Microlocal characterization of Lusztig sheaves for affine quivers and 𝑔-loops quivers

Hennecart, Lucien

doi:10.1090/ert/595

Microlocal characterization of Lusztig sheaves for affine quivers and $g$-loops quivers
HTML articles powered by AMS MathViewer

by Lucien Hennecart PDF

Represent. Theory 26 (2022), 17-67 Request permission

Abstract:

We prove that for extended Dynkin quivers, simple perverse sheaves in Lusztig category are characterized by the nilpotency of their singular support. This proves a conjecture of Lusztig in the case of affine quivers. For cyclic quivers, we prove a similar result for a larger nilpotent variety and a larger class of perverse sheaves. We formulate conjectures concerning similar results for quivers with loops, for which we have to use the appropriate notion of nilpotent variety, due to Bozec, Schiffmann and Vasserot. We prove our conjecture for $g$-loops quivers ($g\geq 2$).

References

Pramod N. Achar, Anthony Henderson, Daniel Juteau, and Simon Riche, Weyl group actions on the Springer sheaf, Proc. Lond. Math. Soc. (3) 108 (2014), no. 6, 1501–1528. MR 3218317, DOI 10.1112/plms/pdt055
A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
Roman Bezrukavnikov and Ivan Losev, Etingof’s conjecture for quantized quiver varieties, Invent. Math. 223 (2021), no. 3, 1097–1226. MR 4213772, DOI 10.1007/s00222-020-01007-z
Joseph Bernstein and Valery Lunts, Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994. MR 1299527, DOI 10.1007/BFb0073549
A. A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of $\textrm {GL}_n$, Duke Math. J. 61 (1990), no. 2, 655–677. MR 1074310, DOI 10.1215/S0012-7094-90-06124-1
Tristan Bozec, Quivers with loops and perverse sheaves, Math. Ann. 362 (2015), no. 3-4, 773–797. MR 3368082, DOI 10.1007/s00208-014-1095-6
Tristan Bozec, Quivers with loops and generalized crystals, Compos. Math. 152 (2016), no. 10, 1999–2040. MR 3569998, DOI 10.1112/S0010437X1600751X
T. Bozec, O. Schiffmann, and E. Vasserot, On the number of points of nilpotent quiver varieties over finite fields, arXiv:1701.01797, 2017.
William Crawley-Boevey, Geometry of the moment map for representations of quivers, Compositio Math. 126 (2001), no. 3, 257–293. MR 1834739, DOI 10.1023/A:1017558904030
Alexandru Dimca, Sheaves in topology, Universitext, Springer-Verlag, Berlin, 2004. MR 2050072, DOI 10.1007/978-3-642-18868-8
Peter Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71–103; correction, ibid. 6 (1972), 309 (German, with English summary). MR 332887, DOI 10.1007/BF01298413
Lutz Hille and José Antonio de la Peña, Stable representations of quivers, J. Pure Appl. Algebra 172 (2002), no. 2-3, 205–224. MR 1906875, DOI 10.1016/S0022-4049(01)00167-0
Seok-Jin Kang, Masaki Kashiwara, and Olivier Schiffmann, Geometric construction of crystal bases for quantum generalized Kac-Moody algebras, Adv. Math. 222 (2009), no. 3, 996–1015. MR 2553376, DOI 10.1016/j.aim.2009.05.015
Masaki Kashiwara and Pierre Schapira, Microlocal study of sheaves, Astérisque 128 (1985), 235 (English, with French summary). Corrections to this article can be found in Astérisque No. 130, p. 209. MR 794557
Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990. With a chapter in French by Christian Houzel. MR 1074006, DOI 10.1007/978-3-662-02661-8
Yiqiang Li and Zongzhu Lin, AR-quiver approach to affine canonical basis elements, J. Algebra 318 (2007), no. 2, 562–588. MR 2371959, DOI 10.1016/j.jalgebra.2007.09.023
Ivan Losev, Wall-crossing functors for quantized symplectic resolutions: perversity and partial Ringel dualities, Pure Appl. Math. Q. 13 (2017), no. 2, 247–289. MR 3858010, DOI 10.4310/PAMQ.2017.v13.n2.a3
