Stratonovich-Weyl correspondence for the Jacobi group (Q2925403)

The Stratonovich-Weyl correspondence for the Jacobi group of index one, \(G^J_1=\mathrm{SU}(1,1)\ltimes H_1\), is studied in detail, where \(H_1\) denotes the 3-dimensional Heisenberg group. The general case of a quasi-Hermitian Lie group was presented in [\textit{B. Cahen}, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 52, No. 1, 35--48 (2013; Zbl 1296.22007)], with a short exemplification for the Jacobi group of index \(n\), \(G^J_n\). The scheme for the construction of a holomorphic representation is taken from [\textit{K.-H. Neeb}, Holomorphy and convexity in Lie theory. Berlin: de Gruyter (1999; Zbl 0936.22001)] and the previous paper of \textit{B. Cahen} [Rend. Semin. Mat. Univ. Padova 129, 277--297 (2013; Zbl 1272.22007)], from where the theorems are extracted which give the Berezin transform \(S_{\chi}\), where the reproducing kernel Hilbert space \(\mathcal{H}_{\chi}\) has the reproducing kernel parametrized by \(\gamma\in\mathbb{R}\) and \(m\in\mathbb{Z}\). The Jacobi group is a group of Harish-Chandra type with associated homogeneous space the Siegel-Jacobi disk \(\mathcal{D}^J_1=\mathbb{C}\times\mathcal{D}_1\), where \(\mathcal{D}_1\) denotes the Siegel disk \(\{z\in\mathbb{C}| |z|<1\}\). The polar decomposition of \(S_{\chi}=(S_{\chi}S_{\chi}^*)^{1/2}W_{\chi}\) gives the Stratonovich-Weyl correspondence \(W_{\chi}\). The diffeomorphism \(\Psi_{\chi}\) from \(\mathcal{D}^J_1\) to a certain Kostant-Kirillov orbit \(\mathcal{O}(\xi_X)\) is described explicitly. The Berezin transform is extended such that it can be applied to the derived representation \(d\pi_{\chi}(X)\) for \(X\) in the Lie algebra of \(G^J_1\). This allows the author to obtain the Stratonovich-Weyl symbols for the derived representation of the Jacobi group \(G^J_1\).