Applications of Toric Geometry to Geometric Representation Theory

We study the algebraic geometry and combinatorics of the affine Grassmannian and affine flag variety, which are infinite-dimensional analogs of the ordinary Grassmannian and flag variety. In particular, we analyze the intersections of Iwahori orbits and semi-infinite orbits in the affine Grassmannian and affine flag variety. These intersections have interesting geometric and topological properties, and are related to representation theory. Moreover, we study the central degeneration (the degeneration that shows up in local models of Shimura varieties and Gaitsgory's central sheaves) of semi-infinite orbits, Mirkovic-Vilonen (MV) Cycles, and Iwahori orbits in the affine Grassmannian of type A, by considering their moment polytopes. We describe the special fiber limits of semi-infinite orbits in the affine Grassmannian by studying the action of a global group scheme. Moreover, we give some bounds for the number of irreducible components for the special fiber limits of Iwahori orbits and MV cycles in the affine Grassmannian. Our results are connected to Gaitsgory's central sheaves, affine Schubert calculus and affine Deligne-Lusztig varieties in number theory