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1. CJM Online first
Lifting Representations of Finite Reductive Groups I: Semisimple Conjugacy Classes 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 Categories:20G15, 20G40, 20C33, 22E35 |
2. CJM 2009 (vol 61 pp. 950)
Infinitesimal Invariants in a Function Algebra Let $G$ be a reductive connected linear algebraic group
over an algebraically closed field of positive
characteristic and let $\g$ be its Lie algebra.
First we extend a well-known result about the Picard group of a
semi-simple group to reductive groups.
Then we prove that if the derived group is simply connected
and $\g$ satisfies a
mild condition, the algebra $K[G]^\g$ of regular functions
on $G$ that are invariant under the action of $\g$ derived
from the conjugation action is a unique factorisation domain.
Categories:20G15, 13F15 |
3. CJM 2004 (vol 56 pp. 945)
Smoothness of Quotients Associated \\With a Pair of Commuting Involutions Let $\sigma$, $\theta$ be commuting involutions of the connected semisimple
algebraic group $G$ where $\sigma$, $\theta$ and $G$ are defined over
an algebraically closed field $\k$, $\Char \k=0$. Let $H:=G^\sigma$
and $K:=G^\theta$ be the fixed point groups. We have an action
$(H\times K)\times G\to G$, where $((h,k),g)\mapsto hgk\inv$, $h\in
H$, $k\in K$, $g\in G$. Let $\quot G{(H\times K)}$ denote the
categorical quotient $\Spec \O(G)^{H\times K}$. We determine when this
quotient is smooth. Our results are a generalization of those of
Steinberg \cite{Steinberg75}, Pittie \cite{Pittie72} and Richardson
\cite{Rich82b} in the symmetric case where $\sigma=\theta$ and $H=K$.
Categories:20G15, 20G20, 22E15, 22E46 |