Geometric realization of PRV components and the Littlewood–Richardson cone

Let X = G/B and let L1 and L2 be two line bundles on X. Consider the cup product map H1 (X,L1)⊗H q2 (X,L2) → H (X,L), where L = L1⊗L2 and q = q1+q2. We find necessary and sufficient conditions for this map to be a nonzero map of G–modules. We also discuss the converse question, i.e. given irreducible G–modules U and V , which irreducible components W of U⊗V may appear in the right hand side of the equation above. The answer is surprisingly elegant — all such W are generalized PRV components of multiplicity one. Along the way we encounter numerous connections of our problem with problems coming from Representation Theory, Combinatorics, and Geometry. Perhaps the most intriguing relations are with questions about the Littlewood–Richardson cone. This article is expository in nature. We announce results, comment on connections between different fields of Mathematics, and state a number of open questions. The proofs will appear in a forthcoming paper.