Singular unitary representations of classical groups (Q909046)

In this interesting paper, the author gives a description of a part of the unitary dual of a type I classical group over a local field of characteristic \(\neq 2\), arising from R. Howe's duality correspondence between dual reductive pairs. Given such a pair \((G,G')\) inside a symplectic group, let \(\bar Sp\) be the metaplectic 2-fold cover and let Z be the kernel of the covering map. Let \(G(\epsilon)\) (resp. \(G'(\epsilon))\) be the subset of the unitary dual of G (resp. \(G')\) which restricts non-trivially to Z. The main theorem in the paper states that in the stable range, Howe's duality gives rise to an injection \(\hat{\bar G}(\epsilon)\to \hat{\bar G}(\epsilon)\). This extends a previous result by \textit{R. Howe}, in the case of the pair \((G,G')=(Sp_{2n}(R),O_{p,q}(R))\) with \(p+q\leq n-2\) [see Proc. Conf. honor G. Mackey 1984, Publ., Math. Sci. Res. Inst. 6, 121-150 (1987; Zbl 0635.22014)]. Furthermore, in a subsequent paper [Compos. Math. 71, 29-48 (1989; see the following preview Zbl 0694.22012)], the author shows that the image of this map can be characterized using the notion of low rank representations, due to R. Howe.