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DOUBLE AFFINE HECKE ALGEBRAS AND BISPECTRAL QUANTUM KNIZHNIK-ZAMOLODCHIKOV EQUATIONS (2008)
| Citations: | 13 - 5 self |
BibTeX
@MISC{Meer08doubleaffine,
author = {Michel van Meer and Jasper V. Stokman},
title = {DOUBLE AFFINE HECKE ALGEBRAS AND BISPECTRAL QUANTUM KNIZHNIK-ZAMOLODCHIKOV EQUATIONS},
year = {2008}
}
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Abstract
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 Φ 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 the Ruijsenaars’ commuting trigonometric q-difference operators. Under this correspondence Φ becomes a self-dual Harish-Chandra series solution Φ + of the bispectral problem. Specializing the central character as above, we recover from Φ + the symmetric self-dual Macdonald polynomials.
Keyphrases
central character q-difference equation compatible system symmetric self-dual macdonald polynomial underlying affine hecke algebra bispectral problem explicit correspondence suitable specialization series representation self-dual harish-chandra series solution meromorphic self-dual solution trigonometric q-difference operator particular bispectral problem type gln double affine hecke algebra bispectral quantum knizhnik-zamolodchikov self-dual laurent polynomial solution quantum kz equation quantum affine kz equation explicit consistent system principal series representation
