THE GEOMETRIC SATAKE CORRESPONDENCE AND MV POLYTOPES

Mirkovic and Vilonen give a proof of the geometric Satake correspondence which provides a natural basis in each representation of a complex reductive group. After briefly reviewing the geometric Satake correspondence, I will discuss the geometry underlying the Mirkovic-Vilonen basis. In the course of this discussion, I will introduce the Anderson-Kamnitzer theory of MV polytopes and explain a surprising connection between MV polytopes and Lusztig’s canonical basis. 1. Geometric Satake correspondence Let G be a Chevalley group, and G ∨ its Langlands dual (corresponding to the dual root datum). Let K be a non-Archimedian local field, and O be its ring of integers (for example, consider K = Qp or K = Fq((t))). Then classical Satake correspondence is an isomorphism Cc[G(O) \G(K)/G(O)] ∼=− → K0 RepG ∨ ⊗ C where the subscript c refers to compactly supported functions. Now focus on the case K = C((t)) and O = C[[t]]. We want to categorify this isomorphism. There is a tensor equivalence of categories Perv(G(O) \G(K)/G(O)) H