Tensoring generalized characters with the Steinberg character

Hiss, G.; Zalesski, A.

doi:10.1090/S0002-9939-10-10322-0

Tensoring generalized characters with the Steinberg character
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by G. Hiss and A. Zalesski PDF

Proc. Amer. Math. Soc. 138 (2010), 1907-1921 Request permission

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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