An example of holomorphically induced representations of exponential solvable lie groups

Abstract

We discuss a holomorphically induced representation ρ = ρ(f, h) of Boidol’s group (split oscillator group) G from a real linear form f of the Lie algebra g of G and a one-dimensional complex subalgebra h of gC given by (2.2) in Sect. 2. ρ is a subrepresentation of the regular representation of G with the Plancherel measure ν. For ν -almost all irreducible representations π of G, the spaces of generalized vectors satisfying the semi-invariance associated with f and h are one-dimensional subspaces. On the other hand, according to the choice of f, there are two cases that (1) ρ vanishes, and (2) ρ is non-zero.