Traces of intertwiners for quantum affine sl_2 and Felder-Varchenko functions
arXiv:1508.03918 · doi:10.1007/s00220-016-2580-4
Abstract
We show that the traces of -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 -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 -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 -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.
57 pages. v2: corrected rationality of intertwiners in Section 2 and its use in Section 5; corrected proof of Proposition 4.7; clarified use of continuation in Section 9; main results are unchanged