Reduction of the principal series representation of Lorentz groups on an Abelian subgroup

Abstract

Let G be a locally compact group and G'CG a subgroup. Suppose that we are given an irreducible unitary representation Γ of G (which is infinite dimensional and is specified by the so-called principal series representation) and we wish to find out how this representation reduces on the subgroup G'. In the special case where the representation Γ of G is an induced representation, then one can first apply Mackey's subgroup theorem. This gives a representation Γ’of G* defined on a Hilbert space 𝓗. However it often happens that Γ is not an irreducible representation of G , nor even a direct sum of irreducible representations, but is a "direct integral" of irreducible representations. In this work, a unitary transformation will be introduced that maps 𝓗’ into another Hilbert space 𝓗’^ which is a direct integral of the Hilbert spaces of the irreducible representation that appear in the reduction of the principal series representation Γ on G’. The group G under consideration is taken to be SL(2,C) which is the universal covering of Lorentz Group , 1 B and G an Abelian subgroup (1,0 β 1)which is called the horospherical subgroup of SL(2,C) and we seek the reduction as mentioned above.