Total positivity, Schubert positivity, and geometric Satake
Seminar Room 1, Newton Institute
Let G be a complex simple simply-connected algebraic group. A theorem proved independently by Ginzburg and Peterson states that the homology H_*(Gr_G) of the affine Grassmannian of G is isomorphic to the ring of functions on the centralizer X of a principal nilpotent in the Langlands dual G^\vee. There is a notion of total positivity on X, using Lusztig's general definitions, and there is also a notion of Schubert positivity, using Schubert classes of Gr_G. We connect the two notions using the geometric Satake correspondence. In addition, we give an explicit parametrization of the positive points of X.
This is joint work with Konstanze Rietsch, generalizing work of hers in type A.