  Confirming a conjecture of Mark Shimozono, we identify polynomial
representatives for the Schubert classes of the affine Grassmannian as the
k-Schur functions in homology and affine Schur functions in cohomology. Our
results rely on Kostant and Kumar's nilHecke ring, work of Peterson on the
homology of based loops on a compact group, and earlier work of ours on
non-commutative k-Schur functions.
