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

TL;DR

This paper extends the concept of stepwise square integrability to nilradicals of all parabolic subgroups in real reductive Lie groups, with applications to harmonic analysis and infinite-dimensional limits.

Contribution

It proves stepwise square integrability for nilradicals of arbitrary parabolics and extends results to infinite-dimensional direct limit parabolics.

Findings

Stepwise square integrability established for all parabolic nilradicals.

Applications to Plancherel and Fourier inversion formulas for maximal subgroups.

Extension of results to infinite-dimensional parabolic limits.

Abstract

In a series of recent papers we 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. Here, 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 $L^2$ space.