Equivariant quantum Schubert calculus

The (small) equivariant quantum cohomology of a homogeneous variety X=G/P is a deformation of both equivariant and quantum cohomology rings of X. It was introduced by A.Givental and B.Kim primarily to study the quantum cohomology of X.

In this talk I will present two results concerning the equivariant quantum cohomology ring when X is a Grassmannian. The first result, a Pieri-Chevalley formula, can be used to compute the structure constants for this ring with respect to the basis of Schubert classes. The second result is a certain positivity property of these structure constants. It holds for any homogeneous space G/P and generalizes the equivariant positivity conjectured by D.Peterson and proved by W.Graham.