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Proceedings of the London Mathematical Society Advance Access originally published online on February 20, 2008
Proceedings of the London Mathematical Society 2008 97(1):60-96; doi:10.1112/plms/pdm055
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© 2008 London Mathematical Society

Macdonald difference operators and Harish-Chandra series

Gail Letzter

Mathematics Department
Virginia Tech
Blacksburg
VA 24061-0123
USA

Jasper V. Stokman

KdV Institute for Mathematics
Universiteit van Amsterdam
Plantage Muidergracht 24
1018 TV Amsterdam
The Netherlands
jstokman@science.uva.nl

Received 21 February 2007. Revision received 12 September 2007.

We analyse the centralizer of the Macdonald difference operator in an appropriate algebra of Weyl group invariant difference operators. We show that it coincides with Cherednik's commuting algebra of difference operators via an analog of the Harish-Chandra isomorphism. Analogs of Harish-Chandra series are defined and realized as solutions to the system of basic hypergeometric difference equations associated to the centralizer algebra. These Harish-Chandra series are then related to both Macdonald polynomials and Chalykh's Baker–Akhiezer functions.

2000 Mathematics Subject Classification 33D67 (primary) 33D80 (secondary).


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