On the Plancherel Formula and the Paley-Wiener Theorem for Spherical Functions on Semisimple Lie Groups

Abstract

One of the difficult points in the proof of Harish-Chandra's Plancherel formula for spherical functions on a semisimple Lie group is to show that an appropriate inversion formula exists for the Fourier transform f f, and that this inversion formula holds for sufficiently many functions in the space of spherical square-integrable functions on G. Briefly, let f be a square integrable spherical function on G for which the Fourier transform f(X)= 5f(x)cp_(x)dx is well-defined. Here, as in [5], qA is the elementary (zonal) G spherical function corresponding to the parameter X. The problem is to show that for f in a L2-dense subspace of square-integrable spherical functions, the following inversion formula holds: