×

zbMATH — the first resource for mathematics

Green functions and Glauberman degree-divisibility. (English) Zbl 07239274
Summary: The Glauberman correspondence is a fundamental bijection in the character theory of finite groups. In 1994, Hartley and Turull established a degree-divisibility property for characters related by that correspondence, subject to a congruence condition which should hold for the Green functions of finite groups of Lie type, as defined by Deligne and Lusztig. Here, we present a general argument for completing the proof of that congruence condition. Consequently, the degree-divisibility property holds in complete generality.
MSC:
20C33 Representations of finite groups of Lie type
20C15 Ordinary representations and characters
20G40 Linear algebraic groups over finite fields
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Be\u{\i}linson, A. A.; Bernstein, J.; Deligne, P., Faisceaux pervers. Analysis and Topology on Singular Spaces, {I}, Ast\'{e}risque, 100, http://www.numdam.org/issues/AST_1982__100__1_0/, 5-171 (1982)
[2] Beynon, W. M.; Spaltenstein, N., Green functions of finite {C}hevalley groups of type {\(E\sb{n} \)} {\((n=6,\,7,\,8)\)}, J. Algebra. Journal of Algebra, 88, 584-614 (1984) · Zbl 0539.20025
[3] Carter, Roger W., Finite Groups of {L}ie Type: Conjugacy {C}lasses and Complex Characters, Wiley Classics Library, xii+544 pp. (1993)
[4] Deligne, P.; Lusztig, G., Representations of reductive groups over finite fields, Ann. of Math. (2). Annals of Mathematics. Second Series, 103, 103-161 (1976) · Zbl 0336.20029
[5] Geck, Meinolf, On the average values of the irreducible characters of finite groups of {L}ie type on geometric unipotent classes, Doc. Math.. Documenta Mathematica, 1, 15-293 (1996) · Zbl 0873.20011
[6] Geck, Meinolf, A first guide to the character theory of finite groups of {L}ie type. Local representation theory and simple groups, EMS Ser. Lect. Math., 63-106 (2018) · Zbl 1430.20014
[7] Geck, Meinolf, On the values of unipotent characters in bad characteristic, Rend. Semin. Mat. Univ. Padova. Rendiconti del Seminario Matematico della Universit\`a di Padova, 141, 37-63 (2019) · Zbl 1472.20096
[8] Geck, Meinolf, Computing {G}reen functions in small characteristic (2020) · Zbl 1472.20096
[9] Geck, Meinolf; Kim, Sungsoon; Pfeiffer, G\"{o}tz, Minimal length elements in twisted conjugacy classes of finite {C}oxeter groups, J. Algebra. Journal of Algebra, 229, 570-600 (2000) · Zbl 1042.20026
[10] Glauberman, George, Correspondences of characters for relatively prime operator groups, Canadian J. Math.. Canadian Journal of Mathematics. Journal Canadien de Math\'{e}matiques, 20, 1465-1488 (1968) · Zbl 0167.02602
[11] Hartley, Brian; Turull, Alexandre, On characters of coprime operator groups and the {G}lauberman character correspondence, J. Reine Angew. Math.. Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle’s Journal], 451, 175-219 (1994) · Zbl 0797.20007
[12] Liebeck, Martin W.; Seitz, Gary M., Unipotent and Nilpotent Classes in Simple Algebraic Groups and {L}ie Algebras, Math. Surveys Monogr., 180, xii+380 pp. (2012) · Zbl 1251.20001
[13] Lusztig, George, Representations of Finite {C}hevalley Groups. ({e}xpository lectures from the CBMS Regional Conference held at Madison, Wis., August 8-12, 1977), CBMS Reg. Conf. Ser. Math., 39, v+48 pp. (1978) · Zbl 0418.20037
