CHAPTER III - REPRESENTATIONS OF THE GROUP OF COMPLEX UNIMODULAR MATRICES IN TWO DIMENSIONS

Abstract

This chapter focuses on the development of the representation theory and harmonic analysis for the group G of two-dimensional complex matrices with determinant αδ – βγ = 1. This group is interesting for several reasons. It is isomorphic to important groups such as the Lobachevskian motions and the Lorentz transformations. It is the simplest of the so-called simple Lie groups, which clearly exhibit the difference between harmonic analysis on the group and harmonic analysis in Euclidean space. There exists a correspondence between the two-dimensional complex unimodular matrices and the proper Lorentz transformations. This correspondence is such that one Lorentz transformation corresponds to each two-dimensional complex unimodular matrix g, and two matrices correspond to each Lorentz transformation, differing only in sign. This correspondence preserves multiplication. The group of complex two-dimensional unimodular matrices is locally isomorphic to the group of Lobachevskian motions. The group of motions on each of these surfaces is transitive, that is, every point of the space can be transformed by some motion to any other point.