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Reflection Subquotients of Unitary Reflection Groups

Published online by Cambridge University Press:  20 November 2018

G. I. Lehrer  and
T. A. Springer
G. I. Lehrer
Affiliation:
School of Mathematics and Statistics, University of Sydney, N.S.W. 2006, Australia email: gusl@maths.usyd.edu.au
T. A. Springer
Affiliation:
Department of Mathematics, University of Utrecht, Budapestlaan 6, Utrecht, The Netherlands email: springer@math.uu.nl
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Abstract

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Let $G$ be a finite group generated by (pseudo-) reflections in a complex vector space and let $g$ be any linear transformation which normalises $G$. In an earlier paper, the authors showed how to associate with any maximal eigenspace of an element of the coset $gG$, a subquotient of $G$ which acts as a reflection group on the eigenspace. In this work, we address the questions of irreducibility and the coexponents of this subquotient, as well as centralisers in $G$ of certain elements of the coset. A criterion is also given in terms of the invariant degrees of $G$ for an integer to be regular for $G$. A key tool is the investigation of extensions of invariant vector fields on the eigenspace, which leads to some results and questions concerning the geometry of intersections of invariant hypersurfaces.

Keywords

51F1520H1520G4020F5514C17
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 6 , 01 December 1999 , pp. 1175 - 1193
Copyright
Copyright © Canadian Mathematical Society 1999

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