The two-dimensional symplectic and metaplectic groups and their Universal cover

Abstract

We give a detailed discussion of the group ${\rm Sp}(2,\bold R)$, organized in such a way as to lead to explicit constructive descriptions of the metaplectic group ${\rm Mp}(2)$ and the universal covering group $\overline{{\rm Sp}(2,\bold R)}$. The aim is to make clear in easily visible fashion the global topological relationships among these groups of physical relevance, and to make practical calculations with them feasible. The properties of one-parameter subgroups and the exponential map, and of the Iwasawa decomposition, are also investigated in detail for these groups.