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Berezin quantization and holomorphic representations

Benjamin Cahen — 2013

Rendiconti del Seminario Matematico della Università di Padova

Déformations éformations formelles de certaines représentations de l'algèbre de Lie d'un groupe de Poincaré généralisé

Benjamin Cahen — 2001

Annales mathématiques Blaise Pascal

Weyl quantization for the semidirect product of a compact Lie group and a vector space

Benjamin Cahen — 2009

Commentationes Mathematicae Universitatis Carolinae

Let $G$ be the semidirect product $V ⋊ K$ where $K$ is a semisimple compact connected Lie group acting linearly on a finite-dimensional real vector space $V$ . Let $𝒪$ be a coadjoint orbit of $G$ associated by the Kirillov-Kostant method of orbits with a unitary irreducible representation $π$ of $G$ . We consider the case when the corresponding little group $H$ is the centralizer of a torus of $K$ . By dequantizing a suitable realization of $π$ on a Hilbert space of functions on $ℂ^{n}$ where $n = dim (K / H)$ , we construct a symplectomorphism between...

Berezin-Weyl quantization for Cartan motion groups

Benjamin Cahen — 2011

Commentationes Mathematicae Universitatis Carolinae

We construct adapted Weyl correspondences for the unitary irreducible representations of the Cartan motion group of a noncompact semisimple Lie group by using the method introduced in [B. Cahen, Weyl quantization for semidirect products, Differential Geom. Appl. 25 (2007), 177--190].

Berezin transform for non-scalar holomorphic discrete series

Benjamin Cahen — 2012

Commentationes Mathematicae Universitatis Carolinae

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...

Invariant symbolic calculus for semidirect products

Benjamin Cahen — 2018

Commentationes Mathematicae Universitatis Carolinae

Let $G$ be the semidirect product $V ⋊ K$ where $K$ is a connected semisimple non-compact Lie group acting linearly on a finite-dimensional real vector space $V$ . Let $π$ be a unitary irreducible representation of $G$ which is associated by the Kirillov-Kostant method of orbits with a coadjoint orbit of $G$ whose little group is a maximal compact subgroup of $K$ . We construct an invariant symbolic calculus for $π$ , under some technical hypothesis. We give some examples including the Poincaré group.

Formal deformations and principal series representations of $SL (2, ℝ)$ and $SL (2, ℂ)$

Benjamin Cahen — 2020

Czechoslovak Mathematical Journal

In this note, we study formal deformations of derived representations of the principal series representations of $SL (2, ℝ)$ . In particular, we recover all the representations of the derived principal series by deforming one of them. Similar results are also obtained for $SL (2, ℂ)$ .

Stratonovich-Weyl correspondence for discrete series representations

Benjamin Cahen — 2011

Archivum Mathematicum

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...

Global Parametrization of Scalar Holomorphic Coadjoint Orbits of a Quasi-Hermitian Lie Group

Benjamin Cahen — 2013

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Let $G$ be a quasi-Hermitian Lie group with Lie algebra $𝔤$ and $K$ be a compactly embedded subgroup of $G$ . Let $ξ_{0}$ be a regular element of $𝔤^{*}$ which is fixed by $K$ . We give an explicit $G$ -equivariant diffeomorphism from a complex domain onto the coadjoint orbit $𝒪 (ξ_{0})$ of $ξ_{0}$ . This generalizes a result of [B. Cahen, Berezin quantization and holomorphic representations, Rend. Sem. Mat. Univ. Padova, to appear] concerning the case where $𝒪 (ξ_{0})$ is associated with a unitary irreducible representation of $G$ which is holomorphically...

Stratonovich-Weyl correspondence for the Jacobi group

Benjamin Cahen — 2014

Communications in Mathematics

We construct and study a Stratonovich-Weyl correspondence for the holomorphic representations of the Jacobi group.

Invariant symbolic calculus for compact Lie groups

Benjamin Cahen — 2019

Archivum Mathematicum

We study the invariant symbolic calculi associated with the unitary irreducible representations of a compact Lie group.

Multiplicities of compact lie group representations via Berezin quantization

Benjamin Cahen — 2008

Matematički Vesnik

Minimal realizations of classical simple Lie algebras through deformations

Didier Arnal; Hádi Benamor; Benjamin Cahen — 1998

Annales de la Faculté des sciences de Toulouse : Mathématiques

Contractions of $S U (2)$ to the Heisenberg group and Berezin calculus. (Contraction de $S U (2)$ vers le groupe de Heisenberg et calcul de Berezin.)

Cahen, Benjamin — 2003

Beiträge zur Algebra und Geometrie

Quantification of coadjoint orbits and the theory of contractions. (Quantification d'orbites coadjointes et théorie des contractions.)

Cahen, Benjamin — 2001

Journal of Lie Theory

Weyl quantization for principal series.

Cahen, Benjamin — 2007

Beiträge zur Algebra und Geometrie

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Currently displaying 1 – 16 of 16

Berezin quantization and holomorphic representations

Déformations éformations formelles de certaines représentations de l'algèbre de Lie d'un groupe de Poincaré généralisé

Weyl quantization for the semidirect product of a compact Lie group and a vector space

Berezin-Weyl quantization for Cartan motion groups

Berezin transform for non-scalar holomorphic discrete series

Invariant symbolic calculus for semidirect products

Formal deformations and principal series representations of $SL (2, ℝ)$ and $SL (2, ℂ)$

Stratonovich-Weyl correspondence for discrete series representations

Global Parametrization of Scalar Holomorphic Coadjoint Orbits of a Quasi-Hermitian Lie Group

Stratonovich-Weyl correspondence for the Jacobi group

Invariant symbolic calculus for compact Lie groups

Multiplicities of compact lie group representations via Berezin quantization

Minimal realizations of classical simple Lie algebras through deformations

Contractions of $S U (2)$ to the Heisenberg group and Berezin calculus. (Contraction de $S U (2)$ vers le groupe de Heisenberg et calcul de Berezin.)

Quantification of coadjoint orbits and the theory of contractions. (Quantification d'orbites coadjointes et théorie des contractions.)

Weyl quantization for principal series.