Stratonovich-Weyl correspondence for discrete series representations

Benjamin Cahen

Archivum Mathematicum (2011)

  • Volume: 047, Issue: 1, page 51-68
  • ISSN: 0044-8753

Abstract

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Let M = G / K be a Hermitian symmetric space of the noncompact type and let π be a discrete series representation of G holomorphically induced from a unitary character of K . Following an idea of Figueroa, Gracia-Bondìa and Vàrilly, we construct a Stratonovich-Weyl correspondence for the triple ( G , π , M ) by a suitable modification of the Berezin calculus on M . We extend the corresponding Berezin transform to a class of functions on M which contains the Berezin symbol of d π ( X ) for X in the Lie algebra 𝔤 of G . This allows us to define and to study the Stratonovich-Weyl symbol of d π ( X ) for X ∈ 𝔤 .

How to cite

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Cahen, Benjamin. "Stratonovich-Weyl correspondence for discrete series representations." Archivum Mathematicum 047.1 (2011): 51-68. <http://eudml.org/doc/116533>.

@article{Cahen2011,
abstract = {Let $M=G/K$ be a Hermitian symmetric space of the noncompact type and let $\pi $ be a discrete series representation of $G$ holomorphically induced from a unitary character of $K$. Following an idea of Figueroa, Gracia-Bondìa and Vàrilly, we construct a Stratonovich-Weyl correspondence for the triple $(G, \pi , M)$ by a suitable modification of the Berezin calculus on $M$. We extend the corresponding Berezin transform to a class of functions on $M$ which contains the Berezin symbol of $d\pi (X)$ for $X$ in the Lie algebra $\mathfrak \{g\}$ of $G$. This allows us to define and to study the Stratonovich-Weyl symbol of $d\pi (X)$ for $X\in \mathfrak \{g\}$.},
author = {Cahen, Benjamin},
journal = {Archivum Mathematicum},
keywords = {Stratonovich-Weyl correspondence; Berezin quantization; Berezin transform; semisimple Lie group; coadjoint orbits; unitary representation; Hermitian symmetric space of the noncompact type; discrete series representation; reproducing kernel Hilbert space; coherent states; Stratonovich-Weyl correspondence; Berezin quantization; semisimple Lie group; coadjoint orbit; discrete series representation; reproducing kernel Hilbert space},
language = {eng},
number = {1},
pages = {51-68},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Stratonovich-Weyl correspondence for discrete series representations},
url = {http://eudml.org/doc/116533},
volume = {047},
year = {2011},
}

TY - JOUR
AU - Cahen, Benjamin
TI - Stratonovich-Weyl correspondence for discrete series representations
JO - Archivum Mathematicum
PY - 2011
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 047
IS - 1
SP - 51
EP - 68
AB - Let $M=G/K$ be a Hermitian symmetric space of the noncompact type and let $\pi $ be a discrete series representation of $G$ holomorphically induced from a unitary character of $K$. Following an idea of Figueroa, Gracia-Bondìa and Vàrilly, we construct a Stratonovich-Weyl correspondence for the triple $(G, \pi , M)$ by a suitable modification of the Berezin calculus on $M$. We extend the corresponding Berezin transform to a class of functions on $M$ which contains the Berezin symbol of $d\pi (X)$ for $X$ in the Lie algebra $\mathfrak {g}$ of $G$. This allows us to define and to study the Stratonovich-Weyl symbol of $d\pi (X)$ for $X\in \mathfrak {g}$.
LA - eng
KW - Stratonovich-Weyl correspondence; Berezin quantization; Berezin transform; semisimple Lie group; coadjoint orbits; unitary representation; Hermitian symmetric space of the noncompact type; discrete series representation; reproducing kernel Hilbert space; coherent states; Stratonovich-Weyl correspondence; Berezin quantization; semisimple Lie group; coadjoint orbit; discrete series representation; reproducing kernel Hilbert space
UR - http://eudml.org/doc/116533
ER -

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