The Penney-Fujiwara Plancherel Formula for Symmetric Spaces

Abstract

I am interested in the direct integral decomposition of the quasi-regular representation τ = Ind G H 1 for symmetric spaces G/H. Suppose that τ is type I and one has an equivalence $$\tau \; \simeq \;\smallint _S^ \oplus \;{n_\tau }(\pi )\pi \;d{\mu _\tau }(\pi ),$$ where µτ is a Borel measure on Ĝ, n τ(π) is a multiplicity function and S ⊂ Ĝ is a minimal closed μτ-co-null set. If we can compute explicitly the triple (S, n τ, μτ), we say that we know the direct integral decomposition of τ. Such computations have been carried out for exponential solvable symmetric spaces [1]. (In that case, the multiplicity function is identically 1.) Actually Benoist’s formula is a special case of direct integral decomposition formulas for general induced representations of nilpotent homogeneous spaces [3] [7] and completely solvable homogeneous spaces [9].