Harmonic analysis in one-parameter metabelian nilmanifolds

DOI10.3842/SIGMA.2011.021zbMATH Open1217.22006arXiv1102.5479MaRDI QIDQ542767

Amira Ghorbel

Publication date: 17 June 2011

Published in: SIGMA. Symmetry, Integrability and Geometry: Methods and Applications (Search for Journal in Brave)

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 $Gamma$ of $G$, we define the quasi-regular representation $au={mind}_{G}amm{a}^{G}1$ of $G$. The basic problem considered in this paper concerns the decomposition of $au$ into irreducibles. We give an orbital description of the spectrum, the multiplicity function and we construct an explicit intertwining operator between $au$ 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.

Full work available at URL: https://arxiv.org/abs/1102.5479

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zbMATH Keywords

intertwining operatorunitary representationnilpotent Lie grouppolarizationnilmanifoldorbitdiscrete subgroupdisintegrationKirillov theory

Mathematics Subject Classification ID

Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) (22E27)

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