  The main classical result of Schubert calculus is that multiplication rules
for the basis of Schubert cycles inside the cohomology ring of the Grassmannian
$G(n,m)$ are the same as multiplication rules for the basis of Schur
polynomials in the ring of symmetric polynomials. In this paper, we explain how
to recover this somewhat mysterious connection by using the geometric Satake
correspondence to put the structure of a representation of $GL_m$ on
$H^\bullet(G(n,m))$ and comparing it to the Fock space representation on
symmetric polynomials. This proof also extends to equivariant Schubert
calculus, and gives an explanation of the relationship between
torus-equivariant cohomology of Grassmannians and double Schur polynomials.
