Graeme Segal once conjectured that there should be a proof of the Kac character formula for affine Kac-Moody algebras using equivariant localization on the affine Grassmannian, analogous to the famous example done by Atiyah and Bott for the Weyl-character formula using Borel-Weil-Bott. He noticed that the idea formally gives the desired answer, modulo several technical considerations such as the infinite dimensionality of the space, singularities, and vanishing of higher Cech cohomology groups. I'll explain how these formulas in type A follow provided one is able to switch two limits in the equivariant K-theory of the infinite dimensional Grassmannian variety into which the affine Grassmannian imbeds. I'll explain a theorem of mine that says gives sufficient (and possibly necessary) conditions in the form of some subtle inequalities for when the limit switching holds, and explain some other examples from Macdonald theory that follow such as the famous constant term formula in type A. I'll then explain the value in studying this question for other varieties, perhaps associated to other root systems.