A new class of Gelfand pairs

The well-known result for rank 1 real symmetric spacesX=G/H, stating the multiplicity-free decomposition of the natural action ofG onL²(X), is extended to the classical rank 1p-adic symmetric spaces. The absence of ap-adic analogue of the Laplace-Beltrami operator causes additional complications and necessitates a global study of the invariant distributions on the quotient space.