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Evaluating characteristic functions of character sheaves at unipotent elements


Author: Jay Taylor
Journal: Represent. Theory 18 (2014), 310-340
MSC (2010): Primary 20C33; Secondary 20G40
DOI: https://doi.org/10.1090/S1088-4165-2014-00457-6
Published electronically: October 17, 2014
MathSciNet review: 3269461
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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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Additional Information

Jay Taylor
Affiliation: FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany
Email: taylor@mathematik.uni-kl.de

DOI: https://doi.org/10.1090/S1088-4165-2014-00457-6
Received by editor(s): February 5, 2014
Received by editor(s) in revised form: September 12, 2014
Published electronically: October 17, 2014
Article copyright: © Copyright 2014 American Mathematical Society