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

 

Tensoring generalized characters with the Steinberg character


Authors: G. Hiss and A. Zalesski
Journal: Proc. Amer. Math. Soc. 138 (2010), 1907-1921
MSC (2010): Primary 20C33, 20C20, 20G05, 20G40
Published electronically: February 16, 2010
MathSciNet review: 2596024
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Abstract: Let $ \mathbf{G}$ be a reductive connected algebraic group over an algebraic closure of a finite field of characteristic $ p$. Let $ F$ be a Frobenius endomorphism on $ \mathbf{G}$ and write $ G := \mathbf{G}^F$ for the corresponding finite group of Lie type.

We consider projective characters of $ G$ in characteristic $ p$ of the form $ St \cdot \beta$, where $ \beta$ is an irreducible Brauer character and $ St$ the Steinberg character of $ G$.

Let $ M$ be a rational $ \mathbf{G}$-module affording $ \beta$ on restriction to $ G$. We say that $ M$ is $ G$-regular if for every $ F$-stable maximal torus $ \mathbf{T}$ distinct weight spaces of $ M$ are non-isomorphic $ \mathbf{T}^F$-modules. We show that if $ M$ is $ G$-regular of dimension $ d$, then the lift of $ St \cdot \beta$ decomposes as a sum of $ d$ regular characters of $ G$.


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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: Departimento di Matematica e Applicazioni, Universitá degli Studi di Milano- Bicocca, via Roberto Cozzi 53, 20125, Milano, Italy
Email: alexandre.zalesski@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10322-0
PII: S 0002-9939(10)10322-0
Keywords: Projective characters, Chevalley groups, Steinberg character
Received by editor(s): January 25, 2009
Published electronically: February 16, 2010
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2010 American Mathematical Society