Harmonic Analysis in One-Parameter Metabelian Nilmanifolds

Amira Ghorbel

Symmetry, integrability and geometry: methods and applications, Tome 7 (2011)

Résumé

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 $\Gamma$ of $G$, we define the quasi-regular representation $\tau=\operatorname{ind}_\Gamma^G1$ of $G$. The basic problem considered in this paper concerns the decomposition of $\tau$ into irreducibles. We give an orbital description of the spectrum, the multiplicity function and we construct an explicit intertwining operator between $\tau$ 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.

Détail

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

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