2003/02/06 by Pavel Etingof, Etingof, Pavel, Alexander Varchenko +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Nonlinear Waves and Solitons #math.QA
paper · pdf · doi:10.48550/arxiv.math/0302071
29 pages, latex
arxiv created 2003/02/06 · arxiv updated 2009/11/30
In our previous paper math.QA/9907181, to every finite dimensional representation V of the quantum group Uq(g), we attached the trace function FV(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=sl2, 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=sln and V=SknCn, 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.