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

1977/03/01 by Jonathan Rosenberg · 33 citations
Mathematics · #Advanced Algebra and Geometry #Mathematical Analysis and Transform Methods #Algebraic and Geometric Analysis #Mathematics #Pure mathematics #Real form #Lie group #Cartan decomposition #Invariant (physics) #Symmetric space #Algebra over a field #Mathematical physics #Affine Lie algebra #Adjoint representation of a Lie algebra #Lie conformal algebra

paper · pdf · doi:10.1090/s0002-9939-1977-0507231-8

published in Proceedings of the American Mathematical Society 63(1), 143-149 (American Mathematical Society)

openalex publication_date 1977/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/21

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."

Citations

• A duality for symmetric spaces with applications to group representations
• Spherical Functions on a Semisimple Lie Group, I

Cited by

• Plancherel theorem for spherical functions [wikipedia]