Weyl quantization for principal series
type: Item
instance of: scholarly article
title: Weyl quantization for principal series
zbMATH Open document ID: 1134.22010
author: Benjamin Cahen
published in: Beiträge zur Algebra und Geometrie
publication date: 2007-05-14T00:00:00Z
full work available at URL: https://eudml.org/doc/224692
full work available at URL: 14.html
review text: Let \(G\) be a connected Lie group. Let \(\pi\) be a unitary irreducible representation of \(G\) on a Hilbert space \(H\). Suppose that the representation \(\pi\) is associated with a coadjoint orbit \(\mathcal O\) of \(G\) by the Kirillov-Kostant method of orbits. A notion of adapted Weyl correspondence was introduced in [\textit{B. Cahen}, Lett. Math. Phys. 36, No. 1, 65--75 (1996; Zbl 0843.22020); see also \textit{B. Cahen}, C. R. Acad. Sci. Paris, Sér. I, Math. 325, No. 7, 803--806 (1997; Zbl 0883.22016)]. Roughly speaking, an adapted Weyl correspondence \(W\) on the orbit \(\mathcal O\) is a one-to-one linear correspondence between a class of functions on the orbit \(\mathcal O\) (called symbols) and a class of operators on \(H\) such that the functions \({\widetilde X}\) (\(X \in {\mathfrak g}\)) are symbols and for each \(v\) in a dense subspace of \(H\) and each \(X \in {\mathfrak g}\) we have \(W(i\widetilde X)\,v=d\pi (X)v\). A more precise definition of an adapted Weyl correspondence inspired by the work of Mark Gotay [\textit{M. Gotay}, Obstructions to Quantization, in: Mechanics: From Theory to Computation (Essays in Honor of Juan-Carlos Simo), J. Nonlinear Science Editors, Springer New York, 171--216 (2000; Zbl 1041.53507)] was proposed in [\textit{B. Cahen}, Differ. Geom. Appl. 25, No. 2, 177--190 (2007; Zbl 1117.81087)]. Our original motivation for constructing adapted Weyl correspondences was to build covariant star-products on coadjoint orbits [\textit{B. Cahen}, Lett. Math. Phys. 36, No. 1, 65--75 (1996; Zbl 0843.22020)]. A more recent motivation is that adapted Weyl correspondences can be used to study contractions of representations of Lie groups in the setting of the Kirillov-Kostant method of orbits. The basic idea is then to interpret contraction results on the symbols of the representation operators [\textit{P. Cotton} and \textit{A. H. Dooley}, J. Lie Theory 7, No. 2, 147--164 (1997; Zbl 0882.22015); \textit{B. Cahen}, J. Lie Theory 11, No.…
reviewed by: Benjamin Cahen
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 81S10
zbMATH DE Number: 5152747
zbMATH Keywords: principal series
zbMATH Keywords: coadjoint orbits
zbMATH Keywords: Weyl quantization
zbMATH Keywords: Berezin quantization
zbMATH Keywords: dequantization
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5152747