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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
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Abstract

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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

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