Berezin transform for non-scalar holomorphic discrete series

Benjamin Cahen

Commentationes Mathematicae Universitatis Carolinae (2012)

  • Volume: 53, Issue: 1, page 1-17
  • ISSN: 0010-2628

Abstract

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Let M = G / K be a Hermitian symmetric space of the non-compact type and let π be a discrete series representation of G which is holomorphically induced from a unitary irreducible representation ρ of K . In the paper [B. Cahen, Berezin quantization for holomorphic discrete series representations: the non-scalar case, Beiträge Algebra Geom., DOI 10.1007/s13366-011-0066-2], we have introduced a notion of complex-valued Berezin symbol for an operator acting on the space of π . Here we study the corresponding Berezin transform and we show that it can be extended to a large class of symbols. As an application, we construct a Stratonovich-Weyl correspondence associated with π .

How to cite

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Cahen, Benjamin. "Berezin transform for non-scalar holomorphic discrete series." Commentationes Mathematicae Universitatis Carolinae 53.1 (2012): 1-17. <http://eudml.org/doc/246625>.

@article{Cahen2012,
abstract = {Let $M=G/K$ be a Hermitian symmetric space of the non-compact type and let $\pi $ be a discrete series representation of $G$ which is holomorphically induced from a unitary irreducible representation $\rho $ of $K$. In the paper [B. Cahen, Berezin quantization for holomorphic discrete series representations: the non-scalar case, Beiträge Algebra Geom., DOI 10.1007/s13366-011-0066-2], we have introduced a notion of complex-valued Berezin symbol for an operator acting on the space of $\pi $. Here we study the corresponding Berezin transform and we show that it can be extended to a large class of symbols. As an application, we construct a Stratonovich-Weyl correspondence associated with $\pi $.},
author = {Cahen, Benjamin},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Berezin quantization; Berezin symbol; Stratonovich-Weyl correspondence; discrete series representation; Hermitian symmetric space of the non-compact type; semi-simple non-compact Lie group; coherent states; reproducing kernel; adjoint orbit; Berezin quantization; Berezin symbol; Stratonovich-Weyl correspondence; coherent state; reproducing kernel; adjoint orbit},
language = {eng},
number = {1},
pages = {1-17},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Berezin transform for non-scalar holomorphic discrete series},
url = {http://eudml.org/doc/246625},
volume = {53},
year = {2012},
}

TY - JOUR
AU - Cahen, Benjamin
TI - Berezin transform for non-scalar holomorphic discrete series
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2012
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 53
IS - 1
SP - 1
EP - 17
AB - Let $M=G/K$ be a Hermitian symmetric space of the non-compact type and let $\pi $ be a discrete series representation of $G$ which is holomorphically induced from a unitary irreducible representation $\rho $ of $K$. In the paper [B. Cahen, Berezin quantization for holomorphic discrete series representations: the non-scalar case, Beiträge Algebra Geom., DOI 10.1007/s13366-011-0066-2], we have introduced a notion of complex-valued Berezin symbol for an operator acting on the space of $\pi $. Here we study the corresponding Berezin transform and we show that it can be extended to a large class of symbols. As an application, we construct a Stratonovich-Weyl correspondence associated with $\pi $.
LA - eng
KW - Berezin quantization; Berezin symbol; Stratonovich-Weyl correspondence; discrete series representation; Hermitian symmetric space of the non-compact type; semi-simple non-compact Lie group; coherent states; reproducing kernel; adjoint orbit; Berezin quantization; Berezin symbol; Stratonovich-Weyl correspondence; coherent state; reproducing kernel; adjoint orbit
UR - http://eudml.org/doc/246625
ER -

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