Rigidity of Bott-Samelson-Demazure-Hansen variety for $PSp(2n,\mathbb{C})$

Abstract

Let $G=PSp(2n,\mathbb{C})(n\ge 3)$ and $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G$. Let $w$ be an element of the Weyl group $W$ and $X(w)$ be the Schubert variety in the flag variety $G/B$ corresponding to $w$. Let $Z(w,\underline{i})$ be the Bott-Samelson-Demazure-Hansen variety (the desingularization of $X(w)$) corresponding to a reduced expression $\underline{i}$ of $w$. In this article, we study the cohomology groups of the tangent bundle on $Z(w_0,\underline{i})$, where $w_0$ is the longest element of the Weyl group $W$. We describe all the reduced expressions $\underline{i}$ of $w_0$ in terms of a Coxeter element such that all the higher cohomology groups of the tangent bundle on $Z(w_0,\underline{i})$ vanish.

37 pages