An abelian formula for the quantum Weyl group action of the coroot lattice

TL;DR

This paper provides an explicit abelian formula for the quantum Weyl group action on finite-dimensional representations of quantum loop algebras, connecting it with Chari-Pressley series and Drinfeld polynomials, and applies it to Nakajima quiver varieties.

Contribution

It introduces a new explicit formula for the quantum Weyl group action of the coroot lattice using commuting generators and establishes a rationality result for the Chari-Pressley series.

Findings

Explicit formula for quantum Weyl group action in terms of commuting generators

Rationality result for Chari-Pressley series derived in the paper

Identification of the lattice action with determinant line bundles on Nakajima quiver varieties

Abstract

Let g be a complex simple Lie algebra and Uq(Lg) its quantum loop algebra, where q is not a root of unity. We give an explicit formula for the quantum Weyl group action of the coroot lattice Q of g on finite-dimensional representations of Uq(Lg) in terms of its commuting generators. The answer is expressed in terms of the Chari-Pressley series, whose evaluation on highest weight vectors gives rise to Drinfeld polynomials. It hinges on a strong rationality result for that series, which is derived in the present paper. As an application, we identify the action of Q on the equivariant K-theory of Nakajima quiver varieties with that of explicitly given determinant line bundles.