Cup product on homogeneous varieties, BWB theorem, and generalized PRV components

Abstract. Let G = GL(n, C) and let B be a Borel subgroup with corresponding homogeneous variety X = G/B. We study the cup product of cohomologies of line bundles on X. Borel-Weil-Bott's theorem implies that the cup product is a G-map from the tensor product of two irreducible G-modules into a third irreducible G-module. In this talk I will present necessary and sufficient conditions for this map to be surjective. It turns out that whenever it is surjective, the target G-module is either trivial, or a generalized PRV component of the tensor product of the other G-modules under consideration. However only generalized PRV components of multiplicity 1 in the tensor product appear in this way. I will recall the objects mentioned in the title, state the results, and formulate a conjecture.