Journal of Lie Theory

no. 3
The Plancherel Formula for Minimal Parabolic Subgroups
Journal of Lie Theory, Volume 24 (2014) no. 3, pp. 791-808
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 $P$.
DOI: 10.5802/jolt.807
Classification: 22E, 43A, 52C
Keywords: Lie group, Plancherel formula, Fourier inversion, parabolic subgroup, Dixmier-Pukanszky operator, square integrable representation, stepwise square integrable representation 
@article{JOLT_2014_24_3_a9,
     author = {J. A. Wolf},
     title = {The {Plancherel} {Formula} for {Minimal} {Parabolic} {Subgroups}},
     journal = {Journal of Lie Theory},
     pages = {791--808},
     year = {2014},
     volume = {24},
     number = {3},
     doi = {10.5802/jolt.807},
     zbl = {1303.22005},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.807/}
}
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J. A. Wolf. The Plancherel Formula for Minimal Parabolic Subgroups. Journal of Lie Theory, Volume 24 (2014) no. 3, pp. 791-808. doi: 10.5802/jolt.807

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