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Lifting Representations of Finite Reductive Groups I: Semisimple Conjugacy Classes

Published online by Cambridge University Press:  20 November 2018

Jeffrey D. Adler  and
Joshua M. Lansky
Jeffrey D. Adler
Affiliation:
Department of Mathematics and Statistics, American University, Washington, DC 20016-8050, USA. e-mail: (Adler) jadler@american.edu (Lansky) lansky@american.edu
Joshua M. Lansky
Affiliation:
Department of Mathematics and Statistics, American University, Washington, DC 20016-8050, USA. e-mail: (Adler) jadler@american.edu (Lansky) lansky@american.edu
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Abstract

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Suppose that $\widetilde{G}$ is a connected reductive group defined over a field $k$, and $\Gamma$ is a finite group acting via $k$-automorphisms of $\widetilde{G}$ satisfying a certain quasi-semisimplicity condition. Then the identity component of the group of $\Gamma$-fixed points in $\widetilde{G}$ is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair $\left( \tilde{G},\Gamma \right)$, and consider any group $G$ satisfying the axioms. If both $\widetilde{G}$ and $G$ are $k$-quasisplit, then we can consider their duals $\widetilde{{{G}^{*}}}$ 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 $\widetilde{{{G}^{*}}}\,(k)$. 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 $\widetilde{G}\,(k)$, one obtains a mapping of such packets.

Keywords

20G1520G4020C3322E35reductive groupliftingconjugacy classrepresentationLusztig series
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 6 , 01 December 2014 , pp. 1201 - 1224
Copyright
Copyright © Canadian Mathematical Society 2014

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