Geodesic
Stratonovich-Weyl correspondence for discrete series representations
Archivum mathematicum, Tome 47 (2011) no. 1, pp. 51-68

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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 $X\in \mathfrak{g}$.
Classification : 22E46, 32M15, 46E22, 81S10
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 
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     author = {Cahen, Benjamin},
     title = {Stratonovich-Weyl correspondence for discrete series representations},
     journal = {Archivum mathematicum},
     pages = {51--68},
     publisher = {mathdoc},
     volume = {47},
     number = {1},
     year = {2011},
     mrnumber = {2813546},
     zbl = {1240.22011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2011__47_1_a4/}
}
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Cahen, Benjamin. Stratonovich-Weyl correspondence for discrete series representations. Archivum mathematicum, Tome 47 (2011) no. 1, pp. 51-68. http://geodesic.mathdoc.fr/item/ARM_2011__47_1_a4/