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On the multiplicity formula of compact nilmanifolds with flat orbits
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Hatem Hamrouni
Published/Copyright:
June 27, 2010
Abstract
Let G be a connected, simply connected nilpotent Lie group and Γ a lattice subgroup of G. The quasi regular representation 
decomposes as a direct sum of unitary irreducible representations of G. The nilmanifold G/Γ is called nilmanifold with flat orbits if the coadjoint orbit corresponding to any element of the spectrum of ℛΓ is flat. In this note, we give a new multiplicity formula related to the decomposition into irreducibles of ℛΓ in the case when G/Γ is a nilmanifold with flat orbits. As an application, we first prove that every nilmanifold with flat orbits satisfies the Moore formula. We also give partial answers to some related questions proposed by Brezin and by Corwin and Greenleaf.
Keywords.: Nilpotent Lie group; lattice subgroup; rational structure; unitary representation; co-adjoint orbit; Kirillov theory
Received: 2009-11-25
Revised: 2010-04-27
Published Online: 2010-06-27
Published in Print: 2011-September
© de Gruyter 2011
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Articles in the same Issue
- Geometric and harmonic analysis on homogeneous spaces
- On the multiplicity formula of compact nilmanifolds with flat orbits
- Hilbert transform and related topics associated with Jacobi–Dunkl operators of compact and noncompact types
- Atomic decomposition of a real Hardy space for Jacobi analysis
- Unitary holomorphic multiplier representations over a homogeneous bounded domain
- A deformation approach of the Kirillov map for exponential groups
- Visible actions on the non-symmetric homogeneous space SO(8, ℂ)/G2(ℂ)
- A Paley–Wiener theorem for some eigenfunction expansions
- Estimate of the Lp-Fourier transform norm for connected nilpotent Lie groups
Keywords for this article
Nilpotent Lie group;
lattice subgroup;
rational structure;
unitary representation;
co-adjoint orbit;
Kirillov theory
Articles in the same Issue
- Geometric and harmonic analysis on homogeneous spaces
- On the multiplicity formula of compact nilmanifolds with flat orbits
- Hilbert transform and related topics associated with Jacobi–Dunkl operators of compact and noncompact types
- Atomic decomposition of a real Hardy space for Jacobi analysis
- Unitary holomorphic multiplier representations over a homogeneous bounded domain
- A deformation approach of the Kirillov map for exponential groups
- Visible actions on the non-symmetric homogeneous space SO(8, ℂ)/G2(ℂ)
- A Paley–Wiener theorem for some eigenfunction expansions
- Estimate of the Lp-Fourier transform norm for connected nilpotent Lie groups
