Modular Representations of Finite Groups of Lie Type in Non-defining Characteristic
Chapter
Abstract
Let us consider a connected reductive algebraic group G, defined over the finite field F q with corresponding Frobenius morphism F. We are concerned here with properties of finite-dimensional modules for the finite group G F over a sufficiently large field k of characteristic ℓ where ℓ is a prime not dividing q.
Keywords
Irreducible Character Levi Subgroup Projective Character Endomorphism Algebra Supercuspidal Representation
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References
- [1]A.M. AUBERT, Formule des traces sur les corps finis, this volume.Google Scholar
- [2]N. BOURBAKI, Groupes et algèbres de Lie, Chap. IV, V, VI, Hermann, Paris, 1968.MATHGoogle Scholar
- [3]M. BROUÉ, G. MALLE, and J. MICHEL, “Représentations unipotentes génériques et blocs des groupes réductifs finis”, Astérisque 212 (1993), 7–92.Google Scholar
- [4]M. BROUÉ and J. MICHEL, “Blocs et séries de Lusztig dans un groupe réductif fini”, J. reine angew. Math. 395 (1989), 56–67.MathSciNetMATHGoogle Scholar
- [56]M. CABANES, “A criterion for complete reducibility and some applications”, in: Représentations Linéaires de Groupes Finis, Luminy, 16–21 mai 1988, Astérisque 181–182 (1990), 93–112Google Scholar
- [6]R. W. CARTER, Finite groups of Lie type: Conjugacy classes and complex characters, Wiley, 1985Google Scholar
- [7]C.W. CURTIS AND I. REINER, Methods of representation theory, vols. 1, 2, Wiley, 1981, 1987MATHGoogle Scholar
- [8]F. DIGNE AND J. MICHEL, Representations of finite groups of Lie type, London Math. Soc. Students Texts 21, Cambridge University Press, 1991Google Scholar
- [9]R. DIPPER, “On quotients of Hom-functors and representations of finite general linear groups I”, J. Algebra 130 (1990), 235–259MathSciNetMATHCrossRefGoogle Scholar
- [10]R. DIPPER AND J. Du, “Harish-Chandra vertices”, J. reine angew. Math. 437 (1993), 101–130MathSciNetMATHGoogle Scholar
- [11]R. DIPPER AND P. FLEISCHMANN, “Modular Harish-Chandra theory I”, Math. Z. 211 (1992), 49–71MathSciNetMATHCrossRefGoogle Scholar
- [12]R. DIPPER AND G.D. JAMES, “The q-Schur algebra”,Proc. London Math. Soc. 59 (1989), 23–50MathSciNetMATHCrossRefGoogle Scholar
- [13]R. DIPPER AND G.D. JAMES, “Representations of Hecke algebras of type Bn”, J. Algebra 146 (1992), 454–481MathSciNetMATHCrossRefGoogle Scholar
- [14]J. Du AND L. SCOTT, “Lusztig conjectures, old and new, I”, J. reine angew. Math. 455 (1994), 141–182MathSciNetMATHCrossRefGoogle Scholar
- [15]P. FONG AND B. SRINIVASAN, “The blocks of finite general linear and unitary groups”, Invent. Math. 69 (1982), 109–153MathSciNetMATHCrossRefGoogle Scholar
- [16]P. FONG AND B. SRINIVASAN, “The blocks of finite classical groups”, J. reine angew. Math. 396 (1989), 122–191MathSciNetMATHGoogle Scholar
- [17]M. GECK, “On the decomposition numbers of the finite unitary groups in non-defining characteristic”, Math. Z. 207 (1991), 83–89MathSciNetMATHCrossRefGoogle Scholar
- [18]M. GECK, “The decomposition numbers of the Hecke algebra of type E 6”, Math. Comp. 61 (1993), 889–899MathSciNetMATHGoogle Scholar
- [19]M. GECK, “Basic sets of Brauer characters of finite groups of Lie type, II”, J. London Math. Soc. 47 (1993), 255–268; III, Manuscripta Math. 85 (1994), 195–216MathSciNetMATHCrossRefGoogle Scholar
- [20]M. GECK AND G. HISS, “Basic sets of Brauer characters of finite groups of Lie type”, J. reine angew. Math. 418 (1991), 173–188MathSciNetMATHGoogle Scholar
- [21]M. GECK, G. HISS, F. LÜBECK, G. MALLE, AND G. PFEIFFER, “CHEVIE—A system for computing and processing generic character tables”, AAECC 7 (1996), 175–210MATHCrossRefGoogle Scholar
- [22]M. GECK, G. HISS, AND G. MALLE, “Cuspidal unipotent Brauer characters”, J. Algebra 168 (1994), 182–220MathSciNetMATHCrossRefGoogle Scholar
- [23]M. GECK, G. HISS, AND G. MALLE, “Towards a classification of the irreducible representations in non-defining characteristic of a finite group of Lie type”, Math. Z. 221 (1996), 353–386MathSciNetMATHGoogle Scholar
