Spherical functions on a real semisimple Lie group. A method of reduction to the complex case

The spherical functions on a real semisimple Lie group (w.r.t. a maximal compact subgroup) are characterized as joint eigenfunctions of certain differential operators on the corresponding complex group. Using this, several results concerning the spherical Fourier transform on the real group are reduced to the corresponding results for the complex group.When the group in question is a normal real form, this leads to new and simpler proofs of such results as the Plancherel formula, the Paley-Wiener theorem and the characterization of the image under the spherical Fourier transform of the L¹- and L²-Schwartz spaces. In these proofs neither any knowledge of Harish-Chandras c-function nor the series expansion for the spherical function are used.For the proof of the main result some analysis of independent interest on pseudo-Riemannian symmetric spaces is developed. Such as a generalized Cartan decomposition and a method of analytic continuation between two “dual” pseudo-Riemannian symmetric spaces.