On the automorphism of a smooth Schubert variety

Let $G$ be a simple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G$. Let $w$ be an element of the Weyl group $W$ and let $X(w)$ be the Schubert variety in $G/B$ corresponding to $w$. Let $\alpha_{0}$ denote the highest root of $G$ with respect to $T$ and $B.$ Let $P$ be the stabiliser of $X(w)$ in $G.$ In this paper, we prove that if $G$ is simply laced and $X(w)$ is smooth, then the connected component of the automorphism group of $X(w)$ containing the identity automorphism equals $P$ if and only if $w^{-1}(\alpha_{0})$ is a negative root ( see Theorem 4.2 ). We prove a partial result in the non simply laced case ( see Theorem 6.6 ).