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Automorphisms of complex reflection groups


Authors: I. Marin and J. Michel
Journal: Represent. Theory 14 (2010), 747-788
MSC (2010): Primary 20F55, 20F28, 20C15
DOI: https://doi.org/10.1090/S1088-4165-2010-00380-5
Published electronically: December 14, 2010
MathSciNet review: 2746138
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Abstract: Let $G\subset \mathrm {GL}(\mathbb {C}^r)$ be a finite complex reflection group. We show that when $G$ is irreducible, apart from the exception $G=\mathfrak {S}_6$, as well as for a large class of non-irreducible groups, any automorphism of $G$ is the product of a central automorphism and of an automorphism which preserves the reflections. We show further that an automorphism which preserves the reflections is the product of an element of $N_{\mathrm {GL}(\mathbb {C}^r)}(G)$ and of a “Galois” automorphism: we show that $\mathrm {Gal}(K/\mathbb {Q})$, where $K$ is the field of definition of $G$, injects into the group of outer automorphisms of $G$, and that this injection can be chosen such that it induces the usual Galois action on characters of $G$, apart from a few exceptional characters; further, replacing $K$ if needed by an extension of degree $2$, the injection can be lifted to $\mathrm {Aut}(G)$, and every irreducible representation admits a model which is equivariant with respect to this lifting. Along the way we show that the fundamental invariants of $G$ can be chosen rational.


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Additional Information

I. Marin
Affiliation: Institut de Mathématiques de Jussieu, Université Paris VII, 175, rue du Chevaleret, 75013 Paris
MR Author ID: 664485
Email: marin@math.jussieu.fr

J. Michel
Affiliation: Institut de Mathématiques de Jussieu, Université Paris VII, 175, rue du Chevaleret, 75013 Paris
MR Author ID: 189248
Email: jmichel@math.jussieu.fr

Received by editor(s): April 8, 2009
Received by editor(s) in revised form: February 1, 2010
Published electronically: December 14, 2010
Additional Notes: I. Marin benefited from the ANR Grant ANR-09-JCJC-0102-01
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.