Integrability and Huygens' principle on symmetric spaces

Abstract

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