On category $\mathcal{O}$ for affine Grassmannian slices and categorified tensor products

TL;DR

This paper establishes a connection between category O over truncated shifted Yangians and categorified tensor products, providing a new algebraic framework and applications to classical representation theory.

Contribution

It introduces the parity KLRW algebra to describe category O for arbitrary parameters and confirms the bijection between highest weights and product monomial crystals.

Findings

Category O is equivalent to a weight space in categorified tensor products.

Introduction of the parity KLRW algebra for arbitrary parameters.

Classification of simple Gelfand-Tsetlin modules for gl(n) and W-algebras.

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 $\lambda$ and $\mu$ for a Lie algebra $\mathfrak{g}$, which we will assume is simply-laced. In this paper, we relate the category $\mathcal{O}$ over truncated shifted Yangians to categorified tensor products: for a generic integral choice of parameters, category $\mathcal{O}$ 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 $\mathcal{O}$ 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 $\mathcal{O}$ 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 $U(\mathfrak{gl}_n)$ and its associated W-algebras.