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Orthogonality and the qKZB-heat equation for traces of \(U_q({\mathfrak g})\)-intertwiners. (English) Zbl 1160.17305

Summary: In our previous paper [Duke Math. J. 104, No. 3, 391–432 (2000; Zbl 1004.17006)], to every finite-dimensional representation \(V\) of the quantum group \(U_q(\mathfrak g)\) we attached the trace function \(F^V(\lambda,\mu)\) with values in \(\text{End}\,V[0]\) 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 \(\mathfrak g=\mathfrak{sl}_2\), this statement is the trigonometric degeneration of a conjecture from [G. Felder and A. N. Varchenko, Adv. Math. 171, No. 2, 228–275 (2002; Zbl 1057.32007)], proved in that paper 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 \(\mathfrak g=\mathfrak{sl}_2\) and \(V=S^{kn}\mathbb C^n\), these functions are known to be Macdonald polynomials of type . 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 I. Cherednik [Int. Math. Res. Not. 1997, No. 10, 449–467 (1997; Zbl 0981.33012)].

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)
33C67 Hypergeometric functions associated with root systems
33D67 Basic hypergeometric functions associated with root systems
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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References:

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