Deformation of the SO(2, C) subgroup of the Lorentz group

Abstract

The two-dimensional complex sphere S12+S22+S32=S2 forms a homogeneous space under the SL(2,C) group. The little group of a point in this space is the SO(2,C) group or the horospheric group T(2), according to whether S ≠ 0 or S = 0. Deformation of the SO(2,C) group into T(2) is investigated and is demonstrated on unitary representations. This deformation is a counterpart to that of the little groups SO(3) or SO(2,1) into E(2). We conclude with a formula relating the matrix elements of unitary representations of the SL(2,C) group in SO(2) × SO(1,1) = SO(2,C) basis to those in horospheric basis.