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

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.