Harmonic Analysis in One-Parameter Metabelian Nilmanifolds

Symmetry Integrability and Geometry-Methods and Applications

article

Amira Ghorbel¹ 

[1] Faculté des Sciences de Sfax, Département de Mathématiques

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

DOI  :  10.3842/SIGMA.2011.021

来源: National Academy of Science of Ukraine

【 摘 要 】

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=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.

【 授权许可】

Unknown