Quantum intertwiners and integrable systems

Abstract

We present a collection of results on the relationship between intertwining operators for quantum groups and eigenfunctions for quantum integrable systems. First, we study the Etingof-Kirillov Jr. expression of Macdonald polynomials as traces of intertwiners of quantum groups in the Gelfand-Tsetlin basis. In the quasiclassical limit, we obtain a new Harish-Chandra type integral formula for Heckman- Opdam hypergeometric functions. This formula is related to an integral formula appearing in recent work of Borodin-Gorin by integration over Liouville tori of the Gelfand-Tsetlin integrable system. At the quantum level, we obtain a new proof of the branching rule for Macdonald polynomials which transparently relates branching of Macdonald polynomials to branching of quantum group representations. Second, we study traces of intertwiners for quantum affine algebras. In the sl2 case, we show that, when valued in the three-dimensional evaluation representation, such traces converge in a certain region of parameters and provide a representation-theoretic construction of Felder-Varchenko's hypergeometric solutions to the q-KZB heat equation. This gives the first proof that such a trace function converges and resolves the first case of a conjecture of Etingof-Varchenko. As an application, we prove Felder-Varchenko's conjecture that their elliptic Macdonald polynomials are related to Etingof-Kirillov Jr.'s affine Macdonald polynomials. In the general case, we modify the setting of the work of Etingof-Schiffmann-Varchenko to show that traces of such intertwiners satisfy four commuting systems of q-difference equations - the Macdonald-Ruijsenaars, dual Macdonald-Ruijsenaars, q-KZB, and dual q-KZB equations.