Berezin quantization for discrete series

Benjamin Cahen

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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