×

Towards spetses. I. (English) Zbl 0972.20024

Rational reflection groups occur naturally as Weyl groups of reductive algebraic groups over finite fields. The authors conjecture that complex reflection groups are similarly related to some as yet undiscovered objects which they call spetses.
To prepare the search for spetses, the authors collect technical results on many subjects like braid groups, generic Hecke algebras, reflection data, cyclotomic specializations of Hecke algebras, spetsial cyclotomic Hecke algebras. They obtain families of unipotent characters for spetses.

MSC:

20F55 Reflection and Coxeter groups (group-theoretic aspects)
20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
20C08 Hecke algebras and their representations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] [Ar] S. Ariki,Representation theory of a Hecke algebra of G(r, p, n), J. Algebra177 (1995), 164-185. · Zbl 0845.20030
[2] [ArKo] S. Ariki and K. Koike,A Hecke algebra of (?/r?)? S n and construction of its irreducible representations, Advances in Math.106 (1994), 216-243. · Zbl 0840.20007
[3] [Ben] M. Benard,Schur indices and splitting fields of the unitary reflection groups, J. Algebra38 (1976), 318-342. · Zbl 0327.20004
[4] [Bes] D. Bessis,Sur le corps de définition d’un groupe de réflexions complexe, Comm. in Algebra25 (8) (1997), 2703-2716. · Zbl 0946.20024
[5] [Bou] N. Bourbaki,Groupes et Algèbres de Lie, Chap. IV, V et VI, Hermann, Paris, 1968. Russian translation: H. ?yp?a??,?pynnbl u arze?pbl ?u, M., M?p, 1972.
[6] [Bou2]?,Algèbre Commutative, Chap. V, Hermann, Paris, 1968. Russian translation: H. ?yp?a??,Ko??ymamu??a? arze?pa, M., M?p, 1971.
[7] [BreMa] K. Bremke and G. Malle,Reduced words and a length function for G(e, 1, n), Indag. Mathem.8 (1997), 453-469. · Zbl 0914.20036
[8] [Bro1] M. Bouré,On Representations of Symmetric Algebras: an Introduction, Notes by Markus Stricker, Forschungsinstitut für Mathematik ETH Zürich (1991).
[9] [Bro2]?,Rickard equivalences and block theory, Groups’93 Galway-St. Andrews, London Math. Soc., Lecture Notes Series 211, Cambridge University Press, Cambridge, U.K., 1995, pp. 58-79. · Zbl 0847.20003
[10] [BrMa1] M. Broué and G. Malle,Théorèmes de Sylow génériques pour les groupes réductifs sur les corps finis, Math. Ann.292 (1992), 241-262. · Zbl 0820.20057
[11] [BrMa2]?,Zyklotomische Heckealgebren, Astérisque212 (1993), 119-189. · Zbl 0835.20064
[12] [BMM1] M. Broué, G. Malle, J. Michel,Generic blocks of finite reductive groups, Astérisque212 (1993), 7-92. · Zbl 0843.20012
[13] [BMM2] M. Broué,Towards spetses II, in preparation (1999). · Zbl 0972.20024
[14] [BMR] M. Broué, G. Malle, R. Rouquier,Complex reflection groups, braid groups, Hecke algebras, J. reine angew. Math.500 (1998), 127-190. · Zbl 0921.20046
[15] [BrMi] M. Broué and J. Michel,Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées, Finite Reductive Groups: Related Structures and Representations (M. Cabanes, ed.), Progress in Math., vol. 141, Birkhäuser, 1997, pp. 73-140.
[16] [Ch] C. Chevalley,Invariants of finite groups generated by reflections, Amer. J. Math.77 (1955), 778-782. · Zbl 0065.26103
[17] [Co] A. M. Cohen,Finite complex reflection groups, Ann. scient. Éc. Norm. Sup.9 (1976), 379-436. · Zbl 0359.20029
