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 $\Psi :X\to \mathrm{Lie}(T{)}^{\ast }$ be the restriction of the moment map relative to the Fubini-Study symplectic form. By a theorem of Atiyah, $\Psi (X)$ is a convex polytope in $\mathrm{Lie}(T{)}^{\ast }$. In this paper we show that all pre-images ${\Psi }^{-1}(\mu )$, $\mu \in \Psi (X)$, 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.

This is a substantially revised version of the paper initially called "On the cohomology of symplectic quotients of Schubert varieties by certain circle actions". It now has 20 pages and 1 figure