INTEGRABILITY AND HUYGENS PRINCIPLE ON SYMMETRICAL SPACES

The explicit formulas for fundamental solutions of the modified wave e quations on certain symmetric spaces are found. These symmetric spaces have the following characteristic property: all multiplicities of the ir restricted roots are even. As a corollary in the odd-dimensional ca se one has that the Huygens' principle in Hadamard's sense for these e quations is fulfilled. We consider also the heat and Laplace equations on such a symmetric space and give explicitly the corresponding funda mental solutions-heat kernel and Green's function. This continues our previous investigations [16] of the spherical functions on the same sy mmetric spaces based on the fact that the radial part of the Laplace-B eltrami operator on such a space is related to the algebraically integ rable case of the generalised Calogero-Sutherland-Moser quantum system . In the last section of this paper we apply the methods of Heckman an d Opdam to extend our results to some other symmetric spaces, in parti cular to complex and quaternian grassmannians.