Stepwise Square Integrable Representations for Locally Nilpotent Lie Groups

TL;DR

This paper extends the concept of stepwise square integrable representations from finite-dimensional nilpotent Lie groups to their direct limits, providing explicit Fourier inversion formulas for infinite-dimensional cases.

Contribution

It introduces a framework for stepwise square integrable representations in direct limit groups and derives explicit Fourier inversion formulas for these infinite-dimensional structures.

Findings

Explicit Fourier inversion formula for infinite-dimensional nilpotent Lie groups.

Extension of stepwise square integrability to direct limits of Lie groups.

Application to nilradicals of classical minimal parabolic subgroups.

Abstract

In a recent paper we found conditions for a nilpotent Lie group $N$ to have a filtration by normal subgroups whose successive quotients have square integrable representations, and such that these square integrable representations fit together nicely to give an explicit construction of Plancherel almost all representations of $N$. 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 to direct limits of stepwise square integrable nilpotent Lie groups. This involves some development of the corresponding Schwartz spaces. The main result is an explicit Fourier inversion formula for that class of infinite dimensional Lie groups. One important consequence is the Fourier inversion formula for nilradicals of classical minimal parabolic subgroups of finitary real reductive Lie groups such as $GL(\infty;R)$, $Sp(\infty;C)$ and $SO(\infty,\infty)$.