Minimal Parabolic subgroups and Automorphism groups of Schubert varieties-II

Let $G$ be a simple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ In this article, we show that $\alpha$ is a co-minuscule root if and only if for any parabolic subgroup $Q$ containing $B$ properly, there is no Schubert variety $X_{Q}(w)$ in $G/Q$ such that the minimal parabolic subgroup $P_{\alpha}$ of $G$ is the connected component, containing the identity automorphism of the group of all algebraic automorphisms of $X_{Q}(w).$