Paley Wiener Theorem on a Reductive Lie Group

Let G be a locally compact group, K a maximal compact subgroup of G and δ on arbitrary class of irreducible unitary representations of K. The spherical Grassmannian Gp,δ is an equivalence class of spherical functions of type δ−positive of height p. In this work, we give an extension of orbital integral with respect to δ, when G is reductive Lie group. Moreover, if the discret serie is not empty, we give an extension of Paley-Wiener theorem using a compact Cartan subgroup of G.