TIFR, Mumbai
October 8, 2020
Automorphism groups of Schubert varieties and rigidity of Bott-Samelson-Demazure-Hansen varieties: Throughout this talk, we assume $G$ to be a simple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, $B$ to be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $G/B$ be the full flag variety consisting of all Borel subgroups of $G.$ For $w$ in $W,$ let $X(w)$ denote the Schubert variety in $G/B$ corresponding to $w.$
In this talk, we discuss the following three problems:
Prob 1: Whether given any parabolic subgroup $P$ of $G$ containing $B$ properly, is there an element $w$ in $W$ such that $P=\mathrm{Aut}^0(X(w))?$
Prob 2: Let $G =PSO(2n+1,\mathbb{C})(n \ge 3).$ Let $w$ be an element of the Weyl group $W$ and $\underline{i}$ be a reduced expression of $w.$ Let $Z(w, \underline{i})$ be the Bott-Samelson-Demazure-Hansen variety (the desingularization of $X(w)$) corresponding to $\underline{i}.$ In this talk, we discuss the cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i}),$ where $w_{0}$ is the longest element of the Weyl group $W$. More precisely, what are all the possible reduced expressions $\underline{i_{0}}$ of $w_{0}$ in terms of a Coxeter element such that all the higher cohomology modules of the tangent bundle on $Z(w_{0}, \underline{i_{0}})$ vanish?
Prob 3: We ask the similar question as that of Prob 2 for the root system of type $F_{4}$ and $G_{2}.$
This talk is based on joint work with S.Senthamarai Kannan.