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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 G L m on H • (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. [ABSTRACT FROM AUTHOR]

The geometric Satake correspondence provides an equivalence of categories between the Satake category of spherical perverse sheaves on the affine Grassmannian and the category of representations of the dual group. In this note, we define a combinatorial version of the Satake category using irreducible components of fibres of the convolution morphism. We then prove an equivalence of coboundary categories between this combinatorial Satake category and the category of crystals of the dual group. [ABSTRACT FROM AUTHOR]

Let G be a connected almost simple algebraic group with a Dynkin automorphism σ . Let G σ be the connected almost simple algebraic group associated with G and σ . We prove that the dimension of the tensor invariant space of G σ is equal to the trace of σ on the corresponding tensor invariant space of G . We prove that if G has the saturation property then so does G σ . As a consequence, we show that the spin group Spin ( 2 n + 1 ) has saturation factor 2, which strengthens the results of Belkale–Kumar [1] and Sam [28] in the case of type B n . [ABSTRACT FROM AUTHOR]

Braverman and Finkelberg recently proposed the geometric Satake correspondence for the affine Kac-Moody group Gaff [Braverman A., Finkelberg M., arXiv:0711.2083]. They conjecture that intersection cohomology sheaves on the Uhlenbeck compactification of the framed moduli space of Gcpt-instantons on $R^4/Z_r$ correspond to weight spaces of representations of the Langlands dual group $G_{aff}^{vee}$ at level $r$. When $G = SL(l)$, the Uhlenbeck compactification is the quiver variety of type $sl(r)_{aff}$, and their conjecture follows from the author's earlier result and I. Frenkel's level-rank duality. They further introduce a convolution diagram which conjecturally gives the tensor product multiplicity [Braverman A., Finkelberg M., Private communication, 2008]. In this paper, we develop the theory for the branching in quiver varieties and check this conjecture for $G = SL(l)$.