Traces of Intertwiners for Quantum Affine sl_2 and FelderVarchenko Functions
Abstract
We show that the traces of {U_q({widehat{sl}_2)}intertwiners of [ESV02] valued in the threedimensional evaluation representation converge in a certain region of parameters and give a representationtheoretic construction of FelderVarchenko's hypergeometric solutions to the qKZB heat equation given in [FV02]. This gives the first proof that such a trace function converges and resolves the first case of the EtingofVarchenko conjecture of [EV00]. As applications, we prove a symmetry property for traces of intertwiners and prove FelderVarchenko'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{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 qWakimoto module of [Mat94], convergence properties given by the theta hypergeometric integrals of [FV02], and rationality properties originating from the representationtheoretic definition of the trace function.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 October 2016
 DOI:
 10.1007/s0022001625804
 arXiv:
 arXiv:1508.03918
 Bibcode:
 2016CMaPh.347..573S
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematical Physics;
 Mathematics  Representation Theory;
 17B37 (primary);
 17B67;
 33C75;
 33D80;
 81R10 (secondary)
 EPrint:
 57 pages. v2: corrected rationality of intertwiners in Section 2 and its use in Section 5