  Given a complex projective algebraic variety, write H(X) for its cohomology
with complex coefficients and IH(X) for its Intersection cohomology. We first
show that, under some fairly general conditions, the canonical map H(X)\to
IH(X) is injective.
  Now let Gr = G((z))/G[[z]] be the loop Grassmannian for a complex semisimple
group G, and let X be the closure of a G[[z]]-orbit in Gr. We prove, using the
general result above, a conjecture of D. Peterson describing the cohomology
algebra H(X) in terms of the centralizer of the principal nilpotent in the
Langlands dual of Lie(G).
  In the last section we give a new "topological" proof of Kostant's theorem
about the polynomial algebra of a semisimple Lie algebra, based on purity of
the equivariant intersection cohomology groups of G[[z]]-orbits on Gr.
