## Duke Mathematical Journal

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

Simon Schieder

#### Abstract

Let $G$ be a reductive group, and let $\operatorname{Bun}_{G}$ denote the moduli stack of $G$-bundles on a smooth projective curve. We begin the study of the singularities of a canonical compactification of $\operatorname{Bun}_{G}$ due to Drinfeld (unpublished), which we refer to as the Drinfeld–Lafforgue–Vinberg compactification $\overline{\operatorname{Bun}}_{G}$. For $G=\operatorname{GL}_{2}$ and $G=\operatorname{GL}_{n}$, 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 $\overline{\operatorname{Bun}}_{G}$ is, however, already singular for $G=\operatorname{SL}_{2}$; questions about its singularities arise naturally in the geometric Langlands program, and form the topic of the present article. Drinfeld’s definition of $\overline{\operatorname{Bun}}_{G}$ 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 $G=\operatorname{SL}_{2}$, where the compactification can alternatively be viewed as a canonical one-parameter degeneration of the moduli space of $\operatorname{SL}_{2}$-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 $\overline{\operatorname{Bun}}_{G}$. We then use this description to determine the intersection cohomology sheaf of $\overline{\operatorname{Bun}}_{G}$ 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 $G=\operatorname{SL}_{2}$ 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

#### 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

#### 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.
• [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.
• [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.
• [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.
• [8] T. Braden, Hyperbolic localization of intersection cohomology, Transform. Groups 8 (2003), 209–216.
• [9] A. Braverman, M. Finkelberg, D. Gaitsgory, and I. Mirković, Intersection cohomology of Drinfeld’s compactifications, Selecta Math. (N.S.) 8 (2002), 381–418.
• [10] A. Braverman and D. Gaitsgory, Geometric Eisenstein series, Invent. Math. 150 (2002), 287–384.
• [11] A. Braverman and D. Gaitsgory, Deformations of local systems and Eisenstein series, Geom. Funct. Anal. 17 (2008), 1788–1850.
• [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.
• [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.
• [17] V. Drinfeld and D. Gaitsgory, Geometric constant term functor(s), Selecta Math. (N.S.) 22 (2016), 1881–1951.
• [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.
• [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.
• [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.
• [22] D. Gaitsgory, On de Jong’s conjecture, Israel J. Math. 157 (2007), 155–191.
• [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].
• [24] L. Lafforgue, Une compactification des champs classifiant les chtoucas de Drinfeld, J. Amer. Math. Soc. 11 (1998), 1001–1036.
• [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.
• [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].
• [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.