[14] Lusztig, George, Characters of Reductive Groups over a Finite Field, Ann. of Math. Stud., 107, xxi+384 pp. (1984) · Zbl 0556.20033
[15] Lusztig, George, Intersection cohomology complexes on a reductive group, Invent. Math.. Inventiones Mathematicae, 75, 205-272 (1984) · Zbl 0547.20032
[16] Lusztig, George, Character sheaves. {I}, Adv. in Math.. Advances in Mathematics, 56, 193-237 (1985) · Zbl 0586.20018
[17] Lusztig, George, Character sheaves. {II}, Adv. in Math.. Advances in Mathematics, 57, 226-265 (1985) · Zbl 0586.20019
[18] Lusztig, George, Character sheaves. {III}, Adv. in Math.. Advances in Mathematics, 57, 266-315 (1985) · Zbl 0594.20031
[19] Lusztig, George, Character sheaves. {IV}, Adv. in Math.. Advances in Mathematics, 59, 1-63 (1986) · Zbl 0602.20035
[20] Lusztig, George, Character sheaves. {V}, Adv. in Math.. Advances in Mathematics, 61, 103-155 (1986) · Zbl 0602.20036
[21] Lusztig, George, Green functions and character sheaves, Ann. of Math. (2). Annals of Mathematics. Second Series, 131, 355-408 (1990) · Zbl 0695.20024
[22] Lusztig, George, Character sheaves on disconnected groups. {IV}, Represent. Theory. Representation Theory. An Electronic Journal of the Amer. Math. Soc., 8, 145-178 (2004) · Zbl 1075.20013
[23] Lusztig, George, On the cleanness of cuspidal character sheaves, Mosc. Math. J.. Moscow Mathematical Journal, 12, 621-631 (2012) · Zbl 1263.20044
[24] Mizuno, Kenzo, The conjugate classes of unipotent elements of the {C}hevalley groups {\(E\sb{7} \)} and {\(E\sb{8} \)}, Tokyo J. Math.. Tokyo Journal of Mathematics, 3, 391-461 (1980) · Zbl 0454.20046
[25] Navarro, Gabriel, Some open problems on coprime action and character correspondences, Bull. London Math. Soc.. The Bulletin of the London Mathematical Society, 26, 513-522 (1994) · Zbl 0829.20013
[26] Navarro, Gabriel, Character Theory and the {M}c{K}ay Conjecture, Cambridge Stud. Adv. Math., 175, xviii+234 pp. (2018) · Zbl 1433.20001
[27] Shoji, Toshiaki, Green functions of reductive groups over a finite field. The {A}rcata {C}onference on {R}epresentations of {F}inite {G}roups, Proc. Sympos. Pure Math., 47, 289-301 (1987)
[28] Shoji, Toshiaki, Character sheaves and almost characters of reductive groups. {I}, Adv. Math.. Advances in Mathematics, 111, 244-313 (1995) · Zbl 0832.20065
[29] Shoji, Toshiaki, Character sheaves and almost characters of reductive groups. {II}, Adv. Math.. Advances in Mathematics, 111, 314-354 (1995) · Zbl 0832.20065
[30] Shoji, Toshiaki, Generalized {G}reen functions and unipotent classes for finite reductive groups. {I}, Nagoya Math. J.. Nagoya Mathematical Journal, 184, 155-198 (2006) · Zbl 1128.20033
[31] Shoji, Toshiaki, Generalized {G}reen functions and unipotent classes for finite reductive groups. {II}, Nagoya Math. J.. Nagoya Mathematical Journal, 188, 133-170 (2007) · Zbl 1133.20036
[32] Shoji, Toshiaki, Generalized {G}reen functions associated to complex reflection groups, J. Algebra. Journal of Algebra, 558, 677-707 (2020) · Zbl 1458.20016
[33] Steinberg, Robert, Endomorphisms of Linear Algebraic Groups, Mem. Amer. Math. Soc., 80, 108 pp. (1968) · Zbl 0164.02902
[34] Taylor, Jay, On unipotent supports of reductive groups with a disconnected centre, J. Algebra. Journal of Algebra, 391, 41-61 (2013) · Zbl 1286.20060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.