Cup products of line bundles on homogeneous varieties and generalized PRV components of multiplicity one

Let X=G/B be a complete flag variety, and L' and L" two line bundles on X. Consider the cup product map H^{d'}(X,L') x H^{d"}(X, L") --> H^{d}(X,L), where L=L' x L" and d=d'+d". We answer two natural questions about the map above: When is it a nonzero map of irreducible G-representations? Conversely, given generic irreducible representations V' and V" of G, which irreducible components of V' x V" may appear in the right hand side of the map above? We also give bounds on the multiplicities appearing in a tensor product, and relate these considerations to the boundary of the Littlewood-Richardson cone.