Unitary highest weight representations via the orbit method. I: The scalar case
The purpose of the paper is to describe realizations of unitary highest weight representations of an Hermitian simple Lie group \(G\) by using the orbit method. Let \(Q\) be a parabolic subgroup associated with an \(sl_2\)-triple \((H,X,Y)\) and a subgroup \(Q_{\text{red}}\) of \(Q\). By the geometrization process the authors associate to the adjoint orbit \({\mathcal O}_X\) a unitary representation of \(Q_{\text{red}}\). Since the orbit is of convex type, i.e. is contained in a pointed convex cone, the Laplace transform can be used to realize this representation in a space \({\mathcal H}\) of holomorphic functions whose reproducing kernel is computed. Further this representation is extended to the subgroup \(Q\). Then one sees that the Hilbert space \({\mathcal H}\) is related to a unitary highest weight representation of \(G\) by comparing the reproducing kernels.
scientific article; zbMATH DE number 943745
scientific article