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International Mathematics Research Notices (2000) 2000:281-323, doi:10.1155/S1073792800000179
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Copyright © 2000 Hindawi Publishing Corporation. All rights reserved.

Commutative rings of difference operators and an adelic flag manifold

Luc Haine and Plamen Iliev

We show that maximal rank 1 commutative rings of difference operators can be systematically constructed from their differential analogues by making an appropriate shift in the variables of the tau function of the KP hierarchy. When the spectrum of the ring is a unicursal curve, the operators in the ring enjoy a bispectral property reminiscent of the familiar bispectral property satisfied by the classical orthogonal polynomials. As an illustration, the tau functions leading to the rings with spectral curves

Formula
are explicitly characterized.


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