## Benjamin Cahen

*Berezin transform for non-scalar holomorphic discrete series*

Comment.Math.Univ.Carolin. 53,1 (2012) 1-17.**Abstract:**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, {\it Berezin quantization for holomorphic discrete series representations: the non-scalar case\/}, Beitr\"age 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~$\pi$.

**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

**AMS Subject Classification:** 22E46 32M10 32M15 81S10

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