Let $G=PSO(2n+1, \mathbb{ C}) (n \ge 3)$ and $B$ be the Borel subgroup of $G$ containing maximal torus $T$ of $G.$ Let $w$ be an element of 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 modules 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 of $w_{ 0}$ in terms of a Coxeter element such that all the higher cohomology modules of the tangent bundle on $Z(w_{ 0}, \underline{ i})$ vanish (see Theorem \ref{ theorem 8.1}).
Categories: math.AG math.CO math.RT