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Computing Green functions in small characteristic. (English) Zbl 07239064
Summary: Let \(G(q)\) be a finite group of Lie type over a field with \(q\) elements, where \(q\) is a prime power. The Green functions of \(G(q)\), as defined by Deligne and Lusztig, are known in almost all cases by work of Beynon-Spaltenstein, Lusztig und Shoji. Open cases exist for groups of exceptional type \({}^2E_6, E_7, E_8\) in small characteristics. We propose a general method for dealing with these cases, which proceeds by a reduction to the case where \(q\) is a prime and then uses computer algebra techniques. In this way, all open cases in type \({}^2E_6, E_7\) are solved, as well as at least one particular open case in type \(E_8\).
Reviewer: Reviewer (Berlin)

MSC:
20C33 Representations of finite groups of Lie type
20G40 Linear algebraic groups over finite fields
Software:
CHEVIE; GAP
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References:
[1] Beynon, W. M.; Spaltenstein, N., Green functions of finite Chevalley groups of type \(E_n (n = 6, 7, 8)\), J. Algebra, 88, 584-614 (1984) · Zbl 0539.20025
[2] Carter, R. W., Simple Groups of Lie Type (1972), Wiley: Wiley New York, reprinted 1989 as Wiley Classics Library Edition · Zbl 0248.20015
[3] Carter, R. W., Finite Groups of Lie Type: Conjugacy Classes and Complex Characters (1985), Wiley: Wiley New York · Zbl 0567.20023
[4] Cohen, A. M.; Murray, S. H.; Taylor, D. E., Computing in groups of Lie type, Math. Comput., 73, 1477-1498 (2004) · Zbl 1062.20049
[5] Deligne, P.; Lusztig, G., Representations of reductive groups over finite fields, Ann. Math., 103, 103-161 (1976) · Zbl 0336.20029
[6] GAP - Groups, Algorithms, and Programming, Version 4.10.0 (2018)
[7] Geck, M., On the average values of the irreducible characters of finite groups of Lie type on geometric unipotent classes, Doc. Math. J. DMV, 1, 293-317 (1996), (electronic) · Zbl 0873.20011
[8] Geck, M., On the construction of semisimple Lie algebras and Chevalley groups, Proc. Am. Math. Soc., 145, 3233-3247 (2017) · Zbl 1419.17018
[9] Geck, M., Minuscule weights and Chevalley groups, (Finite Simple Groups: Thirty Years of the Atlas and Beyond (Celebrating the Atlases and Honoring John Conway, November 2-5, 2015 at Princeton University). Finite Simple Groups: Thirty Years of the Atlas and Beyond (Celebrating the Atlases and Honoring John Conway, November 2-5, 2015 at Princeton University), Contemporary Math., vol. 694 (2017), Amer. Math. Soc.), 159-176
[10] Geck, M., ChevLie — constructing Lie algebras and Chevalley groups in GAP (July 2016), available at
[11] Geck, M., On the values of unipotent characters in bad characteristic, Rend. Semin. Mat. Univ. Padova, 141, 37-63 (2019) · Zbl 07083123
[12] Geck, M., Green functions and Glauberman degree-divisibility (April 2019), preprint, see
[13] Geck, M.; Hiss, G.; Lübeck, F.; Malle, G.; Pfeiffer, G., CHEVIE—a system for computing and processing generic character tables, Appl. Algebra Eng. Commun. Comput., 7, 175-210 (1996), electronically available at · Zbl 0847.20006
[14] Lawther, R., Jordan block sizes of unipotent elements in exceptional algebraic groups, Commun. Algebra, 23, 4125-4156 (1995) · Zbl 0880.20034
[15] Liebeck, M. W.; Seitz, G. M., Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras, Math. Surveys and Monographs, vol. 180 (2012), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1251.20001
[16] Lusztig, G., Representations of Finite Chevalley Groups, C.B.M.S. Regional Conference Series in Mathematics, vol. 39 (1977), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0372.20033
[17] Lusztig, G., Characters of Reductive Groups Over a Finite Field, Ann. Math. Studies, vol. 107 (1984), Princeton U. Press · Zbl 0556.20033
[18] Lusztig, G., Intersection cohomology complexes on a reductive group, Invent. Math., 75, 205-272 (1984) · Zbl 0547.20032
[19] Lusztig, G., Character sheaves IV, Adv. Math., 59, 1-63 (1986) · Zbl 0602.20035
[20] Lusztig, G., Character sheaves V, Adv. Math., 61, 103-155 (1986) · Zbl 0602.20036
