Evaluating Characteristic Functions of Character Sheaves at Unipotent Elements

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.

29 pages. Parts of this article first appeared in arXiv:1306.5882. This is an expanded and generalised of version of what appears there. (v2): 30 pages. Final version post referees report. Referenced work of Digne-Lehrer-Michel who also independently obtained Theorem 7.7

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