Spherical characters on P-Adic symmetric spaces

Symmetric spaces over local fields, and the harmonic analysis of their class one representations, arise as the local calculation of Jacquet's theory of the relative trace formula. There is an extensive literature at the real place, but few general results for p-adic fields are known. The objective here is to carry over to symmetric spaces as much as possible of Queens and the prerequisite results of Howe, and to provide counterexamples for those things which do not generalize. We derive the Weyl integration formula, local constancy of the spherical character on the θ-regular set, Howe's conjecture for θ-groups, the germ expansion for spherical characters at the origin, and a spherical version of Howe's Kirillov theory for compact p-adic groups. We find that the density property of regular orbital integrals fails. Some of the basic ideas are nascent in Hakim's thesis (written under Hervé Jacquet).