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

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Abstract

We show that the traces of \({U_q({\widehat{\mathfrak{sl}}}_2)}\)-intertwiners of [ESV02] 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 given in [FV02]. This gives the first proof that such a trace function converges and resolves the first case of the Etingof–Varchenko conjecture of [EV00]. As applications, we prove a symmetry property for traces of intertwiners and prove Felder–Varchenko’s conjecture in [FV04] 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 in [EK95]. In the trigonometric and classical limits, we recover results of [EK94,EV00]. Our method relies on an interplay between the method of coherent states applied to the free field realization of the q-Wakimoto module of [Mat94], convergence properties given by the theta hypergeometric integrals of [FV02], and rationality properties originating from the representation-theoretic definition of the trace function.

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  1. MIT, Cambridge, MA, USA

    Yi Sun

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  1. Yi Sun
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Communicated by A. Borodin

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Sun, Y. Traces of Intertwiners for Quantum Affine \({\mathfrak{sl}}_2\) and Felder–Varchenko Functions. Commun. Math. Phys. 347, 573–653 (2016). https://doi.org/10.1007/s00220-016-2580-4

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