Degenerate Representations of the Symplectic Groups II. The Noncompact Group Sp(p, q)

Abstract

Using a method based on geometrical properties of homogeneous spaces of rank one homeomorphic to coset spaces of Lie groups, a series of degenerate unitary irreducible representations of the noncompact symplectic group Sp(p, q) is investigated. The representation spaces for a discrete series determined by two integer numbers and a continuous series determined by one real and one integer parameter are given, the corresponding basis functions being formed by the linear combinations of eigenfunctions of the Laplace-Beltrami operator of the considered space. Explicit formulas for the action of generators of Sp(p, q) in these representations are obtained. The results provide a deeper insight into the structure of the two-parameter ``not most degenerate'' unitary representations.