Automorphism group of a Bott-Samelson-Demazure-Hansen variety

Abstract: Let $G$ be a simple, adjoint, algebraic group over the field of complex numbers, $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G$, $w$ be an element of the Weyl group $W$ and $X(w)$ be the Schubert variety in $G/B$ corresponding to $w$. Let $Z(w,\underline i)$ be the Bott-Samelson-Demazure-Hansen variety (the desingularization of the Schubert variety $X(w)$) corresponding to a reduced expression $\underline i$ of $w$. In this article, we compute the connected component $Aut^0(Z(w, \underline i))$ of the automorphism group of $Z(w,\underline i)$ containing the identity automorphism. We show that $Aut^0(Z(w, \underline i))$ contains a closed subgroup isomorphic to $B$ if and only if $w^{-1}(\alpha_0)<0$, where $\alpha_0$ is the highest root. If $w_0$ denotes the longest element of $W$, then we prove that $Aut^0(Z(w_0, \underline i))$ is a parabolic subgroup of $G$. It is also shown that this parabolic subgroup depends very much on the chosen reduced expression $\underline i$ of $w_0$ and we describe all parabolic subgroups of $G$ that occur as $Aut^0(Z(w_0, \underline i))$. If $G$ is simply laced, then we show that for every $w\in W$ and for every reduced expression $\underline i$ of $w$, $ Aut^0(Z(w, \underline i))$ is a quotient of the parabolic subgroup $Aut^0(Z(w_0, \underline j))$ of $G$ for a suitable choice of a reduced expression $\underline j$ of $w_0$. We also prove that the Bott-Samelson-Demazure-Hansen varieties are rigid for simply laced groups and their deformations are unobstructed in general.