15 May 2005 Orthogonality and the qKZB-heat equation for traces of U q g -intertwiners
P. Etingof, A. Varchenko
Duke Math. J. 128(1): 83-117 (15 May 2005). DOI: 10.1215/S0012-7094-04-12814-3


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 EndV 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 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].


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P. Etingof. A. Varchenko. "Orthogonality and the qKZB-heat equation for traces of U q g -intertwiners." Duke Math. J. 128 (1) 83 - 117, 15 May 2005. https://doi.org/10.1215/S0012-7094-04-12814-3


Published: 15 May 2005
First available in Project Euclid: 17 May 2005

zbMATH: 1160.17305
MathSciNet: MR2137950
Digital Object Identifier: 10.1215/S0012-7094-04-12814-3

Primary: 17B37
Secondary: 33D52

Rights: Copyright © 2005 Duke University Press


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Vol.128 • No. 1 • 15 May 2005
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