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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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