Spherical Functions on a Semi-Simple Lie Group

Abstract

Let G be a connected semi-simple Lie group with finite center, G = KA _P N⁺ an Iwasawa decomposition for G; let I²(G) be the space of K-biinvariant square integrable functions on G — then, according to well-known generalities, a given element f in I²(G) can be ‘expanded’ in terms of zonal spherical functions of positive type. More precisely, let I^∞ _c (G) denote the space of K-biinvariant compactly supported C^∞ functions on G; let P be the set of positive definite zonal spherical functions on G — then, by the spherical Fourier transform f̂ of f (f ∊ I^∞ _c (G)), we shall understand the function on P defined by the rule

$${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f}}(\phi )=\int_{G} {f(x)\phi{d_{G}}(x)\quad(\phi\in P)}$$

This being so, the abstract Plancherel Theorem for I²(G) asserts that there exists on P a unique positive measure μ (the Plancherel measure for I²(G)) such that

$$\int_{P} {{{\left| {f(x)} \right|}^{2}}{d_{G}}(x) = \int_{G} {{{\left| {\hat{f}(\phi )} \right|}^{2}}d} \mu (\phi )\quad (all\;f \in I_{c}^{\infty }(G))}$$

The problem is to compute μ explicitly, i.e. to relate μ in a satisfactory manner to the structure of G.