Bounding Harish-Chandra series

Dudas, Olivier; Malle, Gunter

doi:10.1090/tran/7600

Bounding Harish-Chandra series

Authors: Olivier Dudas and Gunter Malle
Journal: Trans. Amer. Math. Soc. 371 (2019), 6511-6530
MSC (2010): Primary 20C33; Secondary 20C08
DOI: https://doi.org/10.1090/tran/7600
Published electronically: October 18, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We use the progenerator constructed in a previous work to give a necessary condition for a simple module of a finite reductive group to be cuspidal, or more generally to obtain information on which Harish-Chandra series it can lie in. As a first application we show the irreducibility of the smallest unipotent character in any Harish-Chandra series. Secondly, we determine a unitriangular approximation to part of the unipotent decomposition matrix of finite orthogonal groups and prove a gap result on certain Brauer character degrees.

[1] Pramod N. Achar and Anne-Marie Aubert, Supports unipotents de faisceaux caractères, J. Inst. Math. Jussieu 6 (2007), no. 2, 173–207 (French, with English and French summaries). MR 2311663, https://doi.org/10.1017/S1474748006000065
[2] Dan Barbasch and David A. Vogan Jr., Unipotent representations of complex semisimple groups, Ann. of Math. (2) 121 (1985), no. 1, 41–110. MR 782556, https://doi.org/10.2307/1971193
[3] Marc Cabanes and Michel Enguehard, Representation theory of finite reductive groups, New Mathematical Monographs, vol. 1, Cambridge University Press, Cambridge, 2004. MR 2057756
[4] Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
[5] François Digne and Jean Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21, Cambridge University Press, Cambridge, 1991. MR 1118841
[6] Olivier Dudas and Gunter Malle, Projective modules in the intersection cohomology of Deligne-Lusztig varieties, C. R. Math. Acad. Sci. Paris 352 (2014), no. 6, 467–471 (English, with English and French summaries). MR 3210126, https://doi.org/10.1016/j.crma.2014.03.011
[7] Olivier Dudas and Gunter Malle, Decomposition matrices for low-rank unitary groups, Proc. Lond. Math. Soc. (3) 110 (2015), no. 6, 1517–1557. MR 3356813, https://doi.org/10.1112/plms/pdv008
[8] Olivier Dudas and Gunter Malle, Decomposition matrices for exceptional groups at 𝑑=4, J. Pure Appl. Algebra 220 (2016), no. 3, 1096–1121. MR 3414409, https://doi.org/10.1016/j.jpaa.2015.08.009
[9] Olivier Dudas and Gunter Malle, Modular irreducibility of cuspidal unipotent characters, Invent. Math. 211 (2018), no. 2, 579–589. MR 3748314, https://doi.org/10.1007/s00222-017-0753-1
[10] Paul Fong and Bhama Srinivasan, The blocks of finite classical groups, J. Reine Angew. Math. 396 (1989), 122–191. MR 988550
[11] Meinolf Geck, Basic sets of Brauer characters of finite groups of Lie type. II, J. London Math. Soc. (2) 47 (1993), no. 2, 255–268. MR 1207947, https://doi.org/10.1112/jlms/s2-47.2.255
[12] Meinolf Geck, On the average values of the irreducible characters of finite groups of Lie type on geometric unipotent classes, Doc. Math. 1 (1996), No. 15, 293–317. MR 1418951
[13] Meinolf Geck and Gerhard Hiss, Basic sets of Brauer characters of finite groups of Lie type, J. Reine Angew. Math. 418 (1991), 173–188. MR 1111205
[14] Meinolf Geck, Gerhard Hiss, and Gunter Malle, Cuspidal unipotent Brauer characters, J. Algebra 168 (1994), no. 1, 182–220. MR 1289097, https://doi.org/10.1006/jabr.1994.1226
[15] Meinolf Geck and Nicolas Jacon, Representations of Hecke algebras at roots of unity, Algebra and Applications, vol. 15, Springer-Verlag London, Ltd., London, 2011. MR 2799052
[16] Meinolf Geck and Gunter Malle, Cuspidal unipotent classes and cuspidal Brauer characters, J. London Math. Soc. (2) 53 (1996), no. 1, 63–78. MR 1362687, https://doi.org/10.1112/jlms/53.1.63
[17] Meinolf Geck and Gunter Malle, On the existence of a unipotent support for the irreducible characters of a finite group of Lie type, Trans. Amer. Math. Soc. 352 (2000), no. 1, 429–456. MR 1475683, https://doi.org/10.1090/S0002-9947-99-02210-2
[18] Meinolf Geck and Götz Pfeiffer, Unipotent characters of the Chevalley groups 𝐷₄(𝑞), 𝑞 odd, Manuscripta Math. 76 (1992), no. 3-4, 281–304. MR 1185021, https://doi.org/10.1007/BF02567762
[19] Frank Himstedt and Felix Noeske, Decomposition numbers of 𝑆𝑂₇(𝑞) and 𝑆𝑝₆(𝑞), J. Algebra 413 (2014), 15–40. MR 3216598, https://doi.org/10.1016/j.jalgebra.2014.04.020
[20] Gordon James, The decomposition matrices of 𝐺𝐿_{𝑛}(𝑞) for 𝑛\le10, Proc. London Math. Soc. (3) 60 (1990), no. 2, 225–265. MR 1031453, https://doi.org/10.1112/plms/s3-60.2.225
[21] N. Kawanaka, Fourier transforms of nilpotently supported invariant functions on a simple Lie algebra over a finite field, Invent. Math. 69 (1982), no. 3, 411–435. MR 679766, https://doi.org/10.1007/BF01389363
[22] Martin W. Liebeck, Permutation modules for rank 3 symplectic and orthogonal groups, J. Algebra 92 (1985), no. 1, 9–15. MR 772468, https://doi.org/10.1016/0021-8693(85)90142-5
[23] George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472
[24] George Lusztig, A unipotent support for irreducible representations, Adv. Math. 94 (1992), no. 2, 139–179. MR 1174392, https://doi.org/10.1016/0001-8708(92)90035-J
[25] G. Lusztig and N. Spaltenstein, Induced unipotent classes, J. London Math. Soc. (2) 19 (1979), no. 1, 41–52. MR 527733, https://doi.org/10.1112/jlms/s2-19.1.41
[26] Gunter Malle, Unipotente Grade imprimitiver komplexer Spiegelungsgruppen, J. Algebra 177 (1995), no. 3, 768–826 (German, with German summary). MR 1358486, https://doi.org/10.1006/jabr.1995.1329
[27] Jean Michel, The development version of the CHEVIE package of GAP3, J. Algebra 435 (2015), 308–336. MR 3343221, https://doi.org/10.1016/j.jalgebra.2015.03.031
[28] Nicolas Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics, vol. 946, Springer-Verlag, Berlin-New York, 1982 (French). MR 672610
[29] N. Spaltenstein, On the generalized Springer correspondence for exceptional groups, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 317–338. MR 803340
[30] Jay Taylor, Generalized Gelfand-Graev representations in small characteristics, Nagoya Math. J. 224 (2016), no. 1, 93–167. MR 3572751, https://doi.org/10.1017/nmj.2016.33