Equivariant $K$-theory ring of the affine Grassmannian as deformation to normal cone

We revisit fixed-point localization techniques in equivariant topological $K$-theory, presenting its fibers over varying points of the equivariant base as cohomology of the corresponding fixed-point schemes. We then employ this framework for an effective description of the equivariant $K$-theory ring of the affine Grassmannian $\mathcal{G}{r}_G$. We do so by a detailed study of the relevant fixed points, extending known results to the action of any $(x,\zeta )\in T\times {\mathbb{G}}_m^{\mathrm{r}\mathrm{o}\mathrm{t}}$. We obtain a description of the complexified topological $K$-theory ring $K_{T\times {\mathbb{G}}_m^{\mathrm{r}\mathrm{o}\mathrm{t}}}^{\mathrm{t}\mathrm{o}\mathrm{p},0}(\mathcal{G}{r}_G;\mathbb{C})$ as the ring of functions on a family of affine blowups, extending and proving a conjecture of Roman Bezrukavnikov. While this ring is fairly big, we provide a preferred infinite set of topological generators.