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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].

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...

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.

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...

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