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Berezin quantization and holomorphic representations. (English) Zbl 1272.22007

This paper is a generalization of previous results about Berezin quantization, obtained by the author for the holomorphic discrete series representations of a semi-simple Lie group. The author considers a quasi-Hermitian Lie group \(G\) with Lie algebra \(g\) and a unitary irreducible representation \(p\). The notion of an adapted Weyl correspondence is then developed with some recalls about the relevant facts. Then, the Berezin symbol map \(S\) and the corresponding Stratonovich-Weyl map \(W\) are introduced and studied. Explicit formulas for the Berezin symbols are obtained. When \(G\) is reductive, the author proves that the Stratonovich-Weyl map \(W\) can be extended to the operators \(d(p)\) with an explicit expression of \(W(d(p))\). The general case is discussed, and the example of the diamond group is also considered.

MSC:

22E46 Semisimple Lie groups and their representations
53D17 Poisson manifolds; Poisson groupoids and algebroids
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
81S10 Geometry and quantization, symplectic methods
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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