The Schubert calculus of the affine Grassmannian is known to be equivalent to the quantum Schubert calculus of the flag variety. In this talk, I focus on the case of the symplectic group. We introduce a class of symmetric functions, called the affine dual Schur functions of symplectic type, whose set forms the Schubert basis for the ring of equivariant K-homology of the affine Grassmannian of the symplectic group. Time permitting, I will also discuss the connection to the equivariant K-theory of the flag variety. This talk is primarily based on joint work with Mark Shimozono and Kohei Yamaguchi.