Rigidity of Bott-Samelson-Demazure-Hansen variety for $PSp(2n, \mathbb C)$

TL;DR

This paper investigates the rigidity of Bott-Samelson-Demazure-Hansen varieties associated with the symplectic group PSp(2n, C), showing that for certain reduced expressions, all higher cohomology groups of the tangent bundle vanish, indicating rigidity.

Contribution

It characterizes all reduced expressions of the longest Weyl group element for which the tangent bundle's higher cohomology vanishes, revealing new rigidity properties.

Findings

All higher cohomology groups of the tangent bundle vanish for specific reduced expressions.

Provides a classification of Coxeter elements related to these expressions.

Establishes conditions for the rigidity of the variety.

Abstract

Let $G=PSp(2n, \mathbb C)(n\geq 3)$ and $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G$. 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 groups 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 $\underline i$ of $w_0$ in terms of a Coxeter element such that all the higher cohomology groups of the tangent bundle on $Z(w_0, \underline i)$ vanish.