"Quantum Weyl reciprocity for cohomology," by B. Parshall and L. Scott 

Department of Mathematics
University of Virginia
Charlottesville, VA 22901

Abstract:
We study the relationship between
the cohomology of $q$-Schur algebras and Hecke algebras in
type $A$.  A key tool, which has independent interest,
involves a specific Hecke algebra resolution, introduced
in a special case by Deodhar. This resolution is somewhat
akin to the Koszul resolution for Lie algebra cohomology,
in that it has an explicit description yet applies for all
modules.