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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Remarks concerning linear characters of reflection groups


Author: G. I. Lehrer
Journal: Proc. Amer. Math. Soc. 133 (2005), 3163-3169
MSC (2000): Primary 20F55; Secondary 14G05, 20G40, 51F15
Published electronically: May 2, 2005
MathSciNet review: 2160177
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Abstract: Let $G$ be a finite group generated by unitary reflections in a Hermitian space $V$, and let $\zeta$ be a root of unity. Let $E$ be a subspace of $V$, maximal with respect to the property of being a $\zeta$-eigenspace of an element of $G$, and let $C$ be the parabolic subgroup of elements fixing $E$ pointwise. If $\chi$ is any linear character of $G$, we give a condition for the restriction of $\chi$ to $C$ to be trivial in terms of the invariant theory of $G$, and give a formula for the polynomial $\sum_{x\in G}\chi(x)T^{d(x,\zeta)}$, where $d(x,\zeta)$ is the dimension of the $\zeta$-eigenspace of $x$. Applications include criteria for regularity, and new connections between the invariant theory and the structure of $G$.


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Additional Information

G. I. Lehrer
Affiliation: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia
Email: gusl@maths.usyd.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07869-X
PII: S 0002-9939(05)07869-X
Received by editor(s): December 12, 2003
Received by editor(s) in revised form: June 8, 2004, and June 14, 2004
Published electronically: May 2, 2005
Communicated by: John R. Stembridge
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.