Distributions de type K-positif sur l'espace tangent

Let K be a compact subgroup of the isometry group of $\text{R}$ⁿ. A distribution T is said to be of K-positif type if it is K-invariant and if $\text{〈T, ? 1}\text{}\text{?}\text{gj〉 = ∝∝ ?(x + y)}\text{?(Y) dT(x) ? 0}$ for every K-invariant $\text{b}$^∞ function ? with compact support. We look for an integral representation of these distributions (i.e., an analog of the Bochner-Schwartz theorem). In this paper we obtain such a representation for distributions with growth of exponential type in the following case: K is the maximal compact subgroup of a semi-simple connected Lie group G with finite center, acting by the adjoint action on the tangent space of $\text{G}\text{K}$. The main step is to prove that it suffices to work with distributions of W-positif type (where W is the Weyl group associated with $\text{G}\text{K}$). This is achieved following ideas of a paper of S. Helgason Advan. in Math.36 (1980) 297]. The end of the proof follows from the case where K is finite N. Bopp, in “Analyse harmonique sur les groupes de Lie,” Lecture Notes in Mathematics No. 739, p. 15, Springer-Verlag, Berlin/New York 1979].