Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 51, No. 2, pp. 301-311 (2010)

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Berezin quantization for discrete series

Benjamin Cahen

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

Abstract: 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 [C]. [C] Cahen, B.: Weyl quantization for semidirect products. Differ. Geom. Appl. {\bf 25} (2007), 177--190.

Keywords: Berezin quantization, Berezin symbol, Hermitian symmetric space of the non-compact type, semi-simple non-compact Lie group, irreducible unitary representation, discrete series representation, coherent states, adapted symbol calculus, adjoint orbit

Classification (MSC2000): 22E46, 32M10, 32M15, 81S10

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Electronic version published on: 24 Jun 2010. This page was last modified: 8 Sep 2010.

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