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Journal of Lie Theory 24 (2014), No. 3, 791--808
Copyright Heldermann Verlag 2014

The Plancherel Formula for Minimal Parabolic Subgroups

Joseph A. Wolf
Department of Mathematics, University of California, Berkeley CA 94720--3840, U.S.A.

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 semi-invariant 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, Dixmier-Pukanszky operator, square integrable representation, stepwise square integrable representation.

MSC: 22E, 43A, 52C

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