On the universal covering group of the real symplectic group (Q441213)

The real symplectic group \(\text{Sp}(2n,\mathbb R)\) has a number of non-trivial central extensions such as the metaplectic double covering \(\text{Mp}(2n,\mathbb R)\), an extension by \(\mathbb Z_2\), the circle extension \(\text{Mp}^c(2n,\mathbb R)\) as the automorphism group of the Heisenberg group, and the universal covering group \(\widetilde{\text{Sp}}(2n,\mathbb R)\) an extension by \(\mathbb Z\). These extensions do not have faithful finite dimensional representations, so there are no nice models for them as Lie groups of matrices. In [Mem. Am. Math. Soc. 410, 92 p. (1989; Zbl 0681.22022)], \textit{P.~L.~Robinson} and the present author used an explicit model for the circle extension \(\text{Mp}^c(2n,\mathbb R)\) of the symplectic group \(\text{Sp}(2n,\mathbb R)\) to facilitate the definition of a theory of symplectic spinors on any symplectic manifold and to extend the Kostant theory of metaplectic half-forms to this case. The group manifold can be described explicitly as a hypersurface in \(\text{Sp}(2n,\mathbb R)\times\mathbb{C}^*\) and the group multiplication given by a single global formula.NEWLINENEWLINEIn this interesting paper, the author shows that the method used can be adapted to other central extensions and does this for the universal covering group \(\widetilde{\text{Sp}}(2n,\mathbb R)\). It is shown how to make models for the universal covering manifold using suitable maps to the circle and to write the group multiplication in terms of an associated cocycle. Then, the author shows how to construct a particularly nice explicit circle map and obtains the formula for its cocycle. As an application, the author gives the universal covering group of \(\text{Sp}(2,\mathbb R)=\text{SL}(2,\mathbb R)\) and the inverse image of \(\text{SL}(2,\mathbb Z)\) in it. It is a theorem of Milnor stating that this extension of \(\text{SL}(2,\mathbb Z)\) is isomorphic to the braid group on three strands. Finally, the author shows how the same methods can be used to construct the universal covering manifold of the Lagrangian Grassmannian and the Maslov Index.