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On the multiplicity formula of compact nilmanifolds with flat orbits

On the multiplicity formula of compact nilmanifolds with flat orbits Let G be a connected, simply connected nilpotent Lie group and ΓΓ a lattice subgroup of G . The quasi regular representation decomposes as a direct sum of unitary irreducible representations of G . The nilmanifold G /ΓΓ is called nilmanifold with flat orbits if the coadjoint orbit corresponding to any element of the spectrum of ℛ ΓΓ is flat. In this note, we give a new multiplicity formula related to the decomposition into irreducibles of ℛ ΓΓ in the case when G /ΓΓ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Pure and Applied Mathematics de Gruyter

On the multiplicity formula of compact nilmanifolds with flat orbits

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References (1)

Publisher
de Gruyter
Copyright
© de Gruyter 2011
ISSN
1867-1152
eISSN
1869-6090
DOI
10.1515/APAM.2010.030
Publisher site
See Article on Publisher Site

Abstract

Let G be a connected, simply connected nilpotent Lie group and ΓΓ a lattice subgroup of G . The quasi regular representation decomposes as a direct sum of unitary irreducible representations of G . The nilmanifold G /ΓΓ is called nilmanifold with flat orbits if the coadjoint orbit corresponding to any element of the spectrum of ℛ ΓΓ is flat. In this note, we give a new multiplicity formula related to the decomposition into irreducibles of ℛ ΓΓ in the case when G /ΓΓ 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.

Journal

Advances in Pure and Applied Mathematicsde Gruyter

Published: Sep 1, 2011

Keywords: Nilpotent Lie group; lattice subgroup; rational structure; unitary representation; co-adjoint orbit; Kirillov theory

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