  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 $SO(\infty,\infty)$.
