Equivariant Pieri Rule for the homology of the affine Grassmannian

Abstract

An explicit rule is given for the product of the degree two class with an arbitrary Schubert class in the torus-equivariant homology of the affine Grassmannian. In addition a Pieri rule (the Schubert expansion of the product of a special Schubert class with an arbitrary one) is established for the equivariant homology of the affine Grassmannians of SL n and a similar formula is conjectured for Sp 2n and SO 2n+1. For SL n the formula is explicit and positive. By a theorem of Peterson these compute certain products of Schubert classes in the torus-equivariant quantum cohomology of flag varieties. The SL n Pieri rule is used in our recent definition of k-double Schur functions and affine double Schur functions.