
Journal of Lie Theory 24 (2014), No. 3, 791808 Copyright Heldermann Verlag 2014 The Plancherel Formula for Minimal Parabolic Subgroups Joseph A. Wolf Department of Mathematics, University of California, Berkeley CA 947203840, U.S.A. jawolf@math.berkeley.edu 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 $P$. Keywords: Lie group, Plancherel formula, Fourier inversion, parabolic subgroup, DixmierPukanszky operator, square integrable representation, stepwise square integrable representation. MSC: 22E, 43A, 52C [ Fulltextpdf (357 KB)] for subscribers only. 