Rigidity of Bott-Samelson-Demazure-Hansen variety for $F_4$ and $G_2$

TL;DR

This paper investigates the rigidity of Bott-Samelson-Demazure-Hansen varieties associated with the longest element in the Weyl group for groups of types F4 and G2, providing classifications based on reduced expressions and cohomology properties.

Contribution

It characterizes all reduced expressions of the longest Weyl group element that yield rigid Bott-Samelson-Demazure-Hansen varieties for type F4 and shows non-existence of such for type G2.

Findings

For type F4, all reduced expressions leading to rigidity are described.

For type G2, no reduced expression results in a rigid variety.

The study links cohomology modules of tangent bundles to the rigidity of these varieties.

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 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 8.1). 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).