Base de la série la plus continue de fonctions généralisées sphériques sur G_C/G_R

Let G be a complex, connected and simply connected semisimple Lie group with Lie algebra g. Let H be a real form of G with Lie algebra h. We suppose that h admits a Cartan subalgebra t of compact type. For a regular element λ of the dual of h, we construct a basis of the space of spherical generalized functions F such that, for all G-invariant differential operator D on G/H, we have D.F=χ_λ(D) F, where χ_λ is the character defined by λ. Then, we study the meromorphic continuation and the growth of the element of this basis. We give these functions in terms of coefficients of representations of G. These results are generalized to other series of spherical generalized functions.