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Representation Theory

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The Weil-Steinberg character of finite classical groups

Authors: G. Hiss and A. Zalesski; with an appendix by Olivier Brunat
Journal: Represent. Theory 13 (2009), 427-459
MSC (2000): Primary 20G40, 20C33
Published electronically: September 24, 2009
Corrigendum: Represent. Theory 15 (2011), 729-732.
MathSciNet review: 2550472
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Abstract: We compute the irreducible constitutents of the product of the Weil character and the Steinberg character in those finite classical groups for which a Weil character is defined, namely the symplectic, unitary and general linear groups. It turns out that this product is multiplicity free for the symplectic and general unitary groups, but not for the general linear groups.

As an application we show that the restriction of the Steinberg character of such a group to the subgroup stabilizing a vector in the natural module is multiplicity free. The proof of this result for the unitary groups uses an observation of Brunat, published as an appendix to our paper.

As our ``Weil character'' for the symplectic groups in even characteristic we use the $ 2$-modular Brauer character of the generalized spinor representation. Its product with the Steinberg character is the Brauer character of a projective module. We also determine its indecomposable direct summands.

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Additional Information

G. Hiss
Affiliation: Lehrstuhl D für Mathematik, RWTH Aachen University, 52056 Aachen, Germany

A. Zalesski
Affiliation: School of Mathematics, University of East Anglia, Norwich, NR47TJ, United Kingdom

Olivier Brunat
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstrasse 150, 44780 Bochum

Keywords: Weil character, Steinberg character, classical groups
Received by editor(s): September 26, 2007
Received by editor(s) in revised form: June 14, 2008
Published electronically: September 24, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.