Tensoring generalized characters with the Steinberg character
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- by G. Hiss and A. Zalesski PDF
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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
- MR Author ID: 86475
- 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
- MR Author ID: 196858
- Email: alexandre.zalesski@gmail.com
- Received by editor(s): January 25, 2009
- Published electronically: February 16, 2010
- Communicated by: Jonathan I. Hall
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 138 (2010), 1907-1921
- MSC (2010): Primary 20C33, 20C20, 20G05, 20G40
- DOI: https://doi.org/10.1090/S0002-9939-10-10322-0
- MathSciNet review: 2596024