbl 1423.11118 [24] L. Lafforgue, Une compactification des champs classifiant les chtoucas de Drinfeld, J. Amer. Math. Soc. 11 (1998), 1001-1036. · Zbl 1045.11041 [25] G. Laumon, “Faisceaux automorphes liés aux séries d’Eisenstein” in Automorphic Forms, Shimura Varieties and $$L$$-Functions, Vol. 1 (Ann Arbor, Mich., 1988), Perspect. Math. 10, Academic Press, Boston, 1990, 227-281. [26] I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2) 166 (2007), 95-143. · Zbl 1138.22013 [27] S. Raskin, The geometric principal series category, in preparation. [28] Y. Sakellaridis, Inverse Satake transforms, preprint, arXiv:1410.2312v2 [math.RT]. [29] Y. Sakellaridis, Non-categorical structures in harmonic analysis, lecture, Math. Sci. Res. Inst., Berkeley, Calif., 21 November 2014, http://www.msri.org/workshops/708/schedules/19168. [30] Y. Sakellaridis and A. Venkatesh, Periods and harmonic analysis on spherical varieties, preprint, arXiv:1203.0039v4 [math.RT]. · Zbl 1479.22016 [31] S. Schieder, Geometric Bernstein asymptotics and the Drinfeld-Lafforgue-Vinberg degeneration for arbitrary reductive groups, preprint, arXiv:1607.00586v2 [math.AG]. [32] S. Schieder, Monodromy and Vinberg fusion for the principal degeneration of the space of $$G$$-bundles, preprint, arXiv:1701.01898v1 [math.AG]. [33] E. B. Vinberg, “On reductive algebraic semigroups” in Lie Groups and Lie Algebras: E. B. Dynkin’s Seminar, Amer. Math. Soc. Transl. Ser. 2 169, Amer. Math. Soc, Providence, 1995, 145-182. [34] J. Wang, Radon inversion formulas over local fields, Math. Res. Lett. 23 (2016), 565-591. · Zbl 1352.43007
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