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Abstract

Abstract. In this paper we construct examples of commutative rings of difference operators with matrix coefficients from representation theory of quantum groups, generalizing the results of our previous paper [ES] to the q-deformed case. A generalized Baker-Akhiezer function is realized as a matrix character of a Verma module and is a common eigenfunction for a commutative ring of difference operators.In particular, we obtain the following result in

Algebro-geometric Schrödinger operators in many dimensions

“…For the root systems, this is related to the operators of MacdonaldRuijsennars type, and their algebraic integrability (in trigonometric case) was established by Etingof & Styrkas (1998) for A n -case and by Chalykh (2000) in general. This has nice applications to the Macdonald theory.…”

“…Finally, it is conceptually clear that there is a q-analogue of this story, related to the commutative rings of partial difference operators, but it is yet to be worked out in detail. For the root systems, this is related to the operators of Macdonald-Ruijsennars type, and their algebraic integrability (in trigonometric case) was established by Etingof & Styrkas (1998) for A n -case and by Chalykh (2000) in general. This has nice applications to the Macdonald theory.…”

Collective field description of spin Calogero - Sutherland models

“…The correspondence between the eigenstates of the spin CSM and the solutions of the Knizhnik-Zamolodchikov equation has been established [25,26]. More recently, Felder and Varchenko [27] (see also [28]) gave some formulae for the eigenstates of the spin CSM. The Dunkl operators [29] or more precisely the representation theory of the degenerate affine Hecke algebra [31] have central rôle in the analysis of the spectrum and integrability of the spin CSM (see also [30]).…”