Equivariant $K$-homology of affine Grassmannian and $K$-theoretic double $k$-Schur functions

TL;DR

This paper explores the structure of the torus equivariant K-homology ring of the affine Grassmannian, introducing new symmetric functions called K-theoretic double k-Schur functions and providing a geometric realization for type A.

Contribution

It introduces equivariantly deformed symmetric functions called K-theoretic double k-Schur functions and offers a Ginzburg-Peterson type realization of the K-homology ring in type A.

Findings

Introduction of K-theoretic double k-Schur functions as Schubert bases

Construction of these functions via Demazure operators

Realization of the K-homology ring as a coordinate ring of a centralizer family

Abstract

We study the torus equivariant K-homology ring of the affine Grassmannian $\mathrm{Gr}_G$ where $G$ is a connected reductive linear algebraic group. In type $A$, we introduce equivariantly deformed symmetric functions called the K-theoretic double $k$-Schur functions as the Schubert bases. The functions are constructed by Demazure operators acting on equivariant parameters. As an application, we provide a Ginzburg-Peterson type realization of the torus-equivariant K-homology ring of $\mathrm{Gr}_{{SL}_n}$ as the coordinate ring of a centralizer family for $PGL_n(\mathbb{C})$.