The orthogonality and qKZB-heat equation for traces of U_q(g)-intertwiners

Abstract

In our previous paper math.QA/9907181, 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], 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 the 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=sl_2, this statement is the trigonometric degeneration of a conjecture of Felder and the second author, proved by them for the 3-dimensional irreducible V. We also establish the orthogonality relation and qKZB-heat equation for trace functions obtained by taking traces in finite dimensional representations (rather than Verma modules). If g=sl_n and V=S^{kn}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, proved by Cherednik.