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

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 $PGL_n(\mathbb{C})$.