Coherent states and Berezin quantization for non-scalar holomorphic representations. (English) Zbl 1319.22008
This paper studies the Berezin quantization map for non-scalar holomorphic representations. Let \(G\) be a quasi-Hermitian Lie group and \(K\) a compactly embedded subgroup of \(G\). Let us recall that each unitary representation of \(G\) is usually realized in a Hilbert space of vector-valued holomorphic functions. In this work, the author introduces a realization in a reproducing kernel Hilbert space. First, the author introduces the notations and some recalls about quasi-Hermitian Lie groups. Then, the coherent spaces are computed. Explicit formulas are also given. The corresponding Berezin transform is finally studied. The construction is illustrated in details in the case of the Heisenberg motion groups.
Reviewer: Angela Gammella-Mathieu (Metz)
MSC:
| 22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |
| 22E10 | General properties and structure of complex Lie groups |
| 32M05 | Complex Lie groups, group actions on complex spaces |
| 32M10 | Homogeneous complex manifolds |
| 46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |
| 81S10 | Geometry and quantization, symplectic methods |
| 81R30 | Coherent states |
Keywords:
Berezin quantization; coherent states; coadjoint orbit; holomorphic functions; Hilbert space; Stratonovich-Weyl correspondence; Heisenberg motion groups; quasi-Hermitian Lie groupReferences:
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