The Plancherel Formula for Minimal Parabolic Subgroups

Abstract

In a recent paper we found conditions for a nilpotent Lie group to be foliated into subgroups that have square integrable unitary representations that fit together to form a filtration by normal subgroups. That resulted in explicit character formulae, Plancherel formulae and multiplicity formulae. We also showed that nilradicals $N$ of minimal parabolic subgroups $P = MAN$ enjoy that "stepwise square integrable" property. Here we extend those results from $N$ to $P$. The Pfaffian polynomials, which give orthogonality relations and Plancherel density for $N$, also give a semiinvariant differential operator that compensates lack of unimodularity for $P$. The result is a completely explicit Plancherel formula for $P$.