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# Lifting Representations of Finite Reductive Groups I: Semisimple Conjugacy Classes

Published:2014-07-24
Printed: Dec 2014
Department of Mathematics and Statistics, American University, Washington, DC 20016-8050, USA
• Joshua M. Lansky,
Department of Mathematics and Statistics, American University, Washington, DC 20016-8050, USA
 Format: LaTeX MathJax PDF

## Abstract

Suppose that $\tilde{G}$ is a connected reductive group defined over a field $k$, and $\Gamma$ is a finite group acting via $k$-automorphisms of $\tilde{G}$ satisfying a certain quasi-semisimplicity condition. Then the identity component of the group of $\Gamma$-fixed points in $\tilde{G}$ is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair $(\tilde{G},\Gamma)$, and consider any group $G$ satisfying the axioms. If both $\tilde{G}$ and $G$ are $k$-quasisplit, then we can consider their duals $\tilde{G}^*$ and $G^*$. We show the existence of and give an explicit formula for a natural map from the set of semisimple stable conjugacy classes in $G^*(k)$ to the analogous set for $\tilde{G}^*(k)$. If $k$ is finite, then our groups are automatically quasisplit, and our result specializes to give a map of semisimple conjugacy classes. Since such classes parametrize packets of irreducible representations of $G(k)$ and $\tilde{G}(k)$, one obtains a mapping of such packets.
 Keywords: reductive group, lifting, conjugacy class, representation, Lusztig series
 MSC Classifications: 20G15 - Linear algebraic groups over arbitrary fields 20G40 - Linear algebraic groups over finite fields 20C33 - Representations of finite groups of Lie type 22E35 - Analysis on $p$-adic Lie groups

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