[18] [Ge] M. Geck,Beiträge zur Darstellungstheorie von Iwahori-Hecke-Algebren, RWTH Aachen, Habilitationsschrift (1993).
[19] [GeRo] M. Geck and R. Rouquier,Centers and simple modules for Iwahori-Hecke algebras, Finite Reductive Groups: Related Structures and Representations (M. Cabanes, ed.), Progr. in Math., vol. 141, Birkhäuser, 1997, pp. 251-272. · Zbl 0868.20013
[20] [Gu] E. A. ?yt??h,Mampuubl, c???ahhbl? c zpynna?u, nopo??ehhbl?u ompa?ehu??u, ?yh??. aha?. ? e?o ?p??o?eh.7 (1973), no. 2, 81-82. English translation: E. A. Gutkin,Matrices connected with groups generated by reflection, Func. Anal. and Appl.7 (1973), 153-154.
[21] [LeSp] G. Lehrer and T.A. Springer,Intersection multiplicities and reflection subquotients of unitary reflection groups I, Preprint (1998). · Zbl 0945.51005
[22] [Lu1] G. Lusztig,Characters of Reductive Groups over a Finite Field, Annals of Mathematical Studies, no 107, Princeton University Press, Princeton, New Jersey, 1984. · Zbl 0556.20033
[23] [Lu2]?,Coxeter groups and unipotent representations, Astérisque212 (1993), 191-203. · Zbl 0853.20030
[24] [Ma1] G. Malle,Unipotente Grade imprimitiver komplexer Spiegelungsgruppen. J. Algebra177 (1995), 768-826. · Zbl 0854.20057
[25] [Ma2] G. Malle,Degrés relatifs des algèbres cyclotomiques associées aux groupes de réflexions complexes de dimension deux, Finite Reductive Groups: Related Structures and Representations (M. Cabanes, ed.), Progress in Math., vol. 141, Birkhäuser, 1997, pp. 311-332.
[26] [Ma3] G. Malle,On the rationality and fake degrees of characters of cyclotomic algebras, submitted (1998).
[27] [Ma4]?,Spetses, Doc. Math. J. DMV Extra Volume ICM II (1998), 87-96.
[28] [Ma5] G. Malle,On the generic degrees of cyclotomic algebras, in preparation (1999).
[29] [MM] G. Malle and A. Mathas,Symmetric cyclotomic Hecke algebras, J. Algebra205 (1998), 275-293. · Zbl 0913.20006
[30] [Op1] E.M. Opdam,A remark on the irreducible characters and fake degrees of finite real reflection groups, Invent. Math.120 (1995), 447-454. · Zbl 0824.20038
[31] [Op2] E.M. Opdam,Complex reflection groups and fake degrees, preprint (1998).
[32] [OrSo1] P. Orlik and L. Solomon,Combinatorics and topology of complements of hyperplanes, Invent. Math.56 (1980), 167-189. · Zbl 0432.14016
[33] [OrSo2]?,Unitary reflection groups and cohomology, Invent. Math.59 (1980), 77-94. · Zbl 0452.20050
[34] [OrTe] P. Orlik and H. Terao,Arrangements of Hyperplanes, Springer, Berlin-Heidelberg, 1992. · Zbl 0757.55001
[35] [Ro] R. Rouquier,Caractérisation des caractères et pseudo-caractères, J. Algebra180 (1996), 571-586. · Zbl 0857.16013
[36] [ShTo] G. C. Shephard and J. A. Todd,Finite unitary reflection groups, Canad. J. Math.6 (1954), 274-304. · Zbl 0055.14305
[37] [Sp] T. A. Springer,Regular elements of finite reflection groups, Invent. Math.25 (1974), 159-198. · Zbl 0287.20043
[38] [St1] R. Steinberg,Differential equations invariant under finite reflection groups, Trans. Amer. Math. Soc.112 (1964), 392-400. · Zbl 0196.39202
[39] [St2]?,Endomorphisms of Linear Algebraic Groups, Memoirs of the AMS, No. 80, AMS, Providence, 1968.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.