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

Abstract Let $\mathfrak{g}$ be a complex simple Lie algebra and $U_q(L\mathfrak{g})$ its quantum loop algebra, where $q\in \mathbb{C}^{\times }$ is not a root of unity. We give an explicit formula for the quantum Weyl group action of the coroot lattice $Q^\vee $ of $\mathfrak{g}$ on finite-dimensional representations of $U_q(L\mathfrak{g})$ 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^\vee $ on the equivariant $K$-theory of Nakajima quiver varieties with that of explicitly given determinant line bundles.