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# Reflection Subquotients of Unitary Reflection Groups

Published:1999-12-01
Printed: Dec 1999
• G. I. Lehrer
• T. A. Springer
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## Abstract

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.
 MSC Classifications: 51F15 - Reflection groups, reflection geometries [See also 20H10, 20H15; for Coxeter groups, see 20F55] 20H15 - Other geometric groups, including crystallographic groups [See also 51-XX, especially 51F15, and 82D25] 20G40 - Linear algebraic groups over finite fields 20F55 - Reflection and Coxeter groups [See also 22E40, 51F15] 14C17 - Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]

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