Equivariant K-theory of affine Grassmannians in representation theory and arithmetic

We develop and employ techniques from equivariant algebraic K-theory and related invariants in the context of geometric representation theory, in both arithmetic and topological situations. We showcase the use of such techniques on the affine Grassmannian Gr, a space of fundamental interest in the geometric Langlands program. It is a deep development of mathematics of the last century that many concrete, yet combina- torially complex algebraic problems may be effectively studied through the lens of algebraic geometry. The objects of interest can be often realized as cohomological invariants of algebraic varieties, and good understanding of their geometry sheds light into the original questions. Such techniques have seen immense applications in the Langlands program, where they go under the label of geometric representation theory. One source of powerful invariants in algebraic geometry comes from algebraic K-theory, Hochschild homology, and their relatives. These localizing invariants contain large amount of information, but are quite hard to compute. For this reason, their usage in geometric representation theory has been limited. The aim of this thesis is to showcase how to control such invariants in the situations of interest and use them to obtain new insights. We start by reinterpreting equivariant Hochschild homology in terms of functions on certain fixed-point schemes, which are of independent interest. We compare it to equivariant K-theory via the trace map. We give new computations and comparisons of such invariants of affine Schubert varieties in Gr, including arithmetic situations. We show that they behave much better than expected. We finally utilize this circle of ideas in a purely topological setting. We describe the varying fixed points of the extended torus action on the affine Grassmannian, and use it to compute its equivariant topological K-theory ring. The answer is nontrivial and verifies an outstanding conjecture in the subject. We compare, partly conjecturally, the resulting K-theory ring to the completed center of an integral even hybrid quantum group and its deformed quantum category O. This gives a genuine application of our computations in pure representation theory.