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ISSN 1088-4165

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
Posted: September 24, 2009
Corrigendum: Represent. Theory 15 (2011), 729-732.
MathSciNet review: 2550472
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Abstract | References | Similar Articles | Additional Information

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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1. R. Carter, Finite groups of Lie type: Conjugacy classes and complex characters, Wiley, 1985. MR 794307 (87d:20060)
2. P. Gérardin, Weil representations associated to finite fields, J. Algebra 46 (1977), 54-101. MR 0460477 (57:470)

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