Sheaves on affine Schubert varieties, modular representations, and Lusztig’s conjecture

Fiebig, Peter

doi:10.1090/S0894-0347-2010-00679-0

Sheaves on affine Schubert varieties, modular representations, and Lusztig’s conjecture
HTML articles powered by AMS MathViewer

by Peter Fiebig

J. Amer. Math. Soc. 24 (2011), 133-181

DOI: https://doi.org/10.1090/S0894-0347-2010-00679-0

Published electronically: September 23, 2010

PDF | Request permission

Abstract:

We relate a certain category of sheaves of $k$-vector spaces on a complex affine Schubert variety to modules over the $k$-Lie algebra (for $\operatorname {char} k>0$) or to modules over the small quantum group (for $\operatorname {char} k=0$) associated to the Langlands dual root datum. As an application we give a new proof of Lusztig’s conjecture on quantum characters and on modular characters for almost all characteristics. Moreover, we relate the geometric and representation-theoretic sides to sheaves on the underlying moment graph, which allows us to extend the known instances of Lusztig’s modular conjecture in two directions: We give an upper bound on the exceptional characteristics and verify its multiplicity-one case for all relevant primes.

References

Sergey Arkhipov, Roman Bezrukavnikov, and Victor Ginzburg, Quantum groups, the loop Grassmannian, and the Springer resolution, J. Amer. Math. Soc. 17 (2004), no. 3, 595–678. MR 2053952, DOI 10.1090/S0894-0347-04-00454-0
H. H. Andersen, J. C. Jantzen, and W. Soergel, Representations of quantum groups at a $p$th root of unity and of semisimple groups in characteristic $p$: independence of $p$, Astérisque 220 (1994), 321 (English, with English and French summaries). MR 1272539
Henning Haahr Andersen, The irreducible characters for semi-simple algebraic groups and for quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 732–743. MR 1403973
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
Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527. MR 1322847, DOI 10.1090/S0894-0347-96-00192-0
Roman Bezrukavnikov, Ivan Mirković, and Dmitriy Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic, Ann. of Math. (2) 167 (2008), no. 3, 945–991. With an appendix by Bezrukavnikov and Simon Riche. MR 2415389, DOI 10.4007/annals.2008.167.945
Tom Braden and Robert MacPherson, From moment graphs to intersection cohomology, Math. Ann. 321 (2001), no. 3, 533–551. MR 1871967, DOI 10.1007/s002080100232
Peter Fiebig, Sheaves on moment graphs and a localization of Verma flags, Adv. Math. 217 (2008), no. 2, 683–712. MR 2370278, DOI 10.1016/j.aim.2007.08.008
Peter Fiebig, The combinatorics of Coxeter categories, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4211–4233. MR 2395170, DOI 10.1090/S0002-9947-08-04376-6
—, The multiplicity one case of Lusztig’s conjecture, Duke Math. J. 153 (2010), no. 3, 551-571.
—, Lusztig’s conjecture as a moment graph problem, to appear in Bull. London Math. Soc., preprint arXiv:0712.3909 (2007).
—, An upper bound on the exceptional characteristics for Lusztig’s character formula, preprint arXiv:0811.1674 (2008).
Peter Fiebig and Geordie Williamson, Parity sheaves, moment graphs and the $p$-smooth locus of Schubert varieties, preprint 2010, arXiv:1008.0719.
Edward Frenkel and Dennis Gaitsgory, Local geometric Langlands correspondence and affine Kac-Moody algebras, Algebraic geometry and number theory, Progr. Math., vol. 253, Birkhäuser Boston, Boston, MA, 2006, pp. 69–260. MR 2263193, DOI 10.1007/978-0-8176-4532-8_{3}
Mark Goresky, Robert Kottwitz, and Robert MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), no. 1, 25–83. MR 1489894, DOI 10.1007/s002220050197
J. E. Humphreys, Modular representations of classical Lie algebras and semisimple groups, J. Algebra 19 (1971), 51–79. MR 283038, DOI 10.1016/0021-8693(71)90115-3
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
Shin-ichi Kato, On the Kazhdan-Lusztig polynomials for affine Weyl groups, Adv. in Math. 55 (1985), no. 2, 103–130. MR 772611, DOI 10.1016/0001-8708(85)90017-9
David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
David Kazhdan and George Lusztig, Schubert varieties and Poincaré duality, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 185–203. MR 573434
D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. I, II, J. Amer. Math. Soc. 6 (1993), no. 4, 905–947, 949–1011. MR 1186962, DOI 10.1090/S0894-0347-1993-99999-X
Masaki Kashiwara and Toshiyuki Tanisaki, Kazhdan-Lusztig conjecture for affine Lie algebras with negative level, Duke Math. J. 77 (1995), no. 1, 21–62. MR 1317626, DOI 10.1215/S0012-7094-95-07702-3
Shrawan Kumar, Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1923198, DOI 10.1007/978-1-4612-0105-2
George Lusztig, Hecke algebras and Jantzen’s generic decomposition patterns, Adv. in Math. 37 (1980), no. 2, 121–164. MR 591724, DOI 10.1016/0001-8708(80)90031-6
George Lusztig, Some problems in the representation theory of finite Chevalley groups, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979) Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, pp. 313–317. MR 604598
George Lusztig, Quantum groups at roots of $1$, Geom. Dedicata 35 (1990), no. 1-3, 89–113. MR 1066560, DOI 10.1007/BF00147341
Wolfgang Soergel, Kategorie $\scr O$, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), no. 2, 421–445 (German, with English summary). MR 1029692, DOI 10.1090/S0894-0347-1990-1029692-5
Wolfgang Soergel, Roots of unity and positive characteristic, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 315–338. MR 1357204
Wolfgang Soergel, Kazhdan-Lusztig-Polynome und eine Kombinatorik für Kipp-Moduln, Represent. Theory 1 (1997), 37–68 (German, with English summary). MR 1445511, DOI 10.1090/S1088-4165-97-00006-X
Wolfgang Soergel, On the relation between intersection cohomology and representation theory in positive characteristic, J. Pure Appl. Algebra 152 (2000), no. 1-3, 311–335. Commutative algebra, homological algebra and representation theory (Catania/Genoa/Rome, 1998). MR 1784005, DOI 10.1016/S0022-4049(99)00138-3