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**Stratonovich-Weyl correspondence for discrete series representations.**
*(English)*
Zbl 1240.22011

The original idea of quantization attempts to assign to real-valued functions (observables) in the phase space \(\mathbb {R}^n \times \mathbb {R}^n\) in a suitably compatible way. A particular version is known as the Weyl correspondence. The Stratonovich-Weyl correspondence is a generalization which equips the phase space with a symmetry group. Specifically, for the Lie group \(G\) with a unitary representation \(\pi \) on a Hilbert space \(\mathcal {H}\) and a homogeneous \(G\)-space \(M\), it provides a suitably compatible isomorphism from a vector space of operators on \(\mathcal {H}\) to a space of (generalized) functions on \(M\) (see [J. M. Gracia-Bondia, Contemp. Math. 134, 93–114 (1992; Zbl 0788.58024); S. T. Ali and M. Engliš, Rev. Math. Phys. 17, No. 4, 391–490 (2005; Zbl 1075.81038)] for the general picture).

The article under review focuses on the case where \(G\) is a connected semisimple noncompact real Lie group with finite center, \(M = G/K\) with \(K \subseteq G\) a maximal compact subgroup and \(\pi \) is a discrete series representation. Then \(M\) is diffeomorphic to the bounded symmetric domain \(\mathcal {D}\), and \(\pi \) can be realized on a Hilbert space \(\mathcal {H}\) of holomorphic functions on \(\mathcal {D}\). Following the compact case, the Stratonovich-Weyl correspondence is then obtained by suitable extension of the Berezin transform. The case of holomorphic discrete series of \(G = SU(1,1)\) is worked out in detail.

The article under review focuses on the case where \(G\) is a connected semisimple noncompact real Lie group with finite center, \(M = G/K\) with \(K \subseteq G\) a maximal compact subgroup and \(\pi \) is a discrete series representation. Then \(M\) is diffeomorphic to the bounded symmetric domain \(\mathcal {D}\), and \(\pi \) can be realized on a Hilbert space \(\mathcal {H}\) of holomorphic functions on \(\mathcal {D}\). Following the compact case, the Stratonovich-Weyl correspondence is then obtained by suitable extension of the Berezin transform. The case of holomorphic discrete series of \(G = SU(1,1)\) is worked out in detail.

Reviewer: Josef Šilhan (Brno)

### MSC:

22E46 | Semisimple Lie groups and their representations |

81S10 | Geometry and quantization, symplectic methods |

46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |

32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |