Rigidity of Bott-Samelson-Demazure-Hansen Variety for PSO(2n+1, C)

Let G = P SO(2n + 1, C) (n 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, i) be the Bott-Samelson-Demazure-Hansen Variety (the desingularization of X(w) ) corresponding to a reduced expression i of w.In this article, we study the cohomology modules of the tangent bundle on Z(w 0 , 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 , i) vanish.