On harmonic analysis of spherical convolutions on semisimple Lie groups

This paper contains a non-trivial generalization of the Harish-Chandra transforms on a connected semisimple Lie group $G,$ with finite center, into what we term spherical convolutions. Among other results we show that its integral over the collection of bounded spherical functions at the identity element $e \in G$ is a weighted Fourier transforms of the Abel transform at $0.$ Being a function on $G,$ the restriction of this integral of its spherical Fourier transforms to the positive-definite spherical functions is then shown to be (the non-zero constant multiple of) a positive-definite distribution on $G,$ which is tempered and invariant on $G=SL(2,\mathbb{R}).$ These results suggest the consideration of a calculus on the Schwartz algebras of spherical functions. The Plancherel measure of the spherical convolutions is also explicitly computed.