Quantum dynamical Weyl groups from quantum loop groups of arbitrary shuffle type

Abstract

We construct quantum dynamical Weyl group elements associated with quantum loop groups of arbitrary shuffle type. Using the construction, we define the quantum lattice operator and the algebraic quantum difference equations for each tensor products of semisimple modules $V$ in category $\mathcal{O}$. We prove that algebraic quantum difference operators form a family of commuting operators, and they also commute with the qKZ operators for the tensor products of modules of the above type in $\mathcal{O}$. This recovers the construction in \cite{OS22} and can be viewed as the difference analog of the trigonmetric Casimir connection when the quantum loop group corresponds to the finite type symmetrisable Cartan matrix.