Total positivity, Schubert positivity, and geometric Satake

Let G be a simple, simply-connected complex algebraic group, and let X⊂G∨ be the centralizer of a principal nilpotent. Ginzburg and Peterson independently related the ring of functions on X with the homology ring of the affine Grassmannian GrG. Peterson furthermore connected X to the quantum cohomology rings of partial flag varieties G/P.In this paper we study three notions of positivity for X: (1) Schubert positivity arising via Peterson's work, (2) Lusztig's total positivity and (3) Mirković-Vilonen positivity obtained from the MV-cycles in GrG. The first main theorem establishes that these three notions of positivity coincide. Our second main theorem proves a parametrization of the totally nonnegative part of X, confirming a conjecture of the second author.In type A the parametrization and relationship with Schubert positivity were proved earlier by the second author. Here we tackle the general type case and also introduce a crucial new connection with the affine Grassmannian and geometric Satake correspondence.