G. Lusztig, Intersection cohomology complexes on a reductive group, Invent. Math. 75 (1984), no. 2, 205–272. MR 732546, DOI 10.1007/BF01388564
George Lusztig, Character sheaves. I, Adv. in Math. 56 (1985), no. 3, 193–237. MR 792706, DOI 10.1016/0001-8708(85)90034-9
G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
G. Lusztig, Canonical bases arising from quantized enveloping algebras. II, Progr. Theoret. Phys. Suppl. 102 (1990), 175–201 (1991). Common trends in mathematics and quantum field theories (Kyoto, 1990). MR 1182165, DOI 10.1143/PTPS.102.175
G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), no. 2, 365–421. MR 1088333, DOI 10.1090/S0894-0347-1991-1088333-2
G. Lusztig, Affine quivers and canonical bases, Inst. Hautes Études Sci. Publ. Math. 76 (1992), 111–163. MR 1215594, DOI 10.1007/BF02699432
G. Lusztig, Tight monomials in quantized enveloping algebras, Quantum deformations of algebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992) Israel Math. Conf. Proc., vol. 7, Bar-Ilan Univ., Ramat Gan, 1993, pp. 117–132. MR 1261904
George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
I. Mirković, Character sheaves on reductive Lie algebras, Mosc. Math. J. 4 (2004), no. 4, 897–910, 981 (English, with English and Russian summaries). MR 2124171, DOI 10.17323/1609-4514-2004-4-4-897-910
I. Mirković and K. Vilonen, Characteristic varieties of character sheaves, Invent. Math. 93 (1988), no. 2, 405–418. MR 948107, DOI 10.1007/BF01394339
Markus Reineke, Quivers, desingularizations and canonical bases, Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000) Progr. Math., vol. 210, Birkhäuser Boston, Boston, MA, 2003, pp. 325–344. MR 1985731
Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589, DOI 10.1007/BFb0072870
Claus Michael Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583–591. MR 1062796, DOI 10.1007/BF01231516
Claus Michael Ringel, The preprojective algebra of a tame quiver: the irreducible components of the module varieties, Trends in the representation theory of finite-dimensional algebras (Seattle, WA, 1997) Contemp. Math., vol. 229, Amer. Math. Soc., Providence, RI, 1998, pp. 293–306. MR 1676227, DOI 10.1090/conm/229/03325
Olivier Schiffmann, Quivers of type $A$, flag varieties and representation theory, Representations of finite dimensional algebras and related topics in Lie theory and geometry, Fields Inst. Commun., vol. 40, Amer. Math. Soc., Providence, RI, 2004, pp. 453–479. MR 2057407, DOI 10.1215/s0012-7094-04-12114-1
Olivier Schiffmann, Lectures on canonical and crystal bases of Hall algebras, Geometric methods in representation theory. II, Sémin. Congr., vol. 24, Soc. Math. France, Paris, 2012, pp. 143–259 (English, with English and French summaries). MR 3202708
Olivier Schiffmann, Lectures on Hall algebras, Geometric methods in representation theory. II, Sémin. Congr., vol. 24, Soc. Math. France, Paris, 2012, pp. 1–141 (English, with English and French summaries). MR 3202707
Ralf Schiffler, Quiver representations, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, Cham, 2014. MR 3308668, DOI 10.1007/978-3-319-09204-1
Ben Webster, Are Lusztig’s perverse sheaves the only equivariant ones with nilpotent characteristic cycle?, MathOverflow, https://mathoverflow.net/questions/80658/are-lusztigs-perverse-sheaves-the-only-equivariant-ones-with-nilpotent-characte (visited on 2020-06-26).
Ben Webster, On generalized category $\mathcal {O}$ for a quiver variety, Math. Ann. 368 (2017), no. 1-2, 483–536. MR 3651581, DOI 10.1007/s00208-016-1438-6