Deformation program for principal series representations
type: Item
instance of: scholarly article
title: Deformation program for principal series representations
zbMATH Open document ID: 0843.22020
DOI: 10.1007/BF00403252
author: Benjamin Cahen
published in: Letters in Mathematical Physics
publication date: 1996-03-11T00:00:00Z
review text: The author begins with a parametrization of the principal series orbits of a semisimple Lie group \(G\) in terms of a cotangent bundle of a nilpotent Lie group and of a coadjoint orbit of a compact Lie group. Then he combines Berezin symbolic calculus on compact orbits and some generalization of Weyl correspondence to a cotangent bundle to obtain a symbolic calculus on the principal series orbits which is adapted to the description of the principal series representations. He shows that the \(*\)-product on principal series orbits associated to this symbolic calculus can be interpreted in terms of a Berezin product and on S. Gutt's star product on the cotangent bundle of a nilpotent Lie group. Finally, the author applies his construction to define the adapted Fourier transform of \(G\) and he gives in the case where \(G\) has a unique class of Cartan subalgebras an elegant retranscription of the Plancherel formula.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 43A30
Mathematics Subject Classification ID: 58H15
zbMATH DE Number: 843629
zbMATH Keywords: parametrization
zbMATH Keywords: principal series orbits
zbMATH Keywords: semisimple Lie group
zbMATH Keywords: cotangent bundle
zbMATH Keywords: nilpotent Lie group
zbMATH Keywords: coadjoint orbit
zbMATH Keywords: compact Lie group
zbMATH Keywords: Berezin symbolic calculus
zbMATH Keywords: Weyl correspondence
zbMATH Keywords: principal series representations
zbMATH Keywords: Gutt's star product
zbMATH Keywords: Fourier transform
zbMATH Keywords: Plancherel formula
reviewed by: Jean Ludwig
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 843629