On the Analytic Structure of Commutative Nilmanifolds

2014/07/01 by Joseph A. Wolf, Wolf, Joseph A.
Mathematics · #22E30 #22E47 #53C60 #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Primary 22E27 #Representation Theory (math.RT) #Secondary 53C35 #math.DG #math.FA #math.RT #msc:22E27 #msc:22E30 #msc:22E47 #msc:53C35 #msc:53C60

paper · pdf · doi:10.48550/arxiv.1407.0399

arxiv created 2014/07/01 · arxiv updated 2014/07/03

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 N₁ \rtimes ℝ or N₁ \rtimes ℂ where the normal subgroup N₁ 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.

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