Affine highest weight categories and quantum affine Schur-Weyl duality of Dynkin quiver types

TL;DR

This paper establishes a deep connection between affine quasi-hereditary algebras from quiver varieties and categories of modules over quantum loop algebras, revealing a new duality in representation theory.

Contribution

It proves that a certain algebra from quiver varieties is affine quasi-hereditary and identifies its module category with a block of Hernandez-Leclerc's category, establishing a new quantum affine Schur-Weyl duality.

Findings

The convolution algebra is affine quasi-hereditary.

The module category matches Hernandez-Leclerc's category $$.

The duality functor provides an equivalence of categories.

Abstract

For a Dynkin quiver $Q$ (of type ADE), we consider a central completion of the convolution algebra of the equivariant K-group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc's monoidal category $\mathcal{C}_Q$ of modules over the quantum loop algebra $U_q(L\mathfrak{g})$ via Nakajima's homomorphism. As an application, we show that Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor gives an equivalence between the category of finite-dimensional modules over the quiver Hecke algebra associated with $Q$ and Hernandez-Leclerc's category $\mathcal{C}_Q$, assuming the simpleness of some poles of normalized R-matrices for type E.