Open Access
2006 Twisted invariant theory for reflection groups
C. Bonnafé, G. I. Lehrer, J. Michel
Nagoya Math. J. 182: 135-170 (2006).


Let $G$ be a finite reflection group acting in a complex vector space $V = \mathbb{C}^{r}$, whose coordinate ring will be denoted by $S$. Any element $\gamma \in \mathrm{GL}(V)$ which normalises $G$ acts on the ring $S^{G}$ of $G$-invariants. We attach invariants of the coset $G\gamma$ to this action, and show that if $G'$ is a parabolic subgroup of $G$, also normalised by $\gamma$, the invariants attaching to $G'\gamma$ are essentially the same as those of $G\gamma$. 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 $G\gamma$ 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.


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C. Bonnafé. G. I. Lehrer. J. Michel. "Twisted invariant theory for reflection groups." Nagoya Math. J. 182 135 - 170, 2006.


Published: 2006
First available in Project Euclid: 20 June 2006

zbMATH: 1172.20030
MathSciNet: MR2235340

Primary: 20F55
Secondary: 20H15 , 51F15

Rights: Copyright © 2006 Editorial Board, Nagoya Mathematical Journal

Vol.182 • 2006
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