On some extension of Paley Wiener theorem

Paley Wiener theorem characterizes the class of functions which are Fourier transforms of C-infinity functions of compact support on R-n by relating decay properties of those functions or distributions at infinity with analyticity of their Fourier transform. The theorem is already proved in classical case : the real case with holomorphic Fourier transform on L-2(R), the case of functions with compact support on R-n from Hormander and the spherical transform on semi simple Lie groups with Gangolli theorem. Let G be a locally compact unimodular group, K a compact subgroup of G, and delta an element of unitary dual (K) over cap of K. In this work, we'll give an extension of Paley-Wiener theorem with respect to delta, a class of unitary irreducible representation of K, where G is either a semi-simple Lie group or a reductive Lie group with non empty discrete series after introducing a notion of delta-orbital integral. If delta is trivial and one dimensional, we obtain the classical Paley-Wiener theorem.