[21] Lusztig, G., Introduction to character sheaves, (The Arcata Conference on Representations of Finite Groups. The Arcata Conference on Representations of Finite Groups, Arcata, Calif., 1986. The Arcata Conference on Representations of Finite Groups. The Arcata Conference on Representations of Finite Groups, Arcata, Calif., 1986, Proc. Sympos. Pure Math., Part 1, vol. 47 (1987), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 164-179
[22] Lusztig, G., Green functions and character sheaves, Ann. Math., 131, 355-408 (1990) · Zbl 0695.20024
[23] Lusztig, G., Character sheaves on disconnected groups, IV, Represent. Theory, 8, 145-178 (2004) · Zbl 1075.20013
[24] Lusztig, G., On the cleanness of cuspidal character sheaves, Mosc. Math. J., 12, 621-631 (2012) · Zbl 1263.20044
[25] Lusztig, G., The canonical basis of the quantum adjoint representation, J. Comb. Algebra, 1, 45-57 (2017) · Zbl 1422.17019
[26] Lusztig, G.; Spaltenstein, N., On the generalized Springer correspondence for classical groups, (Algebraic Groups and Related Topics. Algebraic Groups and Related Topics, Adv. Stud. Pure Math., vol. 6 (1985), North Holland and Kinokuniya), 289-316 · Zbl 0579.20035
[27] Malle, G., Die unipotenten Charaktere von \({}^2F_4( q^2)\), Commun. Algebra, 18, 2361-2381 (1990) · Zbl 0721.20008
[28] Malle, G., Green functions for groups of type \(F_4\) and \(E_6\) in characteristic 2, Commun. Algebra, 21, 747-798 (1993) · Zbl 0815.20033
[29] Marcelo, R. M.; Shinoda, K., Values of the unipotent characters of the Chevalley group of type \(F_4\) at unipotent elements, Tokyo J. Math., 18, 303-340 (1995) · Zbl 0869.20005
[30] Michel, J., The development version of the CHEVIE package of GAP3, J. Algebra, 435, 308-336 (2015), Webpage at
[31] Mizuno, K., The conjugate classes of Chevalley groups of type \(E_6\), J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math., 24, 525-563 (1977) · Zbl 0399.20044
[32] Mizuno, K., The conjugate classes of unipotent elements of the Chevalley groups \(E_7\) and \(E_8\), Tokyo J. Math., 3, 391-461 (1980)
[33] Porsch, U., Die Greenfunktionen der endlichen Gruppen \(E_6(q), q = 3^n (1993)\), Universität Heidelberg, Diplomarbeit
[34] Shoji, T., The conjugacy classes of Chevalley groups of type \(( F_4)\) over finite fields of characteristic \(p \neq 2\), J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math., 21, 1-17 (1974) · Zbl 0279.20038
[35] Shoji, T., On the Green polynomials of a Chevalley group of type \(F_4\), Commun. Algebra, 10, 505-543 (1982) · Zbl 0485.20031
[36] Shoji, T., Green functions of reductive groups over a finite field, (The Arcata Conference on Representations of Finite Groups. The Arcata Conference on Representations of Finite Groups, Arcata, Calif., 1986. The Arcata Conference on Representations of Finite Groups. The Arcata Conference on Representations of Finite Groups, Arcata, Calif., 1986, Proc. Sympos. Pure Math., Part 1, vol. 47 (1987), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 289-302
[37] Shoji, T., Character sheaves and almost characters of reductive groups, Adv. Math., 111, 244-313 (1995) · Zbl 0832.20065
[38] Shoji, T., Character sheaves and almost characters of reductive groups, II, Adv. Math., 111, 314-354 (1995) · Zbl 0832.20065
[39] Shoji, T., Generalized Green functions and unipotent classes for finite reductive groups, I, Nagoya Math. J., 184, 155-198 (2006) · Zbl 1128.20033
[40] Shoji, T., Generalized Green functions and unipotent classes for finite reductive groups, II, Nagoya Math. J., 188, 133-170 (2007) · Zbl 1133.20036
[41] Spaltenstein, N., On the generalized Springer correspondence for exceptional groups, (Algebraic Groups and Related Topics. Algebraic Groups and Related Topics, Adv. Stud. Pure Math., vol. 6 (1985), North Holland and Kinokuniya), 317-338 · Zbl 0574.20029
[42] Steinberg, R., Lectures on Chevalley groups (1967), Department of Math., Yale University, now available as vol. 66 of the University Lecture Series, Amer. Math. Soc., Providence, R.I., 2016
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