A generalization of Fulton's Conjecture for arbitrary groups

(Joint work with Shrawan Kumar and Nicolas Ressayre) It is an interesting open problem to connect the structure constants in the cohomology ring of homogeneous spaces (in the Schubert basis), and those in the invariant theory of groups (generalizing the two appearances of Littlewood Richardson coefficients: in the cohomology of Grassmannians and the invariant theory of $GL_n$). I will talk about a generalization of a conjecture of Fulton which is a step in this direction. It connects multiplicity one properties in the cohomology of homogeneous spaces $G/P$ to rigidity properties in the representation theory of the Levi subgroup $L$ of $P$.