Cohomology of flag bundles over compact Hermitian locally symmetric spaces

Let E→ 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_Γ:=Γ X, E=Y_Γ:=Γ 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⊂ G, a maximal compact subgroup, H=ZK(S), the centralizer in K of a toral subgroup S⊆ K containing $Z(K)$, the center of K and Γ⊂ G, a uniform and torsionless lattice in G. We also obtain a description of the Picard group of Y_Γ and XΓ, for which the complexification of G need not be simply connected. Moreover when G is simple, we obtain the values of q for which Hp,q(X_Γ) vanishes when $p=0,1$. This extends the results of R. Parthasarathy from $1980$, who considered (partially) the case $p=0$.