Reductive dual pair correspondence for complex groups
The authors give an explicit determination of the representation correspondence for complex reductive dual pairs \(G_1\), \(G_2\). The case of orthogonal/symplectic pairs over \(\mathbb{R}\) was treated by \textit{C. Moeglin} [ibid. 85, 1-85 (1989; Zbl 0729.22017)]. She first determined the correspondence for certain basic cases, which are discrete series in which the minimal \(K\)-type is also degree-lowest, and then used the induction principle to extend the results to a large class of representations. The authors treat the complex case by using one-dimensional representations of \(G_1\) for the basic set instead of the discrete series, and obtain the full correspondence from the one for the basic cases by application of the induction principle.
scientific article; zbMATH DE number 790373