Frobenius splitting of cotangent bundles of flag varieties and geometry of nilpotent cones

TL;DR

This paper demonstrates Frobenius splitting of cotangent bundles of flag varieties using G-invariant forms, leading to new results on the geometry of nilpotent cones and filtrations of line bundle sections.

Contribution

It extends Frobenius splitting techniques to all groups in good prime characteristics, generalizing previous results limited to the general linear group.

Findings

Vanishing theorem for pullbacks of line bundles

Normality and rational singularities of subregular nilpotent varieties

Existence of good filtrations for cohomology of line bundle pullbacks

Abstract

We use the G-invariant non-degenerate form on the Steinberg module to Frobenius split the cotangent bundle of a flag variety in good prime characteristics. This was previously only known for the general linear group. Applications are a vanishing theorem for pull back of line bundles to the cotangent bundle (proved for the classical groups and G_2 by Andersen and Jantzen and in characteristic zero by B. Broer (for all groups)), normality and rational singularities for the subregular nilpotent variety and good filtrations of the global sections of pull backs of line bundles to the cotangent bundle, which in turn implies good filtrations of cohomology of induced representations.