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Invariant harmonic analysis on split rank one groups with applications. (English) Zbl 0572.22005

Let G be a real connected noncompact semisimple Lie group with finite center. We assume that if \(G_{{\mathbb{C}}}\) is the simply connected complex analytic Lie group with Lie algebra \({\mathfrak g}_{{\mathbb{C}}}\), then \(G\subset G_{{\mathbb{C}}}\). Fix a maximal compact subgroup K of G. Assume further that \(rk(G/K)=1\). This paper has two principal sections. In Section I we characterize the invariant transforms of functions in \({\mathcal C}^ p(G:F)\) (F\(\subset K\), \(| F| <\infty)\); Section II deals with the characterization of the orbital integrals of such functions.

MSC:

22E30 Analysis on real and complex Lie groups
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
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