The distribution-theoretic
version of the Plancherel formula—known as the Penney-Fujiwara Plancherel
Formula—for the decomposition of the quasi-regular representation of a
Lie group G on L2(G∕H) is considered. Attention is focused on the case
that the spectrum consists of irreducible representations induced from a
finite-dimensional representation. This happens with great regularity for Strichartz
homogeneous spaces wherein G and H are semidirect products of normal
abelian subgroups by a reductive Lie group. The results take an especially
simple form if G∕H is symmetric. Criteria for finite multiplicity and for
multiplicity-free spectrum are developed. In the case that G is a motion
group—the original situation stressed by Strichartz—the results are particularly
striking.
