An abelian formula for the quantum Weyl group action of the coroot lattice
S. Gautam, V. Toledano-Laredo
TL;DR
The paper derives an explicit abelian formula for the quantum Weyl group action of the coroot lattice $Q^{\vee}$ on finite-dimensional representations of the quantum loop algebra $U_q(L\mathfrak{g})$, expressing lattice actions through the commuting lattice generators via the Chari–Pressley series. Central to the construction is a rationality result for the series $\mathcal{P}_i^{\pm}(z)$, whose $z$-expansions recover Drinfeld polynomials on highest-weight spaces; the main lattice action is captured by the limit of these series, $\lambda_{V,o}(L_i)=(o(i)q_i)^{\mathcal{H}_i}\cdot \mathrm{C}_i^{-1}$. The authors develop a rank-one reduction and straightening identities to prove these results, and they extend the framework to describe the action on the equivariant $K$-theory of Nakajima quiver varieties as tensoring with determinant line bundles. They further compare with CKL’s categorical action, showing compatibility after suitable adjustments, and discuss potential categorifications of the braid action. Together, the results provide explicit, computable formulas for lattice actions in both algebraic and geometric settings with broad implications for representation theory and quiver variety geometry.
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.