Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 51, No. 1, pp. 930 (2010) 

On some symplectic quotients of Schubert varietiesAugustinLiviu MareDepartment of Mathematics and Statistics, University of Regina, Regina SK, Canada S4S0A2; email: mareal@math.uregina.caAbstract: 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 $G=K^{\bC}$. The maximal torus $T$ of $K$ acts linearly on the projective space and it leaves $X$ invariant: let $\Psi: X \to {\rm Lie}(T)^*$ be the restriction of the moment map relative to the FubiniStudy symplectic form. By a theorem of Atiyah, $\Psi(X)$ is a convex polytope in ${\rm Lie}(T)^*$. In this paper we show that all preimages $\Psi^{1}(\mu)$, $\mu\in \Psi(X)$, are connected subspaces of $X$. We then consider a onedimensional subtorus $S\subset T$, and the map $f: X\to \bR$, which is the restriction of the $S$ moment map to $X$. We study quotients of the form $f^{1}(r)/S$, where $r\in \bR$. 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 $G_2(\bC^4)$ of 2 planes in $\bC^4$ is discussed in detail. Classification (MSC2000): 53D20, 14L24 Full text of the article:
Electronic version published on: 27 Jan 2010. This page was last modified: 28 Jan 2013.
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