Evaluating characteristic functions of character sheaves at unipotent elements

Taylor, Jay

doi:10.1090/S1088-4165-2014-00457-6

Evaluating characteristic functions of character sheaves at unipotent elements
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by Jay Taylor

Represent. Theory 18 (2014), 310-340

DOI: https://doi.org/10.1090/S1088-4165-2014-00457-6

Published electronically: October 17, 2014

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Abstract:

Assume $\mathbf {G}$ is a connected reductive algebraic group defined over an algebraic closure $\mathbb {K} = \overline {\mathbb {F}}_p$ of the finite field of prime order $p>0$. Furthermore, assume that $F : \mathbf {G} \to \mathbf {G}$ is a Frobenius endomorphism of $\mathbf {G}$. In this article we give a formula for the value of any $F$-stable character sheaf of $\mathbf {G}$ at a unipotent element. This formula is expressed in terms of class functions of $\mathbf {G}^F$ which are supported on a single unipotent class of $\mathbf {G}$. In general these functions are not determined, however, we give an expression for these functions under the assumption that $Z(\mathbf {G})$ is connected, $\mathbf {G}/Z(\mathbf {G})$ is simple and $p$ is a good prime for $\mathbf {G}$. In this case our formula is completely explicit.

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