q-KP hierarchy, bispectrality and Calogero-Moser systems

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 Sate 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 is an element of Gr(q)(ad) we construct a bispectral commutative algebra A(W)(q), Of q-difference operators, which extends to the case q not equal 1 all rank one solutions to the bispectral problem. The common eigenfunction Psi(x, z) for the operators from A(W)(q) 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. (C) 2000 Elsevier Science B.V. All rights reserved.