Evaluating characteristic functions of character sheaves at unipotent elements

Author:

Taylor Jay

Abstract

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

Publisher

American Mathematical Society (AMS)

Subject

Mathematics (miscellaneous)

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