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Berezin quantization for discrete series. (English) Zbl 1342.22022

Summary: Let \(M=G/K\) be a Hermitian symmetric space of the non-compact type and \(\pi\) be a discrete series representation of \(G\) which is holomorphically induced from a unitary character of \(K\). We give explicit formulas for the Berezin symbols of the operators \(\pi (g)\) (\(g\in G\)) and \(d\pi (X)\) (\(X\) in the Lie algebra of \(G\)). We show that the Berezin quantization on \(G/K\) provides an adapted symbol calculus in the sense of [B. Cahen, Weyl quantization for semidirect products. Differ. Geom. Appl. 25, No. 2, 177–190 (2007; Zbl 1117.81087)].

MSC:

22E46 Semisimple Lie groups and their representations
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
81S10 Geometry and quantization, symplectic methods

Citations:

Zbl 1117.81087
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