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Lifting Representations of Finite Reductive Groups I: Semisimple Conjugacy Classes

  Published:2014-07-24
 Printed: Dec 2014
  • Jeffrey D. Adler,
    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
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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 reductive group, lifting, conjugacy class, representation, Lusztig series
MSC Classifications: 20G15, 20G40, 20C33, 22E35 show english descriptions Linear algebraic groups over arbitrary fields
Linear algebraic groups over finite fields
Representations of finite groups of Lie type
Analysis on $p$-adic Lie groups
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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