On Harish-Chandra's Plancherel theorem for Riemannian symmetric spaces

In this article we give an overview of the Plancherel theory for Riemannian symmetric spaces Z = G/K. In particular we illustrate recently developed methods in Plancherel theory for real spherical spaces by explicating them for Riemannian symmetric spaces, and we explain how Harish-Chandra’s Plancherel theorem for Z can be proven from these methods.

💡 Research Summary

This paper provides a modern reinterpretation of Harish‑Chandra’s Plancherel theorem for Riemannian symmetric spaces (Z=G/K) by employing techniques that have recently been developed for real spherical spaces. The authors begin with a concise historical overview, noting that Harish‑Chandra’s original proof (completed in the 1960s) relied on deep analysis of the Harish‑Chandra (c)-function and on intricate asymptotic estimates. Subsequent work by Gindikin‑Karpelovich, Harish‑Chandra himself, and Rosenberg‑Gaave simplified parts of the argument, but the overall structure remained rather specialized to symmetric spaces.

The first technical section establishes the notation and normalizations that will be used throughout. After fixing a real reductive group (G) and a maximal compact subgroup (K), the authors recall the Iwasawa decomposition (G=KAN) and introduce a system of Haar measures normalized so that \