Traces of Intertwiners for Quantum Affine $${\mathfrak{sl}}_2$$ and Felder–Varchenko Functions

Abstract

We show that the traces of $U_q(\widehat{\mathfrak{sl}}_2)$-intertwiners of Etingof-Schiffmann-Varchenko valued in the three-dimensional evaluation representation converge in a certain region of parameters and give 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 the Etingof-Varchenko conjecture. As applications, we prove a symmetry property for traces of intertwiners and prove Felder-Varchenko's conjecture that their elliptic Macdonald polynomials are related to the affine Macdonald polynomials defined as traces over irreducible integrable $U_q(\widehat{\mathfrak{sl}}_2)$-modules by Etingof-Kirillov Jr. In the trigonometric and classical limits, we recover results of Etingof-Kirillov Jr. and Etingof-Varchenko. Our method relies on an interplay between the method of coherent states applied to the free field realization of the $q$-Wakimoto module of Matsuo, convergence properties given by the theta hypergeometric integrals of Felder-Varchenko, and rationality properties originating from the representation-theoretic definition of the trace function.