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

DOI:
https://doi.org/10.1090/S0002-9939-05-07869-X

Published electronically:
May 2, 2005

MathSciNet review:
2160177

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a finite group generated by unitary reflections in a Hermitian space , and let be a root of unity. Let be a subspace of , maximal with respect to the property of being a -eigenspace of an element of , and let be the parabolic subgroup of elements fixing pointwise. If is any linear character of , we give a condition for the restriction of to to be trivial in terms of the invariant theory of , and give a formula for the polynomial , where is the dimension of the -eigenspace of . Applications include criteria for regularity, and new connections between the invariant theory and the structure of .

**[Gu]**E.A. Gutkin, ``Matrices connected with groups generated by reflections'',*Funkcional Anal. Appl.*(1973), 153-154; translated from**7***Funktsional Anal. i Prilozhen*(1973), 81-82. MR**7****0314956 (47:3505)****[Le]**G.I. Lehrer, ``Rational tori, semisimple orbits and the topology of hyperplane complements'',*Comment. Math. Helv.*(1992), 226-251.MR**67****1161283 (93e:20065)****[LM]**G.I. Lehrer and J. Michel, ``Invariant theory and eigenspaces for unitary reflection groups.''*C. R. Acad. Sc. Paris, Ser. I*(2003), 795-800. MR**336****1990017 (2004d:13005)****[LS1]**G.I. Lehrer and T.A. Springer, ``Intersection multiplicities and reflection subquotients of unitary reflection groups I'',*Geometric group theory down under (Canberra, 1996)*, 181-193, de Gruyter, Berlin, 1999.MR**1714845 (2000i:20064)****[LS2]**G.I. Lehrer and T.A. Springer, ``Reflection subquotients of unitary reflection groups.''*Canad. J. Math.*(1999), 1175-1193. MR**51****1756877 (2001f:20082)****[OS]**P. Orlik and L. Solomon, ``Unitary reflection groups and cohomology'',*Inv. Math.*(1980), 77-94. MR**59****0575083 (81f:32017)****[OT]**P. Orlik and H. Terao, ``Arrangements of hyperplanes.''*Grundlehren der Mathematischen Wissenschaften,*, Springer-Verlag, Berlin, 1992.MR**300****1217488 (94e:52014)****[PW]**A. Pianzola and A. Weiss, ``Monstrous 's and a generalization of a theorem of L. Solomon'',*C. R. Math. Rep. Acad. Sci. Canada*(1989), 189-194. MR**11****1010926 (90k:22007)****[Sh]**Anne V. Shepler, ``Semi-invariants of finite reflection groups'',*J. Alg.*, (1999), 314-326. MR**220****1714136 (2000g:20069)****[Sh04]**Anne V. Shepler, ``Generalized exponents and forms'', to appear,*J. Alg. Comb.*.**[Sp]**T. Springer, `` Regular elements of finite reflection groups'',*Invent. Math.*(1974), 159-198. MR**25****0354894 (50:7371)****[St]**R. Steinberg, ``Differential equations invariant under finite reflection groups'',*Trans. Amer. Math. Soc.*(1964), 392-400.MR**112****0167535 (29:4807)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
20F55,
14G05,
20G40,
51F15

Retrieve articles in all journals with MSC (2000): 20F55, 14G05, 20G40, 51F15

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:
https://doi.org/10.1090/S0002-9939-05-07869-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.