On some symplectic quotients of Schubert varieties

Abstract

Let $G/P$ be a generalized flag variety, where $G$ is a complex semisimple connected Lie group and $P\subset G$ a parabolic subgroup. Let also $X\subset G/P$ be a Schubert variety. We consider the canonical embedding of $X$ into a projective space, which is obtained by identifying $G/P$ with a coadjoint orbit of the compact Lie group $K$, where $G=K^{\mathbb C}$. The maximal torus $T$ of $K$ acts linearly on the projective space and it leaves $X$ invariant: let $Ψ: X \to {\rm Lie}(T)^*$ be the restriction of the moment map relative to the Fubini-Study symplectic form. By a theorem of Atiyah, $Ψ(X)$ is a convex polytope in ${\rm Lie}(T)^*$. In this paper we show that all pre-images $Ψ^{-1}(μ)$, $μ\in Ψ(X)$, are connected subspaces of $X$. We then consider a one-dimensional subtorus $S\subset T$, and the map $f: X\to {\mathbb R}$, which is the restriction of the $S$ moment map to $X$. We study quotients of the form $f^{-1}(r)/S$, where $r\in {\mathbb R}$. We show that under certain assumptions concerning $X$, $S$, and $r$, these symplectic quotients are (new) examples of spaces for which the Kirwan surjectivity theorem and Tolman and Weitsman's presentation of the kernel of the Kirwan map hold true (combined with a theorem of Goresky, Kottwitz, and MacPherson, these results lead to an explicit description of the cohomology ring of the quotient). The singular Schubert variety in the Grassmannian $G_2({\mathbb C}^4)$ of 2 planes in ${\mathbb C}^4$ is discussed in detail.