On the Analytic Structure of Commutative Nilmanifolds

Abstract

In the classification theorems of Vinberg and Yakimova for commutative nilmanifolds, the relevant nilpotent groups have a very surprising analytic property. The manifolds are of the form \(G/K = N\rtimes K/K\) where, in all but three cases, the nilpotent group \(N\) has irreducible unitary representations whose coefficients are square integrable modulo the center \(Z\) of \(N\). Here we show that, in those three “exceptional” cases, the group \(N\) is a semidirect product \(N_{1}\rtimes \mathbb {R}\) or \(N_{1}\rtimes \mathbb {C}\) where the normal subgroup \(N_{1}\) contains the center \(Z\) of \(N\) and has irreducible unitary representations whose coefficients are square integrable modulo \(Z\). This leads directly to explicit harmonic analysis and Fourier inversion formulae for commutative nilmanifolds.