Positive definite distributions on K\G/K

Abstract

Let G be a semisimple noncompact Lie group with finite center and let K be a maximal compact subgroup. Then W. H. Barker has shown that if T is a positive definite distribution on G, then T extends to Harish-Chandra's Schwartz space C¹(G). We show that the corresponding property is no longer true for the space of double cosets K\G/K. If G is of real-rank 1, we construct liner functionals T_pε(C_c^∞(K\G/K))' for each p, 0 < p ≤ 2, such that T_p(f^∗f^∗)≥0, ∀fε C_c^∞(K\G/K) but T_p does not extend to a continuous functional on C^p(K\G/K). In particular, if p≤1, T_v does not extend to a continuous functional on C¹(K\G/K). We use this to answer a question (in the negative) raised by Barker whether for a K-bi-invariant distribution T on G to be positive definite it is enough to verify that T(f^∗f^∗)≥0, ∀fε C_c^∞(K\G/K). The main tool used is a theorem of Trombi-Varadarajan.