Geodesic
Reflection Subquotients of Unitary Reflection Groups
Canadian journal of mathematics, Tome 51 (1999) no. 6, pp. 1175-1193

Voir la notice de l'article provenant de la source Cambridge University Press

Let $G$ be a finite group generated by (pseudo-) reflections in a complex vector space and let $g$ be any linear transformation which normalises $G$ . In an earlier paper, the authors showed how to associate with any maximal eigenspace of an element of the coset $gG$ , a subquotient of $G$ which acts as a reflection group on the eigenspace. In this work, we address the questions of irreducibility and the coexponents of this subquotient, as well as centralisers in $G$ of certain elements of the coset. A criterion is also given in terms of the invariant degrees of $G$ for an integer to be regular for $G$ . A key tool is the investigation of extensions of invariant vector fields on the eigenspace, which leads to some results and questions concerning the geometry of intersections of invariant hypersurfaces.
DOI : 10.4153/CJM-1999-052-4
Mots-clés : 51F15, 20H15, 20G40, 20F55, 14C17 
Lehrer, G. I.; Springer, T. A. Reflection Subquotients of Unitary Reflection Groups. Canadian journal of mathematics, Tome 51 (1999) no. 6, pp. 1175-1193. doi: 10.4153/CJM-1999-052-4
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     year = {1999},
     volume = {51},
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     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1999-052-4/}
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[B] [B] Bourbaki, N., Groupes et algèbres de Lie. Hermann, Paris, 1968, Chs. 4, 5, 6. Google Scholar

[BM1] [BM1] Broué, M. and Malle, G., Théorèmes de Sylow génériques pour les groupes réductifs sur les corps finis. Math. Ann. 292(1992), 241–262. Google Scholar

[BM2] [BM2] Broué, M. and Malle, G., Zyklotomische Heckealgebren. Astérisque 212(1993), 119–189. Google Scholar

[BMM] [BMM] Broué, M., Malle, G. and Michel, J., Generic blocks of finite reductive groups. Astérisque 212(1993), 7–92. Google Scholar

[BrM] [BrM] Broué, M. and Michel, J., Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne-Lusztig associées. In: Finite reductive groups (Luminy, 1994), Progr. Math. 141, Birkhäuser, Boston, MA, 1997, 73–139. Google Scholar

[Co] [Co] Cohen, A. M., Finite complex reflection groups. Ann. Sci. École. Norm. Sup. 9(1976), 379–436. Google Scholar

[CN] [CN] Cowsik, R. C. and Nori, M. V., On Cohen-Macaulay rings. J. Algebra 38(1976), 536–538. Google Scholar

[DL] [DL] Denef, J. and Loeser, F., Regular elements and monodromy of discriminants of finite reflection groups. Indag.Math. (N. S.) 6(1995), 129–143. Google Scholar

[F] [F] Fulton, W., Intersection Theory. Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin-Heidelberg-New York-Tokyo, 1984. Google Scholar

[G] [G] Grothendieck, A., Éléments de Géometrie Algébrique IV Étude Locale des Schémas et des Morphismes de Schémas. Inst. Hautes Études Sci. Publ. Math. 32(1967), 5–343. Google Scholar

[GD] [GD] Grothendieck, A. and Dieudonné, J. A., Éléments de Géometrie Algébrique I. Grundlehren Math. Wiss. 166, Springer-Verlag, Berlin-Heidelberg New York, 1971. Google Scholar

[HL] [HL] Howlett, R. and Lehrer, G. I., Induced cuspidal representations and generalised Hecke rings. Invent.Math. 58(1980), 37–64. Google Scholar

[L] [L] Lehrer, G. I., Poincaré series for unitary reflection groups. Invent.Math. 120(1995), 411–425. Google Scholar

[LS1] [LS1] Lehrer, G. I. and Springer, T. A., Intersection multiplicities and reflection subquotients of unitary reflection groups, I. In: Geometric group theory down under (ed. Cossey, J.), W. de Gruyter, 1999, 181–193. Google Scholar

[M] [M] Malle, G., Unipotente Grade imprimitiver komplexer Spiegelungsgruppen. J. Algebra 177(1995), 768–826. Google Scholar

[Mu] [Mu] Mumford, D., The Red Book of Varieties and Schemes. Lecture Notes in Math. 1358, Springer-Verlag, Berlin-New York, 1988. Google Scholar

[OS] [OS] Orlik, P. and Solomon, L., Unitary reflection groups and cohomology. Invent.Math. 59(1980), 77–94. Google Scholar

[Sp] [Sp] Springer, T. A., Regular elements of finite reflection groups. Inv.Math. 25(1974), 159–198. Google Scholar

[St1] [St1] Steinberg, R., Differential equations invariant under finite reflection groups. Trans. Amer. Math. Soc. 112(1964), 392–400. Google Scholar

[St2] [St2] Steinberg, R., Invariants of finite reflection groups. Canad. J. Math. 12(1960), 616–61. Google Scholar

[ST] [ST] Shephard, G. C. and Todd, J. A., Finite unitary reflection groups. Canad. J. Math. 6(1954), 274–304. Google Scholar

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