Homological methods in certain Picard group computations

Let G be a connected complex semisimple Lie group, Γ cocompact, irreducible and torsionless lattice in K maximal compact subgroup of G. Assume acts by left multiplication right on MΓ=Γ﹨G, X=G/K XΓ=Γ﹨X. In this article we prove that for any n≥0, the composition Hn(XΓ,C)→Hn(MΓ,C)→Hn(MΓ,OMΓ) is an isomorphism. As application when simply connected, compute Picard group MΓ cases rank(G) =1,2. More precisely show if =1, Pic(MΓ)=(Cr/Zr)⊕A =2, then Pic(MΓ)≅A via first Chern class map, where A torsion H2(MΓ,Z) r rank Γ/[Γ,Γ].