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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 mapof irreducible G-representations? Conversely, given generic irreduciblerepresentations V' and V" of G, which irreducible components of V' x V" mayappear in the right hand side of the map above? We also give bounds on themultiplicities appearing in a tensor product, and relate these considerationsto the boundary of the Littlewood-Richardson cone.

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