STEPWISE SQUARE INTEGRABLE REPRESENTATIONS FOR LOCALLY NILPOTENT LIE GROUPS

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 for almost all representations of N. The prototype for this sort of group is the group of upper triangular real matrices with 1’s down the diagonal. More generally, this class of groups contains the nilradicals of minimal parabolic subgroups of all (finite-dimensional) reductive real or complex Lie groups, in other words, all groups N in Iwasawa decompositions of reductive real or complex Lie groups.

The construction of stepwise square integrable representations resulted in explicit character formulae, Plancherel formulae and multiplicity formulae. Here we extend those results to direct limits of stepwise square integrable nilpotent Lie groups. There are two keys to this extension. The first is to set up the corresponding direct system so that it respects the construction at every finite level. In the case of simple (or more generally reductive) groups this means that the restricted root Dynkin diagrams increase in a particular manner. The second is to follow Schwartz space theory through the direct limit process, develop a Schwartz space theory for certain direct limit nilpotent groups, and use it to study stepwise square integrability for coefficients of direct limits of stepwise square integrable nilpotent Lie groups. This leads to the main result, 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(∞; ℝ), Sp(∞; ℂ) and SO(∞, ∞).