Cohomology of Flag Varieties and the Brylinski-Kostant Filtration

TL;DR

This paper generalizes the connection between cohomology of G-equivariant line bundles on cotangent bundles of flag varieties and the Brylinski-Kostant filtration, extending previous results to a broader class of sl_2 triples.

Contribution

It extends Brylinski's results to a larger class of sl_2 triples and generalizes Broer's results on cohomology of G-equivariant bundles on cotangent bundles of G/P.

Findings

Generalized Brylinski-Kostant filtration to more sl_2 triples

Extended cohomology results to various parabolics P

Provided new insights into G-equivariant bundle cohomology

Abstract

Let G be a semisimple complex algebraic group with Borel subgroup B and let P be a parabolic subgroup of G. Let T*(G/P) denote the cotangent bundle of G/P. Ranee Brylinski discovered a connection between cohomology of G-equivariant line bundles on T*(G/B) and the so-called Brylinski-Kostant filtration, which describes the action of principal sl_2 triples on G-representations. In this paper we generalize these results to a larger class of sl_2 triples. Along the way we also obtain generalizations of results due to Broer on cohomology of G-equivariant bundles on T*(G/P) for various parabolics P.