Articles written in Proceedings – Mathematical Sciences

    • Rigidity of Bott–Samelson–Demazure–Hansen variety for $F_{4}$ and $G_{2}$


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      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).

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