A resolution of singularities of Drinfeld compactification with an Iwahori structure

Yang, Ruotao

doi:10.1090/ert/673

A resolution of singularities of Drinfeld compactification with an Iwahori structure
HTML articles powered by AMS MathViewer

by Ruotao Yang;

Represent. Theory 28 (2024), 366-380

DOI: https://doi.org/10.1090/ert/673

Published electronically: September 13, 2024

HTML | PDF | Request permission

Abstract:

The Drinfeld compactification $\overline {\operatorname {Bun}}{}_B’$ of the moduli stack $\operatorname {Bun}_B’$ of Borel bundles on a curve $X$ with an Iwahori structure is important in the geometric Langlands program. It is closely related to the study of representation theory. In this paper, we construct a resolution of singularities of it using a modification of Justin Campbell’s construction of the Kontsevich compactification. Furthermore, the moduli stack ${\operatorname {Bun}}_B’$ admits a stratification indexed by the Weyl group. For each stratum, we construct a resolution of singularities of its closure. Then we use this resolution of singularities to prove a universally local acyclicity property, which is useful in the quantum local Langlands program.

References

Dan Abramovich and Frans Oort, Stable maps and Hurwitz schemes in mixed characteristics, Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) Contemp. Math., vol. 276, Amer. Math. Soc., Providence, RI, 2001, pp. 89–100. MR 1837111, DOI 10.1090/conm/276/04513
A. Braverman, M. Finkelberg, D. Gaitsgory, and I. Mirković, Intersection cohomology of Drinfeld’s compactifications, Selecta Math. (N.S.) 8 (2002), no. 3, 381–418. MR 1931170, DOI 10.1007/s00029-002-8111-5
A. Braverman and D. Gaitsgory, Geometric Eisenstein series, Invent. Math. 150 (2002), no. 2, 287–384. MR 1933587, DOI 10.1007/s00222-002-0237-8
Justin Campbell, A resolution of singularities for Drinfeld’s compactification by stable maps, J. Algebraic Geom. 28 (2019), no. 1, 153–167. MR 3875364, DOI 10.1090/jag/727
L. Chen and Y. Fu, An extension of the Kazhdan-Lusztig equivalence, arXiv:2111.14606, 2021.
E. Frenkel, D. Gaitsgory, and K. Vilonen, Whittaker patterns in the geometry of moduli spaces of bundles on curves, Ann. of Math. (2) 153 (2001), no. 3, 699–748. MR 1836286, DOI 10.2307/2661366
D. Gaitsgory, Twisted Whittaker model and factorizable sheaves, Selecta Math. (N.S.) 13 (2008), no. 4, 617–659. MR 2403306, DOI 10.1007/s00029-008-0053-0
Takeshi Saito, The characteristic cycle and the singular support of a constructible sheaf, Invent. Math. 207 (2017), no. 2, 597–695. MR 3595935, DOI 10.1007/s00222-016-0675-3
Ruotao Yang, Twisted Whittaker category on affine flags and the category of representations of the mixed quantum group, Compos. Math. 160 (2024), no. 6, 1349–1417. MR 4744874, DOI 10.1112/S0010437X24007139