Equivariant Schubert calculus and geometric Satake
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 give a new explanation of 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 new proof also extends to equivariant Schubert calculus, and gives an explanation of the relationship between torus-equivariant cohomology of Grassmannians and double Schur polynomials.