Nagoya Mathematical Journal

Twisted invariant theory for reflection groups

C. Bonnafé, G. I. Lehrer, and J. Michel

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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.

Article information

Nagoya Math. J., Volume 182 (2006), 135-170.

First available in Project Euclid: 20 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]
Secondary: 20H15: Other geometric groups, including crystallographic groups [See also 51-XX, especially 51F15, and 82D25] 51F15: Reflection groups, reflection geometries [See also 20H10, 20H15; for Coxeter groups, see 20F55]


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

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