Reduced standard modules, filtrations, and cohomology
Tuesday April 1, 2008
03:30PM - 04:30PM
Speakers:
Brian Parshall
Abstract:
Let $G$ be a semisimple, simply
connected algebraic group defined over an algebraically closed field
$k$ of positive characteristic $p$. A reduced standard module
$\rDelta(\lambda)$ is a rational $G$-module obtained by reduction
from a minimal lattice for the irreducible representation of high
weight $\lambda$ of the corresponding quantum enveloping algebra at
a $p$th root of unity. The dual notion of a reduced costandard
module $\rnabla(\lambda)$ can also be defined. Thus, these modules
are, in some sense, very similar to standard modules
$\Delta(\lambda)$ and costandard modules $\nabla(\lambda)$ for $G$
obtained from the complex semisimple Lie algebra (as in Steinberg's
notes). And, like $\Delta(\lambda)$ and $\nabla(\lambda)$, there are
several equivalent definitions of $\rDelta(\lambda)$ and
$\rnabla(\lambda)$, one due to Z. Lin. We discuss the homological
properties of these modules, filtration questions involving them and
corresponding interesting highest weight categories, and
applications to cohomology of finite groups. This is joint work with
E. Cline and L. Scott.
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