Double Affine Hecke Algebras and Bispectral Quantum Knizhnik-Zamolodchikov Equations

作者：van Meer Michel^*; Stokman Jasper

来源：International Mathematics Research Notices, 2010, (6): 969-1040.

DOI：10.1093/imrn/rnp165

We use 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.

• 出版日期2010