Positive definite distributions on K\G/K

Sitaram, Alladi (1978) Positive definite distributions on K\G/K Journal of Functional Analysis, 27 (2). pp. 179-184. ISSN 0022-1236

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Official URL: http://www.sciencedirect.com/science/article/pii/0...

Related URL: http://dx.doi.org/10.1016/0022-1236(78)90025-3


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 Tpε(Cc(K\G/K))' for each p, 0 < p ≤ 2, such that Tp(ff)≥0, ∀fε Cc(K\G/K) but Tp does not extend to a continuous functional on Cp(K\G/K). In particular, if p≤1, Tv 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(ff)≥0, ∀fε Cc(K\G/K). The main tool used is a theorem of Trombi-Varadarajan.

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