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 $Au{t}^0(Z(w,\underline{i}))$ of the automorphism group of $Z(w,\underline{i})$ containing the identity automorphism. We show that $Au{t}^0(Z(w,\underline{i}))$ 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 $Au{t}^0(Z(w_0,\underline{i}))$ 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 $Au{t}^0(Z(w_0,\underline{i}))$. If $G$ is simply laced, then we show that for every $w\in W$ and for every reduced expression $\underline{i}$ of $w$, $Au{t}^0(Z(w,\underline{i}))$ is a quotient of the parabolic subgroup $Au{t}^0(Z(w_0,\underline{j}))$ 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.

arXiv comment34 pages, to appear in Transformation Groups