Amira Ghorbel | Searcharxiv

arXiv2011

Harmonic Analysis in One-Parameter Metabelian Nilmanifolds

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 $τ= {\rm ind}_Γ^G 1$ 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.

math.GR↗

arXiv2009

Discrete Cocompact Subgroups of the Five-Dimensional Connected and Simply Connected Nilpotent Lie Groups

The discrete cocompact subgroups of the five-dimensional connected, simply connected nilpotent Lie groups are determined up to isomorphism. Moreover, we prove if $G=N\times A$ is a connected, simply connected, nilpotent Lie group with an Abelian factor $A$, then every uniform subgroup of $G$ is the direct product of a uniform subgroup of $N$ and ${\mathbb Z}^r$ where $r=\dim A$.

math.GR↗