Metadata
Abstract
In this paper we prove a character formula expressing the classes of simple representations in the principal block of a simply-connected semisimple algebraic group in terms of baby Verma modules, under the assumption that the characteristic of the base field is bigger than , where is the Coxeter number of . This provides a replacement for Lusztig’s conjecture, valid under a reasonable assumption on the characteristic.
References
[ACR18] The parabolic exotic t-structure, Épijournal de Géom. Algébr., EPIGA, Volume 2 (2018), 8, 31 pages | MR | Zbl
[AJS94] Representations of quantum groups at a -th root of unity and of semisimple groups in characteristic : independence of , Astérisque, 220, Société Mathématique de France, 1994 | Numdam | Zbl
[AMRW19] Koszul duality for Kac–Moody groups and characters of tilting modules, J. Am. Math. Soc., Volume 32 (2019) no. 1, pp. 261-310 | DOI | MR | Zbl
[And98] Tilting modules for algebraic groups, Algebraic groups and their representations (Cambridge, 1997) (NATO ASI Series. Series C. Mathematical and Physical Sciences), Volume 517, Kluwer Academic Publishers, 1998, pp. 25-42 | DOI | MR | Zbl
[AR19] Dualité de Koszul formelle et théorie des représentations modulaire des groupes algébriques réductifs, SMF 2018 : Congrès de la Société Mathématique de France (Séminaires et Congrès), Volume 33, Société Mathématique de France (2019), pp. 83-172 | Zbl
[BBM04] Some results about geometric Whittaker model, Adv. Math., Volume 186 (2004) no. 1, pp. 143-152 | DOI | MR | Zbl
[BGM + 19] An Iwahori–Whittaker model for the Satake category, J. Éc. Polytech., Math., Volume 6 (2019), pp. 707-735 | DOI | Numdam | MR | Zbl
[BNPS20] Counterexamples to the tilting and -filtration conjectures, J. Reine Angew. Math., Volume 767 (2020), pp. 193-202 | DOI | MR | Zbl
[BY13] On Koszul duality for Kac–Moody groups, Represent. Theory, Volume 17 (2013), pp. 1-98 | DOI | MR | Zbl
[Don93] On tilting modules for algebraic groups, Math. Z., Volume 212 (1993) no. 1, pp. 39-60 | DOI | MR | Zbl
[Fie10] Lusztig’s conjecture as a moment graph problem, Bull. Lond. Math. Soc., Volume 42 (2010) no. 6, pp. 957-972 | DOI | MR | Zbl
[Fie11] Sheaves on affine Schubert varieties, modular representations, and Lusztig’s conjecture, J. Am. Math. Soc., Volume 24 (2011) no. 1, pp. 133-181 | DOI | MR | Zbl
[Fie12] An upper bound on the exceptional characteristics for Lusztig’s character formula, J. Reine Angew. Math., Volume 673 (2012), pp. 1-31 | DOI | MR | Zbl
[Hum71] Modular representations of classical Lie algebras and semisimple groups, J. Algebra, Volume 19 (1971), pp. 51-79 | DOI | MR | Zbl
[IM65] On some Bruhat decomposition and the structure of the Hecke rings of -adic Chevalley groups, Publ. Math., Inst. Hautes Étud. Sci., Volume 25 (1965), pp. 5-48 | DOI | Numdam | Zbl
[Jan80] Darstellungen halbeinfacher Gruppen und ihrer Frobenius–Kerne, J. Reine Angew. Math., Volume 317 (1980), pp. 157-199 | MR | Zbl
[Jan03] Representations of algebraic groups, Mathematical Surveys and Monographs, 107, American Mathematical Society, 2003 | Zbl
[JMW14] Parity sheaves, J. Am. Math. Soc., Volume 27 (2014) no. 4, pp. 1169-1212 | DOI | MR | Zbl
[JMW16] Parity sheaves and tilting modules, Ann. Sci. Éc. Norm. Supér., Volume 49 (2016) no. 2, pp. 257-275 | DOI | MR | Zbl
[JW17] The -canonical basis for Hecke algebras, Categorification and higher representation theory (Beliakova, Anna et al., eds.) (Contemporary Mathematics), Volume 683, American Mathematical Society, 2017, pp. 333-361 | DOI | MR | Zbl
[KL93a] Tensor structures arising from affine Lie algebras I, J. Am. Math. Soc., Volume 6 (1993) no. 4, pp. 905-947 | DOI | MR | Zbl
[KL93b] Tensor structures arising from affine Lie algebras II, J. Am. Math. Soc., Volume 6 (1993) no. 4, pp. 949-1011 | DOI | Zbl
[KL94a] Tensor structures arising from affine Lie algebras III, J. Am. Math. Soc., Volume 7 (1994) no. 2, pp. 335-381 | DOI | MR | Zbl
[KL94b] Tensor structures arising from affine Lie algebras IV, J. Am. Math. Soc., Volume 7 (1994) no. 2, pp. 383-453 | DOI | MR | Zbl
[KT95] Kazhdan–Lusztig conjecture for affine Lie algebras with negative level, Duke Math. J., Volume 77 (1995) no. 1, pp. 21-62 | MR | Zbl
[KT96] Kazhdan–Lusztig conjecture for affine Lie algebras with negative level II: Non-integral case, Duke Math. J., Volume 84 (1996) no. 3, pp. 771-813 | Zbl
[Lus80a] Hecke algebras and Jantzen’s generic decomposition patterns, Adv. Math., Volume 37 (1980), pp. 121-164 | DOI | MR | Zbl
[Lus80b] Some problems in the representation theory of finite Chevalley groups, The Santa Cruz Conference on Finite Groups (Proceedings of Symposia in Pure Mathematics), Volume 37, American Mathematical Society (1980), pp. 313-317 | DOI | MR | Zbl
[Lus94] Monodromic systems on affine flag manifolds, Proc. R. Soc. Lond., Volume 445 (1994) no. 1923, pp. 231-246 Errata in Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences vol. 450, (1995), 731–732 | MR | Zbl
[MR18] Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirković–Vilonen conjecture, J. Eur. Math. Soc., Volume 20 (2018) no. 9, pp. 2259-2332 | DOI | Zbl
[Ric] Geometric Representation Theory in positive characteristic (habilitation thesis, available at https://tel.archives-ouvertes.fr/tel-01431526)
[RW18] Tilting modules and the -canonical basis, Astérisque, 397, Société Mathématique de France, 2018 | Zbl
[RW20] Smith–Treumann theory and the linkage principle (2020) (https://arxiv.org/abs/2003.08522)
[Sob20] On character formulas for simple and tilting modules, Adv. Math., Volume 369 (2020), 107172 | MR
[Soe97] Kazhdan–Lusztig polynomials and a combinatorics for tilting modules, Represent. Theory, Volume 1 (1997), pp. 83-114 | DOI | MR | Zbl
[Spr82] Quelques applications de la cohomologie d’intersection, Séminaire N. Bourbaki, Vol. 1981/82 (Astérisque), Volume 92–93, Springer, 1982, pp. 249-273 | Numdam | Zbl
[Wil17] Schubert calculus and torsion explosion, J. Am. Math. Soc., Volume 30 (2017) no. 4, pp. 1023-1046 | DOI | MR | Zbl