Distributions de type K-positif sur l'espace tangent

Abstract

Let K be a compact subgroup of the isometry group of R n . A distribution T is said to be of K -positif type if it is K -invariant and if 〈T, ϑ ∗ \ ̃ gj〉 = ∝∝ ϑ(x + y) ϑ(Y) dT(x) ⩾ 0 for every K -invariant b ∞ function ϑ 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 G 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 G 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].