Equivariant $K$-theory, affine Grassmannian and perfection

We study torus-equivariant algebraic K -theory of affine Schubert varieties in the perfect affine Grassmannians over \mathbb{F}_{p} . We further compare it to the torus-equivariant Hochschild homology of perfect complexes, which has a geometric description in terms of global functions on certain fixed-point schemes. We prove that \mathbb{F}_{p} -linearly, this comparison is an isomorphism. Our approach is quite constructive, resulting in new computations of these K -theory rings. We establish various structural results for equivariant perfect algebraic K -theory on the way; we believe these are of independent interest.