Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-06-09T09:24:07.536Z Has data issue: false hasContentIssue false
  • English
  • Français

Twisted Invariant Theory for Reflection Groups

Published online by Cambridge University Press:  11 January 2016

C. Bonnafé ,
G. I. Lehrer  and
J. Michel
C. Bonnafé
Affiliation:
UFR ST, Laboratoire de Mathématiques de Besançon, CNRS (UMR 6623), 16 Route de Gray, 25000 Besançon, France, bonnafe@math.univ-fcomte.fr
G. I. Lehrer
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia, gusl@maths.usyd.edu.au
J. Michel
Affiliation:
Institut de Mathématiques, Université Paris VII, 175 rue du Chevaleret, 75013 Paris, France, jmichel@math.jussieu.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let G be a finite reflection group acting in a complex vector space V = ℂr, whose coordinate ring will be denoted by S. Any element γ ∈ GL(V) which normalises G acts on the ring SG of G-invariants. We attach invariants of the coset Gγ to this action, and show that if G′ is a parabolic subgroup of G, also normalised by γ, the invariants attaching to G′γ are essentially the same as those of Gγ. Four applications are given. First, we give a generalisation of a result of Springer-Stembridge which relates the module structures of the coinvariant algebras of G and G′ and secondly, we give a general criterion for an element of Gγ to be regular (in Springer’s sense) in invariant-theoretic terms, and use it to prove that up to a central element, all reflection cosets contain a regular element. Third, we prove the existence in any well-generated group, of analogues of Coxeter elements of the real reflection groups. Finally, we apply the analysis to quotients of G which are themselves reflection groups.

Keywords

20F5520H1551F15
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 182: Special Issue Celebrating the 60th Birthday of George Lusztig , June 2006 , pp. 135 - 170
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2006

References

[B] Bessis, D., Zariski theorems and diagrams for braid groups, Inv. Math., 145 (2001), 487–507.Google Scholar
[B2] Bessis, D., Topology of complex reflection arrangements, math.GT/0411645, to appear.Google Scholar
[BBR] Bessis, D., Bonnafé, C. and Rouquier, R., Quotients et extensions de groupes de r éflexion, Math. Ann., 323 (2002), 405–436.Google Scholar
[BL] Blair, J. and Lehrer, G. I., Cohomology actions and centralisers in unitary reflection groups, Proc. Lond. Math. Soc. (3), 83 (2001), 582–604.Google Scholar
[Br] Broué, M., Reflection groups, braid groups, Hecke algebras, finite reductive groups, Current developments in mathematics, 2000, Harvard university & M.I.T. international press, Boston (2001), pp. 1–103.Google Scholar
[BMM] Broué, M., Malle, G. and Michel, J., Towards Spetses I, Transf. Groups, 4 (1999), 157–218.Google Scholar
[CHEVIE] see www.math.jussieu.fr/~jmichel/chevie.Google Scholar
[L] Lehrer, G. I., A new proof of Steinberg’s fixed-point theorem, Int. Math. Res. Not., 28 (2004), 1407–1411.Google Scholar
[LM] Lehrer, G. I. and Michel, J., Invariant theory and eigenspaces for unitary reflection groups, C.R.A.S., 336 (2003), 795–800.Google Scholar
[LS1] Lehrer, G. I. and Springer, T. A., Intersection multiplicities and reflection subquotients of unitary reflection groups I, Geometric group theory down under (Canberra, 1996), de Gruyter, Berlin (1999), pp. 181–193.Google Scholar
[LS2] Lehrer, G. I. and Springer, T. A., Reflection subquotients of unitary reflection groups, Canad. J. Math., 51 (1999), 1175–1193.Google Scholar
[Ma] Malle, G., Splitting fields for extended complex reflection groups and Hecke algebras, Transformation groups, to appear.Google Scholar
[MN] Morita, H. and Nakajima, T., The coinvariant algebra of the symmetric group as a direct sum of induced modules, Osaka J. Math., 42 (2005), 217–231.Google Scholar
[OS] Orlik, P. and Solomon, L., Unitary reflection groups and cohomology, Inv. Math., 59 (1980), 77–94.Google Scholar
[OT] Orlik, P. and Terao, H., Arrangements of hyperplanes, Grunlehren der math. Wissenschaften 300, Springer-Verlag, 1991.Google Scholar
[Sp] Springer, T., Regular elements of finite reflection groups, Invent. Math., 25 (1974), 159–198.Google Scholar
[Ste] Stembridge, J. R., On the eigenvalues of representations of reflection groups and wreath products, Pacific J. Math., 140 (1989), 353–396.Google Scholar