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Search: MSC category 20H15 ( Other geometric groups, including crystallographic groups [See also 51-XX, especially 51F15, and 82D25] )

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1. CJM 1999 (vol 51 pp. 1175)

Lehrer, G. I.; Springer, T. A.
Reflection Subquotients of Unitary Reflection Groups
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.

Categories:51F15, 20H15, 20G40, 20F55, 14C17

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