Lifting Representations of Finite Reductive Groups I: Semisimple Conjugacy Classes
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 $\widetild...
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Veröffentlicht in: | Canadian journal of mathematics 2014-12, Vol.66 (6), p.1201-1224 |
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Hauptverfasser: | , |
Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | 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. |
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ISSN: | 0008-414X 1496-4279 |
DOI: | 10.4153/CJM-2014-013-6 |