A geometric Schur functor

Abstract

We give geometric descriptions of the category \(C_k(n,d)\) of rational polynomial representations of \(GL_n\) over a field \(k\) of degree \(d\) for \(d\le n\), the Schur functor and Schur–Weyl duality. The descriptions and proofs use a modular version of Springer theory and relationships between the equivariant geometry of the affine Grassmannian and the nilpotent cone for the general linear groups. Motivated by this description, we propose generalizations for an arbitrary connected complex reductive group of the category \(C_k(n,d)\) and the Schur functor.