The Unitary Irreducible Representations of SL(2,C)

The Lie group SL(2,C) is the simply-connected cover of the Lorentz group. Understanding its representation theory is therefore vital to understanding Lorentz-invariant theories. The standard irreps that one encounters in QFT are indexed by two half integers, which correspond to left and right chirality. These irreps have one major drawback, namely that they aren't unitary. There do actually exist unitary irreps of SL(2,C). These are the infinite dimensional Principal and Supplementary Series representations. In this talk I will show how they can be induced from representations of the parabolic subgroups of SL(2,C) using Mackey analysis.