  Let $E\to B$ be a complex analytic fiber bundle with fiber $F$, a flag
variety over a compact complex manifold $B$. We shall obtain a description of
the cohomology of $E$ when $B=X_\Gamma:=\Gamma\backslash X,
E=Y_\Gamma:=\Gamma\backslash Y$ and $F=K/H$, a flag variety, where $Y=G/H$ and
$X=G/K$, a Hermitian globally symmetric space of non-compact type with $G$
being a real, connected, non-compact, semisimple linear Lie group with no
compact factors and simply connected complexification, $K\subset G$, a maximal
compact subgroup, $H=Z_K(S)$, the centralizer in $K$ of a toral subgroup
$S\subseteq K$ containing $Z(K)$, the center of $K$ and $\Gamma\subset G$, a
uniform and torsionless lattice in $G$. We also obtain a description of the
Picard group of $Y_\Gamma$ and $X_{\Gamma}$, for which the complexification of
$G$ need not be simply connected. Moreover when $G$ is simple, we obtain the
values of $ q$ for which $H^{p,q}(X_\Gamma)$ vanishes when $p=0,1$. This
extends the results of R. Parthasarathy from $1980$, who considered (partially)
the case $p=0$.
