Transfers of \(K\)-types on local theta lifts of characters and unitary lowest weight modules
The authors study representations of the indefinite orthogonal group \(O(n,m)\) which are local theta lifts of one-dimensional characters or unitary lowest weight modules of the double covers of the symplectic groups. They apply the transfer of \(K\)-types on these representations of \(O(n,m)\). Introducing the Harish-Chandra module \(\theta_p^{n,m}(1)\) of \(O(n,m)\), they consider \(K_{n,m}=O(n)\times O(m)\), a maximal compact subgroup of \(O(n,m)\), and the \(K_{n,m}\)-types of \(\theta_p^{n,m}(1)\). Another ingredient is the subgroup \(O(n)\times O(r)\otimes O(m-r)\) of \(K_{n,m}\) and the restriction of \(\theta_p^{n,m}(1)\). With these objects, they apply the transfer of \(K\)-types due to Enright and Wallach (see, e.g., [\textit{T. Enright} et al., Prog. Math. 40, 97--143 (1983; Zbl 0535.22012)] and study their effects on the dual pair correspondences. Finally, they also use these results to study subquotients of some cohomologically induced modules. The techniques presented in this paper can be applied to \(U(n,m)\) and \(Sp(n,m)\).
scientific article; zbMATH DE number 6361292
scientific article