Algèbres commutatives d'opérateurs aux q-différences et systèmes de Calogero—Moser

We define a subspace Gr(q)(ad) of Sato's Grassmannian Gr, which is a q-deformation of Wilson's adelic Grassmannian Gr(ad). From each plane W is an element of Gr(q)(ad) lye constrict a bispectral commutative algebra A(W)(q) of q-difference operators. The common eigenfunction Psi (x, z) for the operators from A(W)(q) is a q-wave (Baker-Akhiezer) function for a rational solution to a q-deformation of the KP hierarchy. The poles of these solutions are governed by a certain q-deformation of the Calogero-Moser hierarchy. (C) 1999 Academie des sciences/Editions scientifiques et medicales Elsevier SAS