Abstract
Let \(G\) be a quasi-Hermitian Lie group and let \(K\) be a maximal compactly embedded subgroup of \(G\). Let \(\pi \) be a unitary representation of \(G\) which is holomorphically induced from a unitary representation \(\rho \) of \(K\). Then \(\pi \) is usually realized in a Hilbert space of vector-valued holomorphic functions. Here we introduce a realization of \(\pi \) in a reproducing kernel Hilbert space of complex-valued functions and we compute the (scalar-valued) coherent states. We study the corresponding Berezin quantization map and, in particular, we show that this map is equivalent to that introduced by Cahen (Rend Istit Mat Univ Trieste 46:157–180, 2014).
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Acknowledgments
I would like to thank the referee for some pertinent remarks and for pointing out the recent book [28] to me.
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Communicated by Salvatore Rionero.
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Cahen, B. Coherent states and Berezin quantization for non-scalar holomorphic representations. Ricerche mat. 64, 115–135 (2015). https://doi.org/10.1007/s11587-015-0223-2
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DOI: https://doi.org/10.1007/s11587-015-0223-2
Keywords
- Berezin quantization
- Berezin transform
- Coherent states
- Quasi-Hermitian Lie group
- Coadjoint orbit
- Unitary holomorphic representation
- Reproducing kernel Hilbert space
- Stratonovich–Weyl correspondence
- Heisenberg motion group