UC Berkeley

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 (Formula presented.) where, in all but three cases, the nilpotent group (Formula presented.) has irreducible unitary representations whose coefficients are square integrable modulo the center (Formula presented.) of (Formula presented.). Here we show that, in those three “exceptional” cases, the group (Formula presented.) is a semidirect product (Formula presented.) or (Formula presented.) where the normal subgroup (Formula presented.) contains the center (Formula presented.) of (Formula presented.) and has irreducible unitary representations whose coefficients are square integrable modulo (Formula presented.). This leads directly to explicit harmonic analysis and Fourier inversion formulae for commutative nilmanifolds.