Plane waves, integrable quantum systems, Green functions and the groups SO( p, q), p q

Plane waves on symmetric spaces (SS) of rank p, , are constructed by realization of the irreducible representations (principal series) of the group SO(p,q) in the space of infinitely differentiable homogeneous vector functions on cones , , with values in the representation space of the stability subgroups SO(p-i,q-i), i = 1,...,p. We define the cones , , corresponding to the SS X related with Cartan involutive automorphism , , where is the metric tensor of the pseudo-Euclidean space . Calculating Harish - Chandra c-functions the orthogonality, completeness conditions and addition theorems for plane waves are derived. The integrable n-body quantum systems related to groups SO(p,q) are considered. The explicit expressions for the Green functions in the case SS X of rank p = 1 and the integral representation in the general case are given.