A quick proof of Harish-Chandra’s Plancherel theorem for spherical functions on a semisimple Lie group

Abstract

Some lemmas of S. Helgason and R. Gangolli, originally conceived for proving an analogue of the Paley-Wiener theorem for symmetric spaces, are used to give a quick proof of Harish-Chandra’s inversion formula and Plancherel theorem for bi-invariant functions on a semisimple Lie group. The method is elementary in that it does not require introduction of Harish-Chandra’s “Schwartz space."