Invariant metric on the extended Siegel-Jacobi upper half space

Abstract

The real Jacobi group $G_n^J(\mathbb{R})$, defined as the semidirect product of the Heisenberg group ${\mathrm{H}}_n(\mathbb{R})$ with the symplectic group ${\mr {Sp}}(n,\mathbb{R})$, admits a matrix embedding in $\text{Sp}(n+1,\mathbb{R})$. The modified pre-Iwasawa decomposition of $\mathrm{Sp}(\mathrm{n},\mathbb{R})$ allows us to introduce a convenient coordinatization $S_n$ of $G_n^J(\mathbb{R})$, which for $G_1^J(\mathbb{R})$ coincides with the $S$-coordinates. Invariant one-forms on $G_n^J(\mathbb{R})$ are determined. The formula of the 4-parameter invariant metric on $G_1^J(\mathbb{R})$ obtained as sum of squares of 6 invariant one-forms is extended to $G_n^J(\mathbb{R})$, $n\in \mathbb{N}$. We obtain a three parameter invariant metric on the extended Siegel-Jacobi upper half space ${\tilde{\mathcal{X}}}_n^J\approx {\mathcal{X}}_n^J\times \mathbb{R}$ by adding the square of an invariant one-form to the two-parameter balanced metric on the Siegel-Jacobi upper half space $ {\mathcal{X}}^J_n =\frac{G^J_n(\mathbb{R})}{\mr{U}(n)\times\mathbb{R}}$.

28 pages, Latex, amsart, AMS fonts

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• A holomorphic representation of the multidimensional Jacobi algebra
• Realization of coherent state Lie algebras by differential operators
• A convenient coordinatization of Siegel-Jacobi domains
• Quantization of Kähler manifolds
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Cited by in corpus (2)

• Hamiltonian systems on almost cosymplectic manifolds
• Geodesics on the extended Siegel-Jacobi upper half-plane