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International Mathematics Research Notices Advance Access first published online on October 23, 2009
This version published online on October 29, 2009
International Mathematics Research Notices, doi:10.1093/imrn/rnp165
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© The Author 2009. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

Double Affine Hecke Algebras and Bispectral Quantum Knizhnik–Zamolodchikov Equations

Michel van Meer and Jasper Stokman

Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands

Correspondence: Correspondence to be sent to: m.vanmeer{at}uva.nl

We use the double affine Hecke algebra of type GLN to construct an explicit consistent system of q-difference equations, which we call the bispectral quantum Knizhnik–Zamolodchikov (BqKZ) equations. BqKZ includes, besides Cherednik’s quantum affine KZ equations associated to principal series representations of the underlying affine Hecke algebra, a compatible system of q-difference equations acting on the central character of the principal series representations. We construct a meromorphic self-dual solution {Phi} of BqKZ which, upon suitable specializations of the central character, reduces to symmetric self-dual Laurent polynomial solutions of quantum KZ equations. We give an explicit correspondence between solutions of BqKZ and solutions of a particular bispectral problem for Ruijsenaars’ commuting trigonometric q-difference operators. Under this correspondence, {Phi} becomes a self-dual Harish-Chandra series solution {Phi}+of the bispectral problem. Specializing the central character as above, we recover from {Phi}+the symmetric self-dual Macdonald polynomials.

Communicated by Prof. Alexei Borodin


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