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_: = X, E=Y_: = 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 G, a uniform and torsionless lattice in G. We also obtain a description of the Picard group of Y_ and X_, 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_) vanishes when p=0, 1. This extends the results of R. Parthasarathy from 1980, who considered (partially) the case p=0.