A simple character formula
Annales Henri Lebesgue, Volume 4 (2021) , pp. 503-535.

Metadata

Keywordsreductive algebraic groups, characters, p-canonical basis

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 G in terms of baby Verma modules, under the assumption that the characteristic of the base field is bigger than 2h-1, where h is the Coxeter number of G. This provides a replacement for Lusztig’s conjecture, valid under a reasonable assumption on the characteristic.


References

[ACR18] Achar, Pramod N.; Cooney, Nicolas; Riche, Simon The parabolic exotic t-structure, Épijournal de Géom. Algébr., EPIGA, Volume 2 (2018), 8, 31 pages | MR 3894857 | Zbl 1408.14064

[AJS94] Andersen, Henning H.; Jantzen, Jens C.; Soergel, Wolfgang Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p: independence of p, Astérisque, 220, Société Mathématique de France, 1994 | Numdam | Zbl 0802.17009

[AMRW19] Achar, Pramod N.; Makisumi, Shotaro; Riche, Simon; Williamson, Geordie Koszul duality for Kac–Moody groups and characters of tilting modules, J. Am. Math. Soc., Volume 32 (2019) no. 1, pp. 261-310 | Article | MR 3868004 | Zbl 1450.20011

[And98] Andersen, Henning H. 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 | Article | MR 1670762 | Zbl 0918.20034

[AR19] Achar, Pramod N.; Riche, Simon 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 (2019), pp. 83-172 | Zbl 07272372

[BBM04] Bezrukavnikov, Roman; Braverman, Alexander; Mirković, Ivan Some results about geometric Whittaker model, Adv. Math., Volume 186 (2004) no. 1, pp. 143-152 | Article | MR 2065510 | Zbl 1071.20039

[BGM + 19] Bezrukavnikov, Roman; Gaitsgory, Dennis; Mirković, Ivan; Riche, Simon; Rider, Laura An Iwahori–Whittaker model for the Satake category, J. Éc. Polytech., Math., Volume 6 (2019), pp. 707-735 | Article | Numdam | MR 4014634 | Zbl 07114040

[BNPS20] Bendel, Christopher P.; Nakano, Daniel K.; Pillen, Cornelius; Sobaje, Paul Counterexamples to the tilting and (p,r)-filtration conjectures, J. Reine Angew. Math., Volume 767 (2020), pp. 193-202 | Article | MR 4160306 | Zbl 07268726

[BY13] Bezrukavnikov, Roman; Yun, Zhiwei On Koszul duality for Kac–Moody groups, Represent. Theory, Volume 17 (2013), pp. 1-98 | Article | MR 3003920 | Zbl 1326.20051

[Don93] Donkin, Stephen On tilting modules for algebraic groups, Math. Z., Volume 212 (1993) no. 1, pp. 39-60 | Article | MR 1200163 | Zbl 0798.20035

[Fie10] Fiebig, Peter Lusztig’s conjecture as a moment graph problem, Bull. Lond. Math. Soc., Volume 42 (2010) no. 6, pp. 957-972 | Article | MR 2740015 | Zbl 1234.20054

[Fie11] Fiebig, Peter Sheaves on affine Schubert varieties, modular representations, and Lusztig’s conjecture, J. Am. Math. Soc., Volume 24 (2011) no. 1, pp. 133-181 | Article | MR 2726602 | Zbl 1270.20053

[Fie12] Fiebig, Peter An upper bound on the exceptional characteristics for Lusztig’s character formula, J. Reine Angew. Math., Volume 673 (2012), pp. 1-31 | Article | MR 2999126 | Zbl 1266.20059

[Hum71] Humphreys, James E. Modular representations of classical Lie algebras and semisimple groups, J. Algebra, Volume 19 (1971), pp. 51-79 | Article | MR 283038 | Zbl 0219.17003

[IM65] Iwahori, Nagayoshi; Matsumoto, Hideya On some Bruhat decomposition and the structure of the Hecke rings of 𝔭-adic Chevalley groups, Publ. Math., Inst. Hautes Étud. Sci., Volume 25 (1965), pp. 5-48 | Article | Numdam | Zbl 0228.20015

[Jan80] Jantzen, Jens C. Darstellungen halbeinfacher Gruppen und ihrer Frobenius–Kerne, J. Reine Angew. Math., Volume 317 (1980), pp. 157-199 | MR 581341 | Zbl 0451.20040