- [24]M. GECK AND G. MALLE, “Cuspidal unipotent classes and cuspidal Brauer characters”, J. London Math. Soc. 53 (1996), 63–78MathSciNetMATHGoogle Scholar
- [25]J. A. GREEN, “On a theorem of Sawada”, J. London Math. Soc. 18 (1978), 247–252MathSciNetMATHCrossRefGoogle Scholar
- [26]J. GRUBER, Cuspidate Untergruppen und Zerlegungszahlen klassischer Gruppen, (PhD-Thesis, Heidelberg, 1995)Google Scholar
- [27]G. Hiss, “Harish-Chandra series of Brauer characters in a finite group with a split BN-pair”, J. London Math. Soc. 48 (1993), 219–228MathSciNetMATHCrossRefGoogle Scholar
- [28]G. Hiss, “Supercuspidal representations of finite reductive groups”, J. Algebra 184 (1996), 839–851MathSciNetMATHCrossRefGoogle Scholar
- [29]G. Hiss, Zerlegungszahlen endlicher Gruppen vom Lie-Typ in nicht-definierender Charakteristik (Habilitationsschrift, RWTH Aachen, 1990)MATHGoogle Scholar
- [30]G. Hiss, “Decomposition numbers of finite groups of Lie type in non-defining characteristic”, in: G. O. Michler and C. M. Ringel, Eds., Representation Theory of Finite Groups and Finite-Dimensional Algebras, Birkhäuser, 1991, pp. 405–418CrossRefGoogle Scholar
- [31]G. Hiss, F. LÜBECK, and G. MALLE, “The Brauer trees of the exceptional Chevalley groups of type E6, Manuscripta Math. 87 (1995), 131–144MathSciNetMATHCrossRefGoogle Scholar
- [32]R. B. HOWLETT AND G. I. LEHRER, “Induced cuspidal representations and generalized Hecke rings”, Invent. Math. 58 (1980), 37–64MathSciNetMATHCrossRefGoogle Scholar
- [33]R. B. HOWLETT AND G. I. LEHRER, “On Harish-Chandra induction for modules of Levi subgroups”, J. Algebra 165 (1994), 172–183MathSciNetMATHCrossRefGoogle Scholar
- [34]G.D. JAMES, “The decomposition matrices of GL n (q) for n ≤ 10”, Proc. London Math. Soc. 60 (1990), 225–265MathSciNetMATHCrossRefGoogle Scholar
- [35]N. KAWANAKA, “Shintani lifting and Gelfand-Graev representations”, Proc. Symp. Pure Math., Amer. Math. Soc. 47 (1986), 575–616MathSciNetGoogle Scholar
- [36]M. LINCKELMANN, Letter to the authors, April 25th, 1994Google Scholar
- [37]G. LUSZTIG, “On the finiteness of the number of unipotent classes”, Invent. Math. 34 (1976), 201–213MathSciNetMATHCrossRefGoogle Scholar
- [38]G. LUSZTIG, Characters of reductive groups over a finite field, Ann. Math. Studies 107, Princeton U. Press, 1984Google Scholar
- [39]G. LUSZTIG, “Character sheaves”, Adv. Math. 56 (1985), 193–237; II, 57 (1985), 226–265; III, 57 (1985), 266–315; IV, 59 (1986), 1–63; V, 61 (1986), 103–155MathSciNetMATHCrossRefGoogle Scholar
- [40]G. LUSZTIG, “On the character values of finite Chevalley groups at unipotent elements”, J. Algebra 104 (1986), 146–194MathSciNetMATHCrossRefGoogle Scholar
- [41]G. LUSZTIG, “Remarks on computing irreducible characters”, J. Amer. Math. Soc. 5 (1992), 971–986MathSciNetMATHCrossRefGoogle Scholar
- [42]G. LUSZTIG, “Green functions and character sheaves”, Annals of Math. 131 (1990), 355–408MathSciNetMATHCrossRefGoogle Scholar
- [43]L. PUIG, “Algèbres de source de certains blocs de groupes de Chevalley”, in: Représentations Linéaires de Groupes Finis, Luminy, 16–21 May 1988, Astérisque 181–182 (1990), 221–236Google Scholar
- [44]T. SHOJI, “Character sheaves and almost characters of reductive groups”, Adv. Math. 111 (1995), 244–313MathSciNetMATHCrossRefGoogle Scholar
- [45]T. SHOJI, “Character sheaves and almost characters of reductive groups, II”, Adv. Math. 111 (1995), 314–354MathSciNetCrossRefGoogle Scholar
- [46]T. SHOJI, “Unipotent characters of finite Chevalley groups”, this volumeGoogle Scholar
- [47]T. SHOJI, “On the computation of unipotent characters of finite classical groups”, AAECC 7 (1996), 165–174MathSciNetMATHCrossRefGoogle Scholar
- [48]M.-F. VIGNERAS, “Sur la conjecture locale de Langlands pour GL(n, F) sur F l, C. R. Acad. Sci. Pans Sér. I Math. 318 (1994),905–908MathSciNetMATHGoogle Scholar
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