Abstract view
Lifting Representations of Finite Reductive Groups I: Semisimple Conjugacy Classes


Published:20140724
Printed: Dec 2014
Jeffrey D. Adler,
Department of Mathematics and Statistics, American University, Washington, DC 200168050, USA
Joshua M. Lansky,
Department of Mathematics and Statistics, American University, Washington, DC 200168050, USA
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 quasisemisimplicity 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
fixedpoint 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.