Irreducibility of some unitary representations of the poincaré group with respect to the Poincaré subsemigroup, II

Abstract

Let P+(3) and P+(3) be the 3-dimensional space-time Poincaré group and the Poincaré subsemigroup, that is, P(3) = R3 × sSU(1, 1) and P+(3) = V+(3)=SSU(1, 1) where The multiplication is defined by the formula (x, g)(x′, g′) = (x + g*−1x′g−1, gg′) for x, x′ ∈ R3 and g, g′ ∈ SU(l, 1). Here x = (x0, x1, x2) is identified with the matrix

Decomposition of representations of SL(2, C) induced by the continuous series of E(2)

“…We refer to [6] (especially, § 3) the notations in the following. As is verified easily, the operators ω 3 corresponding to the representation ϋ^'…”

“…for fe H* with respect to the representation U** e . Quite similarly to the §4 of[6], we obtain the following table.…”

“…As is known (see, for example, § 3 of [6]), the problem to decompose a unitary representation of SL{2, C) into irreducible ones can be reduced to the problem to specify the spectral type of certain selfadjoint operators. In our case we must deal with ordinary differential operators Lg and L p k 0>0 and k = 1/2,1,3/2, •••):…”

“…A representation (C7, £Γ) of G determines a sequence of mutually singular σ-finite measures {σ u σ 2 , , σ TO } on G = J; =o ^T O (see the preceding paragraph of Lemma 2 of [6]) such that…”

“…(see the preceding paragraph of Lemma 3 of [6]). Denoting by σ jtaCtP the sum of the absolutely continuous and purely discontinuous parts in Lebesgue's decomposition of the measure σ j9 a representation U aCtP of G is understood to be one which is unitarily equivalent to…”