Mirkovic-Vilonen cycles and polytopes

We will start by quickly reviewing the (classical) Satake isomorphism and then the geometric Satake correspondence. Under this correspondence the IC sheaves (intersection homology) of affine Schubert cells are associated with irreducible representations   of the Langlands dual group of G. The Mirkovic-Vilonen cycles are subvarieties in the affine Schubert cells,which give bases in the intersection homologies of those cells, thus playing a fundamental role in the correspondence. Similar to the Weyl (weight) polytopes, the polytopes arising as the images of Mirkovic-Vilonen cycles under the moment map w.r.t. the action of maximal compact torus T allow to (combinatorially) read off some information on representations of G (dimension of weight spaces, multiplicities of irreducibles in tensor products, etc.)  I will assume only basic knowledge on affine Grassmannians (the notes https://sites.pitt.edu/~bdt18/NotesIntroAffGrassmannian.pdf are more than sufficient). Examples will be given.