An analogue of Bott's theorem for Schubert varieties-related to torus semistable points

TL;DR

This paper establishes conditions under which higher cohomologies vanish for tangent bundle restrictions on Schubert varieties and describes the global sections as the adjoint representation in simply laced cases.

Contribution

It provides a criterion for cohomology vanishing on Schubert varieties and characterizes global sections as the adjoint representation for simply laced groups.

Findings

Higher cohomologies vanish under specific conditions.

Global sections correspond to the adjoint representation in simply laced cases.

Provides an analogue of Bott's theorem for Schubert varieties.

Abstract

Let $G$ be a simple, simply connected algebraic group over the field of complex numbers. We give a necessary and a sufficient condition for a Schubert variety $X(\tau)$ for which all the higher cohomologies $H^{i}(X(\tau), E)$ vanish for the restriction $E$ of the tangent bundle of $G/B$ to X(\tau)$. We further show that the global sections $H^{0}(X(\tau), E)$ is the adjoint representation of $G$ when $G$ is simply laced.