La relation entre \(Sp_{\infty}\), revêtement universel du groupe symplectique Sp, et Sp\(\times {\mathbb{Z}}\). (The relation between \(Sp_{\infty}\), the universal covering of the symplectic group Sp, and Sp\(\times {\mathbb{Z}})\). (French) Zbl 0732.22001

de Gosson, Maurice

C. R. Acad. Sci., Paris, Sér. I 310, No. 5, 245-248 (1990).

This note contains the proof of a relation between the universal covering \(Sp_{\infty}\) of the symplectic group Sp and Sp\(\times {\mathbb{Z}}\). The relation has been stated without proof in G. Lion and M. Vergne’s book “The Weil representation, Maslov index and Theta series” (1980; Zbl 0444.22005) and the present justification is based on the definition and properties of the Maslov index given in a previous note of the author [C. R. Acad. Sci., Paris, Sér. I 310, No.5, 279-282 (1990; Zbl 0705.22012)].
A multiplication rule is defined in the set \((Sp\times {\mathbb{Z}})_{\ell}\) and a topology is introduced so that it becomes a topological group \(\tilde G_{\ell}\), for each \(\ell \in \Lambda\) (the Grassmannian Lagrangian). Then, it is proved that the group \([Sp\times {\mathbb{Z}}]_{\ell}\) (isomorphic to \(Sp_{\infty})\) is a connected subgroup of \(\tilde G_{\ell}\) and contains the unit element. It is shown also that \(\tilde G_{\ell}\) has four connected homeomorphic components. The component including the unit element coincides with \(Sp_{\infty}\) and \(\tilde G_{\ell}/Sp_{\infty}={\mathbb{Z}}_ 4\).

Reviewer: G.Zet (Iaşi)

MSC:

22E10	General properties and structure of complex Lie groups
53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
53D50	Geometric quantization
22E45	Representations of Lie and linear algebraic groups over real fields: analytic methods

Keywords:

universal covering; symplectic group; Maslov index; topological group; Grassmannian Lagrangian; connected subgroup

Citations:

Zbl 0444.22005; Zbl 0705.22012