Weil Representation and Metaplectic Groups over an Integral Domain
Given F a locally compact, nondiscrete, non-archimedean field of characteristic ≠ 2 and R an integral domain such that a non-trivial smooth character χ: F → R × exists, we construct the (reduced) metaplectic group attached to χ and R. We show that it is in the expected cases a double cover of the symplectic group over F. Finally, we define a faithful infinite dimensional R-representation of the metaplectic group analogue to the Weil representation in the complex case.