Q-KP hierarchy, bispectrality and Calogero–Moser systems

Abstract

We show that there is a one-to-one correspondence between the q-tau functions of a q-deformation of the KP hierarchy and the planes in Sato Grassmannian Gr. Using this correspondence, we define a subspace Gr q ad of Gr, which is a q-deformation of Wilson’s adelic Grassmannian Gr ad. From each plane W∈ Gr q ad we construct a bispectral commutative algebra A q W of q-difference operators, which extends to the case q≠1 all rank one solutions to the bispectral problem. The common eigenfunction Ψ( x, z) for the operators from A q W is a q-wave (Baker–Akhiezer) function for a rational (in x) solution to the q-KP hierarchy. The poles of these solutions are governed by a certain q-deformation of the Calogero–Moser hierarchy.