Rigidity of Bott–Samelson–Demazure–Hansen variety for ${\varvec{F}}_{\varvec{4}}$ and ${\varvec{G}}_{\varvec{2}}$

Abstract

Let G be a simple algebraic group of adjoint type over ${\mathbb {C}},$ whose root system is of type $F_{4}.$ Let T be a maximal torus of G and B be a Borel subgroup of G containing T. 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 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 $Z(w_{0}, {\underline{i}})$ is rigid (see Theorem 7.1). Further, if G is of type $G_{2},$ there is no reduced expression ${\underline{i}}$ of $w_{0}$ for which $Z(w_{0}, {\underline{i}})$ is rigid (see Theorem 8.2).