The dual pair (U(3),U(1)) over a p-adic field
This paper treats some interesting aspects of Howe's correspondence for the dual reductive pair (U(3),U(1)) over a p-adic field, with p odd. Roughly speaking, the idea is this: one embeds the group U(3) (with U(1) embedded as the center) inside the rank 3 symplectic group \(Sp_ 3\), and considers the restriction of the canonical metaplectic representation \(\omega_{\psi}\) of \(Sp_ 3\) to this unitary subgroup; according to Howe's general theory, we have a decomposition \[ \omega_{\psi}|_{U(3)}=\oplus_{\chi}\pi_{\chi} \] where \(\chi\) runs through the character group of U(1) (which is isomorphic to the norm 1 group of the quadratic extension E over which U(1) and U(3) are defined), and \(\pi_{\chi}\) is an irreducible admissible representation of U(3). The problem considered in this paper is to describe \(\pi_{\chi}\) as explicitly as possible, in terms of \(\chi\). First it is shown that \(\pi_{\chi}\) is supercuspidal (not a constituent of any induced representation ind \(\nu)\) if and only if \(\chi\) is ``ramified'' with respect to the additive character \(\psi\) defining \(\omega_{\psi}\) (the precise condition is that \(\chi\) should not occur in the restriction to U(1) - embedded as a compact Cartan subgroup - of the canonical Weil representation \(\omega^ 1_{\psi}\) of \(SL_ 2(F))\). In case \(\pi_{\chi}\) is not supercuspidal, the author gives an explicit description of the character \(\nu^*\) of \(E^ x\) such that \(\pi_{\chi}\) is an irreducible constituent of Ind \(\nu^*| |^{1/2}\). On the other hand, if \(\pi_{\chi}\) is supercuspidal, then the author explicitly constructs some representation \(\sigma\) of some maximal compact subgroup of U(3) such that \(\pi_{\chi}=ind \sigma\); here it seems that the group over the ring of integers does not suffice - in almost all cases of E and \(\chi\), the other class of maximal compact subgroups is required. Reviewer's Remarks. The Weil representation \(\omega_{\psi}|_{U(3)}\) is actually obtained in two states. First U(3) is embedded inside the rank 3 symplectic group Sp(V); then an appeal is made to the (non-trivial) fact that the two-cocycle determined by the canonical Weil representation of Sp(V) is cohomologically trivial on U(3), provided cohomology is consided with values in the torus \(T=\{z\in {\mathbb{C}}:\) \(| z| =1\}\). It is the reviewer's impression that the author incorrectly applied this last fact, since he assumed a splitting in \(\{\pm 1\}\) rather than T; it would therefore follow that some of the author's explicit results on principal series embeddings need to be rewritten.
scientific article; zbMATH DE number 4105259