On spherical distributions in abstract harmonic analysis

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I need a detailed and comprehensive guidance or possibly relevant materials that can help me in my research on spherical distributions on a locally compact group. Jacques Faraut wrote on spherical distributions on hyperbolic spaces in a famous paper in 1978, " Distributions sphériques sur les espaces hyperboliques..." Helminck A.G. and Helminck G.F. did the same in 2002 and 2005. However I am still confused as per the true and explicit picture of a spherical distribution in abstract harmonic analysis.