\(\Theta\)-correspondences \((U(1),U(2))\). II: Ramified case (Q2568212)

This paper is a continuation of Part I [J. Number Theory 111, No. 2, 287--317 (2005; Zbl 1084.11018)]. Here the author considers the case when \(E\) is a \textbf{ramified} quadratic extension of \(F\) and proves the following: {\(\bullet\)} Suppose \(-1\) is a square in \(F\). Then 1 all, but one, smooth characters of \(U(1)\) (resp. \(U(2)\)) occur in the Weil representation; 2 any smooth irreducible representation \(\rho_{\alpha,\varphi,1}\) of \(U(2)\) such that \(\alpha=a^2\pi^{-2r-1}\), with \(a\in{\mathfrak o}_F^\times\) and \(r\geq 1\) an integer, occurs in the Weil representation; 3 any smooth irreducible representation \(\rho_{-\alpha,\bar\varphi,\bar\varphi}\) of \(U(2)\) such that \(\alpha=a\pi^{-2r-1}\), with \(a\) a non-square element in \({\mathfrak o}_F^\times\), \(r\geq 1\) an integer, occurs in the Weil representation; 4 the trivial character of \(U(2)\) occurs in the Weil representation; 5 none of the other representations of \(U(2)\) occur in the Weil representation. {\(\bullet\)} Suppose \(-1\) is not a square in \(F\). Then 1 all, but one, smooth characters of \(U(1)\) (resp. \(U(2)\)) occur in the Weil representation; 2 any smooth irreducible representation \(\rho_{\alpha,\varphi,1}\) of \(U(2)\) such that \(\alpha=a^2\pi^{-2r-1}\), with \(a\in{\mathfrak o}_F^\times\) and \(r\geq 1\) an even integer, occurs in the Weil representation; 3 any smooth irreducible representation \(\rho_{\alpha,\varphi,1}\) of \(U(2)\) such that \(\alpha=a\pi^{-2r-1}\), with \(a\) a non-square element in \({\mathfrak o}_F^\times\) and \(r\geq 1\) an odd integer, occurs in the Weil representation; 4 any smooth irreducible representation \(\rho_{-\alpha,\bar\varphi,\bar\varphi}\) of \(U(2)\) such that \(\alpha=a\pi^{-2r-1}\), with \(a\) a non-square element in \({\mathfrak o}_F^\times\), \(r\geq 2\) an even integer, occurs in the Weil representation; 5 any smooth irreducible representation \(\rho_{-\alpha,\bar\varphi,\bar\varphi}\) of \(U(2)\) such that \(\alpha=a^2\pi^{-2r-1}\), with \(a\in{\mathfrak o}_F^\times\), \(r\geq 2\) an odd integer, occurs in the Weil representation; 6 the trivial character of \(U(2)\) occurs in the Weil representation; 7 none of the other representations of \(U(2)\) occur in the Weil representation.