Geometric realization of Dynkin quiver type quantum affine Schur-Weyl duality

TL;DR

This paper constructs a geometric bimodule linking quantum loop algebras and quiver Hecke algebras for ADE type quivers, confirming a conjecture on R-matrix poles and advancing the understanding of quantum affine Schur-Weyl duality.

Contribution

It introduces a new geometric construction of a bimodule that realizes quantum affine Schur-Weyl duality for ADE quivers, connecting various algebraic and geometric frameworks.

Findings

Constructed a bimodule via equivariant K-theory for ADE quivers.

Identified the functor with a known duality functor by Kang-Kashiwara-Kim.

Verified a conjecture on the simplicity of poles of normalized R-matrices.

Abstract

For a Dynkin quiver $Q$ of type ADE and a sum $\beta$ of simple roots, we construct a bimodule over the quantum loop algebra and the quiver Hecke algebra of the corresponding type via equivariant K-theory, imitating Ginzburg-Reshetikhin-Vasserot's geometric realization of the quantum affine Schur-Weyl duality. Our construction is based on Hernandez-Leclerc's isomorphism between a certain graded quiver variety and the space of representations of the quiver $Q$ of dimension vector $\beta$. We identify the functor induced from our bimodule with Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor. As a by-product, we verify a conjecture by Kang-Kashiwara-Kim on the simpleness of some poles of normalized R-matrices for any quiver $Q$ of type ADE.