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 $U_q\left(\mathfrak{g}\right)$ we attached the trace function $F^V\left(\lambda ,\mu \right)$ with values in $\mathrm{End}V\left[0\right]$ 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 $\mathfrak{g}=s{l}_2$, 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 $\mathfrak{g}={}_n$ 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 that was proved by Cherednik [Ch2].