  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
$\Psi: X \to {\rm Lie}(T)^*$ 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
$\Psi^{-1}(\mu)$, $\mu\in \Psi(X)$, 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.
