Geodesic
Berezin transform for non-scalar holomorphic discrete series
Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 1, pp. 1-17

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Let $M=G/K$ be a Hermitian symmetric space of the non-compact type and let $\pi$ be a discrete series representation of $G$ which is holomorphically induced from a unitary irreducible representation $\rho$ 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 $\pi$. Here we study the corresponding Berezin transform and we show that it can be extended to a large class of symbols. As an application, we construct a Stratonovich-Weyl correspondence associated with $\pi$.
Classification : 22E46, 32M10, 32M15, 81S10
Keywords: Berezin quantization; Berezin symbol; Stratonovich-Weyl correspondence; discrete series representation; Hermitian symmetric space of the non-compact type; semi-simple non-compact Lie group; coherent states; reproducing kernel; adjoint orbit 
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Cahen, Benjamin. Berezin transform for non-scalar holomorphic discrete series. Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 1, pp. 1-17. http://geodesic.mathdoc.fr/item/CMUC_2012__53_1_a0/