Eigenvalue Problem and a New Product in Cohomology of Flag Varieties

This is a report on my joint work with P. Belkale. We define a new commutative and associative product in the cohomology of any flag variety G/P (which still satisfies the Poincaré duality) and use this product to generate certain inequalities which solves the analog of the classical Hermitian eigenvalue problem for any complex semisimple group G. Our recipe provides considerable improvement, in general, over the set of inequalities defined by Berenstein-Sjamaar. In fact, our set of inequalities form an irredundant system of inequalities. The talk should be accessible to general mathematical audience.