On some symplectic quotients of Schubert varieties (Q846751)

Let \(K\) be a compact Lie group, \(G=K^{\mathbb{C}}\) a semisimple connected Lie group and \(P\subset G\) a parabolic subgroup. Let \(X\subset G/P\) be a Schubert variety being canonically embedded into a projective space via the identification of \(G/P\) with a co-adjoint orbit of \(K\). The maximal torus \(T\) of \(K\) acts linearly on the projective space leaving \(X\) invariant. Let \(\psi:X\to \text{Lie}(T)^\ast\) be the restriction of the moment map relative to the Fubini--Study symplectic form. It is proved that all pre-images \(\psi^{-1}(\mu), \mu\in \psi(X)\), are connected subspaces of \(X\). Let \(S\subset T\) be a one-dimensional subtorus. Let \(f:X\to\mathbb{R}\) be the restriction of the \(S\) moment map to \(X\). Quotients of the form \(f^{-1}(r)/S, r\in \mathbb{R}\), are studied. It is proved that one obtains examples for which the Kirwan surjectivity theorem and Tolman and Weitsman's presentation of the kernel of the Kirwan map hold. The singular Schubert variety in the Grassmannian of planes in \(\mathbb{C}^4\) is discussed in detail.