  Let G be a simple and simply-connected complex algebraic group, and let X
\subset G^\vee be the centralizer subgroup of a principal nilpotent element.
Ginzburg and Peterson independently related the ring of functions on X with the
homology ring of the affine Grassmannian Gr_G. Peterson furthermore connected
this ring to the quantum cohomology rings of partial flag varieties G/P.
  The first aim of this paper is to study three different notions of positivity
on X: (1) Schubert positivity arising via Peterson's work, (2) total positivity
in the sense of Lusztig, and (3) Mirkovic-Vilonen positivity obtained from the
MV-cycles in Gr_G. Our first main theorem establishes that these three notions
of positivity coincide. The second aim of this paper is to parametrize the
totally nonnegative part of X, confirming a conjecture of the second author.
  In type A a substantial part of our results were previously established by
the second author. The crucial new component of this paper is the connection
with the affine Grassmannian and the geometric Satake correspondence.
