Let G be a semisimple algebraic group, B a Borel subgroup, and X = G/B. If L is a line bundle on X then by the Borel-Weil-Bott theorem there is at most one value of d for which the cohomology group H^d(X, L) is nonzero. Given two line bundles L_1 and L_2 with nonzero cohomologies in degrees d_1 and d_2, it is natural to look at the cup-product map H^{d_1}(X, L_1) x H^{d_2}(X, L_2) ----> H^{d}(X, L) where d = d_1+d_2 and L is the tensor product L_1 x L_2.
The cup product map is either surjective or zero, again by the Borel-Weil-Bott theorem, but it was not known how to tell which occurs.
The talk will give a complete answer in the A_n, B_n, C_n, and D_n cases, although most of the discussion will be type independent. A large part of the talk will be an exposition of the Borel-Weil-Bott theorem and some of the geometry of the varieties G/B. If there is time we will also address the representation theoretic question of which components of a tensor product of irreducible representations can be realized via cup product, and the connection of the question with the generalized Horn problem.
This is a joint project with Ivan Dimitrov.