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Journal of Lie Theory 27 (2017), No. 3, 845--886
Copyright Heldermann Verlag 2017

Stepwise Square Integrability for Nilradicals of Parabolic Subgroups and Maximal Amenable Subgroups

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

In a series of recent papers:
(1) Plancherel Formulae associated to Filtrations of Nilpotent Lie Groups, arXiv 1212.1908,
(2) Stepwise Square Integrable Representations of Nilpotent Lie Groups, Mathematische Annalen 357 (2013) 895-914,
(3) Principal series representations of infinite dimensional Lie groups, I: Minimal parabolic subgroups, Progress in Mathematics 257 (2014) 519--538,
(4) The Plancherel Formula for minimal parabolic subgroups, J. Lie Theory 24 (2014) 791--808,
(5) Stepwise square integrable representations for locally nilpotent Lie groups, Transformation Groups 20 (2015) 863--879,
the author extended the notion of square integrability, for representations of nilpotent Lie groups, to that of stepwise square integrability. There we discussed a number of applications based on the fact that nilradicals of minimal parabolic subgroups of real reductive Lie groups are stepwise square integrable. In Part I we prove stepwise square integrability for nilradicals of arbitrary parabolic subgroups of real reductive Lie groups. This is technically more delicate than the case of minimal parabolics. We further discuss applications to Plancherel formulae and Fourier inversion formulae for maximal exponential solvable subgroups of parabolics and maximal amenable subgroups of real reductive Lie groups. Finally, in Part II, we extend a number of those results to (infinite dimensional) direct limit parabolics. These extensions involve an infinite dimensional version of the Peter-Weyl Theorem, construction of a direct limit Schwartz space, and realization of that Schwartz space as a dense subspace of the corresponding L2 space.

Keywords: Parabolic subgroups, nilradicals, stepwise square integrable representations, Dixmier-Pukanszky operators, Plancherel formulae, Fourier inversion formulae.

MSC: 22E25, 22E45, 22E65, 22E66; 44A80

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