Macdonald difference operators and Harish-Chandra series

We analyse the centralizer of the Macdonald difference operatorin an appropriate algebra of Weyl group invariant differenceoperators. We show that it coincides with Cherednik's commutingalgebra of difference operators via an analog of the Harish-Chandraisomorphism. Analogs of Harish-Chandra series are defined andrealized as solutions to the system of basic hypergeometricdifference equations associated to the centralizer algebra.These Harish-Chandra series are then related to both Macdonaldpolynomials and Chalykh's Baker–Akhiezer functions.