  In this paper, we will look at actions on complex flag varieties $G/P$ of the
torus $\hat{T}=\bigcap\limits_{\alpha\in\Delta\setminus\Delta_P}\ker(\alpha)$,
and under reasonable assumptions, we will give a description of the set
$X^{us}$ of unstable points for $\hat{T}$-linearized invertible sheaves. We
will investigate the case where $P$ is a maximal parabolic subgroup, and show
that $X^{us}$ can be written as a disjoint union of a Schubert variety and an
opposite Schubert variety, and we deduce the vanishing of cohomology groups
$H^i(Y,{\cal M})$ for invertible sheaves ${\cal M}$ on the quotient variety $Y$
for $i$ in a range given by the codimension of $X^{us}$.
