Lifting representations of finite reductive groups I: Semisimple conjugacy classes

Details Lifting representations of finite reductive groups I: Semisimple conjugacy classes Lifting representations of finite reductive groups I: Semisimple conjugacy classes

2011-06-04

arxiv.org/abs/1106.0786v3

Mathematics

Representation Theory

Added a paragraph drawing a contrast between our map and some others out there

Scientific paper

Suppose that $\tilde{G}$ is a connected reductive group defined over a field $k$, $\Gamma$ is a group of $k$-automorphisms of $\tilde{G}$ satisfying a quasi-semisimplicity condition, and $G$ is the connected part of the group of fixed points. Then $G$ is reductive. If both $\tilde{G}$ and $G$ are $k$-quasisplit, then we can consider their duals $\tilde{G}^*$ and $G^*$. We show the existence and give an explicit formula for a natural map from stable conjugacy classes in $G^*(k)$ to those in $\tilde{G}^*(k)$. If $k$ is finite, then our groups are automatically quasisplit, and our result specializes to give a map from semisimple conjugacy classes in $G^*(k)$ to those in $\tilde{G}^*(k)$. Since such classes parametrize packets of irreducible representations of $G(k)$ and $\tilde{G}(k)$, one obtains a mapping of such packets.

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