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

Rosenberg, Jonathan

doi:10.1090/S0002-9939-1977-0507231-8

A quick proof of Harish-Chandra’s Plancherel theorem for spherical functions on a semisimple Lie group
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Proc. Amer. Math. Soc. 63 (1977), 143-149 Request permission

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."

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Harmonic analysis on semisimple Lie groups