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A parabolic subgroup of a reductive Lie group is called "good" if the center of the universal enveloping algebra of its nilradical contains an element that is semi-invariant of weight proportional to the modular function. The "good" case is characterized here by invariance of the set of simple roots defining the parabolic, under the negative of the opposition element of the Weyl group. In the "good" case, the unbounded Dixmier-Pukanszky operator of the parabolic subgroup is described, the conditions under which it is a differential operator rather than just a pseudodifferential operator are specified, and an explicit Plancherel formula is derived for that parabolic. 1. Introduction. This paper continues our work on the Plancherel formula for parabolic subgroups of reductive Lie groups. Earlier, we considered maximal parabolic subgroups ((7) and (11)), minimal parabolic subgroups ((4) and (6)), and parabolic subgroups whose unipotent radical has square-integrable representations modulo its center ((7) and (12)). In each case we described the representation theory, the Plancherel measure, and the Dixmier-Pukanszky operator that appears in the Plancherel formula to compensate for nonunimodularity. We now consider arbitrary parabolic subgroups. We isolate the common features of all previously treated cases which enabled us to describe the fundamental semi-invariant polynomial that gives rise to the Dixmier-Pukanszky operator. That is our first main result here. The second main result is a simple characterization of which parabolics have the requisite property, and the third is the construction in that case of the canonical semi-invariant polynomial. We explicitly compute the Dixmier-Pukanszky operator and the Plancherel formula for parabolics that have the requisite property. Here is a brief summary of the contents of this paper. In §2 we describe the key property (see Definition 2.1) for parabolic groups that guarantees the existence of a unique appropriate semi-invariant (Theorem 2.2) in the center of the universal enveloping algebra of the nilradical. We call such parabolics "good". Let P be a good parabolic and P = NAM its Langlands decomposition. We prove (Proposition 2.4) that the generic stability groups Si for

Canonical semi-invariants and the Plancherel formula for parabolic groups