Torus quotients of homogeneous spaces-minimal dimensional Schubert Variety admitting semi-stable points

TL;DR

This paper classifies minimal Schubert varieties with semistable points under torus actions in certain algebraic groups and describes Coxeter elements with similar properties.

Contribution

It provides a comprehensive description of minimal Schubert varieties and Coxeter elements admitting semistable points for torus actions in specific algebraic groups.

Findings

Identifies all minimal Schubert varieties with semistable points in $G/P$ for types $B_n,C_n,D_n$.

Characterizes Coxeter elements with semistable points in $G/B$.

Provides criteria for semistability in these geometric settings.

Abstract

In this paper, for any simple, simply connected algebraic group $G$ of type $B_n,C_n$ or $D_n$ and for any maximal parabolic subgroup $P$ of $G$, we describe all minimal dimensional Schubert varieties in $G/P$ admitting semistable points for the action of a maximal torus $T$ with respect to an ample line bundle on $G/P$. In this paper, we also describe, for any semi-simple simply connected algebraic group $G$ and for any Borel subgroup $B$ of $G$, all Coxeter elements $\tau$ for which the Schubert variety $X(\tau)$ admits a semistable point for the action of the torus $T$ with respect to a non-trivial line bundle on $G/B$.