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- W1636741572 abstract "Let $G/P$ be a generalized flag variety, where $G$ is a complex semisimple connected Lie group and $Psubset G$ a parabolic subgroup. Let also $Xsubset G/P$ 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 $Psi: X to {rm Lie}(T)^*$ 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 ${rm Lie}(T)^*$. In this paper we show that all pre-images $Psi^{-1}(mu)$, $muin Psi(X)$, are connected subspaces of $X$. We then consider a one-dimensional subtorus $Ssubset T$, and the map $f: Xto {mathbb R}$, which is the restriction of the $S$ moment map to $X$. We study quotients of the form $f^{-1}(r)/S$, where $rin {mathbb R}$. 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." @default.
- W1636741572 created "2016-06-24" @default.
- W1636741572 creator A5016248898 @default.
- W1636741572 date "2006-06-19" @default.
- W1636741572 modified "2023-09-27" @default.
- W1636741572 title "On some symplectic quotients of Schubert varieties" @default.
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- W1636741572 hasPublicationYear "2006" @default.
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