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Berezin quantization for holomorphic discrete series representations: the non-scalar case

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Abstract

Let M = G/K be a Hermitian symmetric space of the non-compact type and let π be a discrete series representation of G which is holomorphically induced from a unitary irreducible representation ρ of K. We define and we study a notion of Berezin symbol for operators acting on the space of π. In particular, we give explicit formulas for the Berezin symbols of the representation operators \({\pi(g) (g\in G)}\) and dπ(X) (X in the Lie algebra of G). Moreover, we construct an adapted Weyl correspondence for π in the sense of Cahen (Differ Geom Appl 25:177–190, 2007). This generalizes the results already obtained in the case when ρ is a unitary character of K, see Cahen (Beiträge Algebra Geom 51:301–311, 2010).

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  1. Département de mathématiques, Université de Metz, UFR-MIM, LMMAS, ISGMP-Bât. A, Ile du Saulcy, 57045, Metz cedex 01, France

    Benjamin Cahen

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  1. Benjamin Cahen
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Correspondence to Benjamin Cahen.

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Cahen, B. Berezin quantization for holomorphic discrete series representations: the non-scalar case. Beitr Algebra Geom 53, 461–471 (2012). https://doi.org/10.1007/s13366-011-0066-2

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