### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 021, 16 pages      arXiv:1102.5479      https://doi.org/10.3842/SIGMA.2011.021

### Harmonic Analysis in One-Parameter Metabelian Nilmanifolds

Amira Ghorbel
Faculté des Sciences de Sfax, Département de Mathématiques, Route de Soukra, B.P. 1171, 3000 Sfax, Tunisie

Received September 02, 2010, in final form February 21, 2011; Published online February 27, 2011

Abstract
Let G be a connected, simply connected one-parameter metabelian nilpotent Lie group, that means, the corresponding Lie algebra has a one-codimensional abelian subalgebra. In this article we show that G contains a discrete cocompact subgroup. Given a discrete cocompact subgroup Γ of G, we define the quasi-regular representation τ=indΓG1 of G. The basic problem considered in this paper concerns the decomposition of τ into irreducibles. We give an orbital description of the spectrum, the multiplicity function and we construct an explicit intertwining operator between τ and its desintegration without considering multiplicities. Finally, unlike the Moore inductive algorithm for multiplicities on nilmanifolds, we carry out here a direct computation to get the multiplicity formula.

Key words: nilpotent Lie group; discrete subgroup; nilmanifold; unitary representation; polarization; disintegration; orbit; intertwining operator; Kirillov theory.

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