Given a finite multiplicity
monomial representation τ of a completely solvable Lie group G, a smooth
decomposition of τ is a concrete direct integral decomposition into irreducibles
parametrized by a manifold Σ, with the property that compactly-supported
elements of ℋτ∞ are mapped to smooth sections on Σ by the intertwining
operator. A natural way of constructing such a decomposition is by means of the
distribution-theoretic Plancherel formula for τ and a cross- section Σ for
coadjoint orbits. However, for irreducible representations πl, l ∈ Σ, the
determination of appropriate distributions βl∈ℋl−∞ is problematic. For the case
where τ is induced from a “Levi” component, we overcome these problems
and give an explicit and natural construction for a smooth decomposition.
In the process we show that in this situation the nilradical of G must be
two-step.
