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Reflection Subquotients of Unitary Reflection Groups |
Canad. J. Math. 51(1999), 1175-1193
Published:1999-12-01
Printed: Dec 1999
Printed: Dec 1999
- G. I. Lehrer
- T. A. Springer
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] |