The Penney-Fujiwara Plancherel formula for abelian symmetric spaces and completely solvable homogeneous spaces

Abstract

A distribution theoretic version of the Plancherel formula for the decomposition of the quasi-regular representation of a Lie group G on L 2 (G/H) is presented. The formula is proven in two situations wherein the irreducible representations that occur in the decomposition are monomial. The intertwining operator that effects the decomposition is derived in terms of integral operators that arise from the distributions.