On category O for affine Grassmannian slices and categorified tensor products

Abstract

Truncated shifted Yangians are a family of algebras which naturally quantize slices in the affine Grassmannian. These algebras depend on a choice of two weights λ and μ for a Lie algebra g, which we will assume is simply laced. In this paper, we relate the category O over truncated shifted Yangians to categorified tensor products: For a generic integral choice of parameters, category O is equivalent to a weight space in the categorification of a tensor product of fundamental representations defined by the third author using KLRW algebras. We also give a precise description of category O for arbitrary parameters using a new algebra which we call the parity KLRW algebra. In particular, we confirm the conjecture of the authors that the highest weights of category O are in canonical bijection with a product monomial crystal depending on the choice of parameters. This work also has interesting applications to classical representation theory. In particular, it allows us to give a classification of simple Gelfand–Tsetlin modules of U ( gl n ) and its associated W-algebras.