Picard groups of certain compact complex parallelizable manifolds and related spaces

Let G be a complex simply connected semisimple Lie group and let Γ torsionless uniform irreducible lattice in G. Then Γ﹨G is compact non-Kähler manifold whose tangent bundle holomorphically trivial. In this note we compute the Picard of when rank(G)≥3. When rank(G)<3, determine Pic0(Γ﹨G)⊂Pic(Γ﹨G) topologically trivial holomorphic line bundles. rank(G)≥2, also show that Pic0(PΓ) isomorphic to Pic0(Y) where PΓ Γ﹨G-bundle associated principal G-bundle over Y, and, rank(G)≥3, Pic(Y)→Pic(PΓ) injective with finite cokernel.