Cohomology of flag varieties and the Brylinski–Kostant filtration

Let G be a reductive algebraic group over C and let N be a G-module. For any subspace M of N, the Brylinski–Kostant filtration on M is defined through the action of a principal nilpotent element in LieG. This filtration is related to a q-analog of weight multiplicity due to Lusztig. We generalize this filtration to other nilpotent elements and show that this generalized filtration is related to “parabolic” versions of Lusztig's q-analog of weight multiplicity. Along the way we also generalize results of Broer on cohomology vanishing of bundles on cotangent bundles of partial flag varieties. We conclude by computing some explicit examples.