Orthogonality and the qKZB-heat equation for traces of U q g -intertwiners

Abstract

In our previous paper [EV2], to every finite-dimensional representation V of the quantum group U q g we attached the trace function F V λ μ with values in End⁡V 0 which was obtained by taking the (weighted) trace in a Verma module of an intertwining operator. We showed that these trace functions satisfy the Macdonald-Ruijsenaars and quantum Knizhnik-Zamolodchikov-Bernard (qKZB) equations, their dual versions, and the symmetry identity. In this paper, we show that the trace functions satisfy the orthogonality relation and the qKZB-heat equation. For g = s l 2 , this statement is the trigonometric degeneration of a conjecture from [FV3], proved in [FV3] for the 3-dimensional irreducible V . We also establish the orthogonality relation and the qKZB-heat equation for trace functions that were obtained by taking traces in finite-dimensional representations (rather than in Verma modules). If g = n and V = S k n ℂ n , these functions are known to be Macdonald polynomials of type A . In this case, the orthogonality relation reduces to the Macdonald inner product identities, and the qKZB-heat equation coincides with the q-Macdonald-Mehta identity that was proved by Cherednik [Ch2].