Structure and matrix elements of the degenerate series representations of U(p+q) and U(p,q) in a U(p)×U(q) basis

Abstract

The representations of the most degenerate series of the group U(p,q) which are induced by the representations of the maximal parabolic subgroup are considered in this article. By making use of the infinitesimal operators of these representations in the U(p)×U(q) basis the conditions are derived which are necessary and sufficient for irreducibility. For the reducible representations we describe their structure (composition series). We select from among the irreducible representations which are obtained in this article all representations of U(p,q) which admit unitarization. As a result we obtain the principal degenerate series, the supplimentary degenerate series, the discrete degenerate series, and the exceptional degenerate series of unitary representations of U(p,q). The U(p)×U(q) spectrum of the representations of U(p+q) with highest weights (λ₁, 0,..., 0, λ₂) is defined. We obtain the integral representation for the matrix elements of the degenerate representations of U(p,q) in the U(p)×U(q) basis. The matrix elements of the irreducible representations of U(p+q) with highest weights (λ ,0,...,0), (0,...,0, λ) are evaluated in the U(p)×U(q) basis.