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 GrG 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 Gr_SLₙ as the coordinate ring of a centralizer family for PGLₙ(C).