The dual pair \((U(1),U(1))\) over a \(p\)-adic field
Let \(F\) denote a \(p\)-adic field with \(p \neq 2\) and \(n\) a natural number. The metaplectic group \(MSL_ n(F)\) is the nontrivial two-sheeted covering group of \(SL_ n(F)\). The metaplectic representation was introduced by A. Weil and it rules the theory of theta series and theta correspondences. A theta correspondence is thus given by the restriction of the metaplectic representation to a dual reductive pair \((G,G')\) in \(MSL_ n (F)\) which defines a correspondence between representations of \(G\) and representations of \(G'\). In the paper under consideration one of the simplest cases: \(n=2\), \(G=G'=U(1)\) is treated and the restriction of the metaplectic representation to \(G\) is explicitly computed.
scientific article; zbMATH DE number 509109