GIT quotient of Schubert varieties modulo one dimensional torus

Abstract

Let $G$ be a simple algebraic group of adjoint type of rank $n$ over $\mathbb{C}$. Let $T$ be a maximal torus of $G$, and $B$ be a Borel subgroup of $G$ containing $T$. Let $W=N_{G}(T)/T$ be the Weyl group of $G$. Let $S=\{\alpha_{1},\ldots,\alpha_{n}\}$ be the set of simple roots of $G$ relative to $(B,T)$. Let $\lambda_{s}$ be the one parameter subgroup of $T$ dual to $\alpha_{s}$. In this paper, we give a criterion for Schubert varieties admitting semistable points for the $\lambda_{s}$-linearized line bundles $\mathcal{L}(\chi)$ associated to every dominant character $\chi$ of $T$. If $\omega_{r}$ is a minuscule fundamental weight and $m\omega_{r}\in X(T)$, then we prove that there is a unique minimal dimensional Schubert variety $X(w_{s,r})$ in $G/P_{S\setminus\{\alpha_{r}\}}$ such that $X(w_{s,r})^{ss}_{\lambda_{s}}(\mathcal{L}(m\omega_{r}))\neq \phi$. Further, we prove that if $G=PSL(n,\mathbb{C})$, and $n\nmid rs$, $m=\frac{n}{(rs,n)}$, and $p=\lfloor\frac{rs}{n}\rfloor$ then the GIT quotient of the minimal dimensional Schubert variety $X(w_{s,r})$ is isomorphic to the projective space $\mathbb{P}(M(s-p, r-p))$, where $M(s-p, r-p)$ is the $(s-p)\times (r-p)$-matrices with complex numbers as entries.