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Keywords:
Stratonovich-Weyl correspondence; Berezin quantization; Berezin transform; semisimple Lie group; coadjoint orbits; unitary representation; Hermitian symmetric space of the noncompact type; discrete series representation; reproducing kernel Hilbert space; coherent states
Stratonovich-Weyl correspondence; Berezin quantization; Berezin transform; semisimple Lie group; coadjoint orbits; unitary representation; Hermitian symmetric space of the noncompact type; discrete series representation; reproducing kernel Hilbert space; coherent states
Summary:
Let $M=G/K$ be a Hermitian symmetric space of the noncompact type and let $\pi $ be a discrete series representation of $G$ holomorphically induced from a unitary character of $K$. Following an idea of Figueroa, Gracia-Bondìa and Vàrilly, we construct a Stratonovich-Weyl correspondence for the triple $(G, \pi , M)$ by a suitable modification of the Berezin calculus on $M$. We extend the corresponding Berezin transform to a class of functions on $M$ which contains the Berezin symbol of $d\pi (X)$ for $X$ in the Lie algebra $\mathfrak{g}$ of $G$. This allows us to define and to study the Stratonovich-Weyl symbol of $d\pi (X)$ for $X\in \mathfrak{g}$.
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