[Jan03] Jantzen, Jens C. Representations of algebraic groups, Mathematical Surveys and Monographs, 107, American Mathematical Society, 2003 | Zbl 1034.20041

[JMW14] Juteau, Daniel; Mautner, Carl; Williamson, Geordie Parity sheaves, J. Am. Math. Soc., Volume 27 (2014) no. 4, pp. 1169-1212 | Article | MR 3230821 | Zbl 1344.14017

[JMW16] Juteau, Daniel; Mautner, Carl; Williamson, Geordie Parity sheaves and tilting modules, Ann. Sci. Éc. Norm. Supér., Volume 49 (2016) no. 2, pp. 257-275 | Article | MR 3481350 | Zbl 1402.14023

[JW17] Jensen, Lars T.; Williamson, Geordie The p-canonical basis for Hecke algebras, Categorification and higher representation theory (Beliakova, Anna, ed.) (Contemporary Mathematics), Volume 683, American Mathematical Society, 2017, pp. 333-361 | Article | MR 3611719 | Zbl 1390.20001

[KL93a] Kazhdan, David; Lusztig, George Tensor structures arising from affine Lie algebras I, J. Am. Math. Soc., Volume 6 (1993) no. 4, pp. 905-947 | Article | MR 1186962 | Zbl 0786.17017

[KL93b] Kazhdan, David; Lusztig, George Tensor structures arising from affine Lie algebras II, J. Am. Math. Soc., Volume 6 (1993) no. 4, pp. 949-1011 | Article | Zbl 0786.17017

[KL94a] Kazhdan, David; Lusztig, George Tensor structures arising from affine Lie algebras III, J. Am. Math. Soc., Volume 7 (1994) no. 2, pp. 335-381 | Article | MR 1239506 | Zbl 0802.17007

[KL94b] Kazhdan, David; Lusztig, George Tensor structures arising from affine Lie algebras IV, J. Am. Math. Soc., Volume 7 (1994) no. 2, pp. 383-453 | Article | MR 1239507 | Zbl 0802.17008

[KT95] Kashiwara, Masaki; Tanisaki, Toshiyuki Kazhdan–Lusztig conjecture for affine Lie algebras with negative level, Duke Math. J., Volume 77 (1995) no. 1, pp. 21-62 | MR 1317626 | Zbl 0829.17020

[KT96] Kashiwara, Masaki; Tanisaki, Toshiyuki 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 0929.17027

[Lus80a] Lusztig, George Hecke algebras and Jantzen’s generic decomposition patterns, Adv. Math., Volume 37 (1980), pp. 121-164 | Article | MR 591724 | Zbl 0448.20039

[Lus80b] Lusztig, George 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 (1980), pp. 313-317 | Article | MR 604598 | Zbl 0453.20005

[Lus94] Lusztig, George 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 1276910 | Zbl 0829.17018

[MR18] Mautner, Carl; Riche, Simon 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 | Article | Zbl 1401.14097

[Ric] Riche, Simon Geometric Representation Theory in positive characteristic (habilitation thesis, available at https://tel.archives-ouvertes.fr/tel-01431526)

[RW18] Riche, Simon; Williamson, Geordie Tilting modules and the p-canonical basis, Astérisque, 397, Société Mathématique de France, 2018 | Zbl 1437.20001

[RW20] Riche, Simon; Williamson, Geordie Smith–Treumann theory and the linkage principle (2020) (https://arxiv.org/abs/2003.08522)

[Sob20] Sobaje, Paul On character formulas for simple and tilting modules, Adv. Math., Volume 369 (2020), 107172 | MR 4092982

[Soe97] Soergel, Wolfgang Kazhdan–Lusztig polynomials and a combinatorics for tilting modules, Represent. Theory, Volume 1 (1997), pp. 83-114 | Article | MR 1444322 | Zbl 0886.05123

[Spr82] Springer, Tonny A. Quelques applications de la cohomologie d’intersection, Séminaire N. Bourbaki, Vol. 1981/82 (Astérisque), Volume 92–93, Springer, 1982, pp. 249-273 | Numdam | Zbl 0526.22014

[Wil17] Williamson, Geordie Schubert calculus and torsion explosion, J. Am. Math. Soc., Volume 30 (2017) no. 4, pp. 1023-1046 | Article | MR 3671935 | Zbl 1380.20015