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The Weil-Steinberg character of finite classical groups
Author(s):
G.
Hiss;
A.
Zalesski;
with an appendix by
Olivier
Brunat
Journal:
Represent. Theory
13
(2009),
427-459.
MSC (2000):
Primary 20G40, 20C33
Posted:
September 24, 2009
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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  -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
Email:
gerhard.hiss@math.rwth-aachen.de
A.
Zalesski
Affiliation:
School of Mathematics, University of East Anglia, Norwich, NR47TJ, United Kingdom
Email:
alexandre.zalesski@gmail.com
Olivier
Brunat
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstrasse 150, 44780 Bochum
Email:
olivier.brunat@ruhr-uni-bochum.de
DOI:
10.1090/S1088-4165-09-00351-3
PII:
S 1088-4165(09)00351-3
Keywords:
Weil character,
Steinberg character,
classical groups
Received by editor(s):
September 26, 2007 and, in revised from, June 14, 2008
Posted:
September 24, 2009
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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