Lifting representations of finite reductive groups I: Semisimple conjugacy classes

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 connected part 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 $G$, not just the $\Gamma$-fixed points of $\tilde{G}$, satisfying the axioms. (In fact, the axioms do not require $\Gamma$ to act on all of $\tilde{G}$.) If both $\tilde{G}$ and $G$ are $k$-quasisplit, then we can consider their duals ${\tilde{G}}^{\ast }$ and $G^{\ast }$. We show the existence of and give an explicit formula for a natural map from semisimple stable conjugacy classes in $G^{\ast }(k)$ to those in ${\tilde{G}}^{\ast }(k)$. If $k$ is finite, then our groups are automatically quasisplit, and our result specializes to give a map from semisimple conjugacy classes in $G^{\ast }(k)$ to those in ${\tilde{G}}^{\ast }(k)$. Since such classes parametrize packets of irreducible representations of $G(k)$ and $\tilde{G}(k)$, one obtains a mapping of such packets.

v6: Emphasized root data more, reductive groups less. Some results renumbered. v5: Some simplifications and corrections. v4: Generalized and rewritten. For example, $G$ no longer needs to be the group of $\Gamma$-fixed points in $\tilde{G}$, and indeed $\Gamma$ need not act on all of $\tilde{G}$

References in corpus (3)

• Lifting tropical intersections
• The diagonal of tropical matroid varieties and cycle intersections
• Genus 0 characteristic numbers of the tropical projective plane

Cited by in corpus (7)

• Brief introduction to tropical geometry
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• A-Tint: A polymake extension for algorithmic tropical intersection theory
• The Schläfli Fan
• Complex Tropical Currents, Extremality, and Approximations
• The Realizability of Curves in a Tropical Plane
• Tropical Graph Curves