Equivariant homology of the symplectic affine Grassmannian and dual affine Schur $P$-functions

Abstract

We study the torus-equivariant homology $H_{\ast }^T({\mathrm{Gr}}_G)$ of the affine Grassmannian ${\mathrm{Gr}}_G$, where $G={\mathrm{Sp}}_{2n}(\mathbb{C})$ is the symplectic group. This homology admits a natural ring structure and a Schubert basis, giving rise to a well-defined Schubert calculus. We realize $H_{\ast }^T({\mathrm{Gr}}_G)$ in terms of symmetric functions. Our first main result introduces a new family of symmetric functions, called the \emph{dual affine Schur $P$-functions}, which represent the Schubert classes. These functions are defined through the action of the affine nil-Hecke algebra, and specialize, in the stable limit as $n\to \infty$, to the dual factorial $P$-functions of Nakagawa and Naruse. Our second main result gives a precise comparison between this symmetric function model and the geometric construction of $H_{\ast }^T({\mathrm{Gr}}_G)$ due to Ginzburg and Peterson, which identifies it with a coordinate ring of a centralizer family in the Langlands dual group.

50 pages, 1 figure