Introduction

This text grew out of an attempt to understand a remark by Harish-Chandra in the introduction of [12]. In that paper and its sequel he determined the Plancherel decomposition for Riemannian symmetric spaces of the non-compact type. The associated Plancherel measure turned out to be related to the asymptotic behavior of the so-called zonal spherical functions, which are solutions to a system of invariant differential eigenequations. Harish-Chandra observed: ‘this is reminiscent of a result of Weyl on ordinary differential equations’, with reference to Hermann Weyl's 1910 paper, [29], on singular Sturm–Liouville operators and the associated expansions in eigenfunctions.

For Riemannian symmetric spaces of rank one the mentioned system of equations reduces to a single equation of the singular Sturm–Liouville type. Weyl's result indeed relates asymptotic behavior of eigenfunctions to the continuous spectral measure but his result is formulated in a setting that does not directly apply.

In [23], Kodaira combined Weyl's theory with the abstract Hilbert space theory that had been developed in the 1930's. This resulted in an efficient derivation of a formula for the spectral measure, previously obtained by Titchmarsh. In the same paper Kodaira discussed a class of examples that turns out to be general enough to cover all Riemannian symmetric spaces of rank 1.

It is the purpose of this text to explain the above, and to describe later developments in harmonic analysis on groups and symmetric spaces where Weyl's principle has played an important role.