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

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.

Added sections on the Drinfeld coproduct of lattice coroot operators (1.9) and the extended affine braid group (1.13). Added two references in 1.7. Minor edits throughout. 31 pages