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Reflection Subquotients of Unitary Reflection Groups

Published online by Cambridge University Press:  20 November 2018

G. I. Lehrer  and
T. A. Springer
G. I. Lehrer
Affiliation:
School of Mathematics and Statistics, University of Sydney, N.S.W. 2006, Australia email: gusl@maths.usyd.edu.au
T. A. Springer
Affiliation:
Department of Mathematics, University of Utrecht, Budapestlaan 6, Utrecht, The Netherlands email: springer@math.uu.nl
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Abstract

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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.

Keywords

51F1520H1520G4020F5514C17
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 6 , 01 December 1999 , pp. 1175 - 1193
Copyright
Copyright © Canadian Mathematical Society 1999

References

[B] Bourbaki, N., Groupes et algèbres de Lie. Hermann, Paris, 1968, Chs. 4, 5, 6.Google Scholar
[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] Broué, M. and Malle, G., Zyklotomische Heckealgebren. Astérisque 212(1993), 119–189.Google Scholar
[BMM] Broué, M., Malle, G. and Michel, J., Generic blocks of finite reductive groups. Astérisque 212(1993), 7–92.Google Scholar
[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] Cohen, A. M., Finite complex reflection groups. Ann. Sci. École. Norm. Sup. 9(1976), 379–436.Google Scholar
[CN] Cowsik, R. C. and Nori, M. V., On Cohen-Macaulay rings. J. Algebra 38(1976), 536–538.Google Scholar
[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] Fulton, W., Intersection Theory. Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin-Heidelberg-New York-Tokyo, 1984.Google Scholar
[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] 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] Howlett, R. and Lehrer, G. I., Induced cuspidal representations and generalised Hecke rings. Invent.Math. 58(1980), 37–64.Google Scholar
[L] Lehrer, G. I., Poincaré series for unitary reflection groups. Invent.Math. 120(1995), 411–425.Google Scholar
[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] Malle, G., Unipotente Grade imprimitiver komplexer Spiegelungsgruppen. J. Algebra 177(1995), 768–826.Google Scholar
[Mu] Mumford, D., The Red Book of Varieties and Schemes. Lecture Notes in Math. 1358, Springer-Verlag, Berlin-New York, 1988.Google Scholar
[OS] Orlik, P. and Solomon, L., Unitary reflection groups and cohomology. Invent.Math. 59(1980), 77–94.Google Scholar
[Sp] Springer, T. A., Regular elements of finite reflection groups. Inv.Math. 25(1974), 159–198.Google Scholar
[St1] Steinberg, R., Differential equations invariant under finite reflection groups. Trans. Amer. Math. Soc. 112(1964), 392–400.Google Scholar
[St2] Steinberg, R., Invariants of finite reflection groups. Canad. J. Math. 12(1960), 616–61.Google Scholar
[ST] Shephard, G. C. and Todd, J. A., Finite unitary reflection groups. Canad. J. Math. 6(1954), 274–304.Google Scholar