CHAPTER VI - HARMONIC ANALYSIS ON SPACES HOMOGENEOUS WITH RESPECT TO THE LORENTZ GROUP

Abstract

This chapter discusses harmonic analysis on spaces that are homogenous to the Lorentz group. A group G is called a transformation group of some space X if to every element g ∈ G, there corresponds a one-to-one bicontinuous transformation of X. If G is continuous, then one also usually requires that the mapping x→ xg be continuous with respect to g for all x. If for every pair of points x, y ∈ X there exists g ∈ G such that y = xg, G is called a transitive group of motions of X, and X itself is said to be homogeneous with respect to G. It is particularly important to study integral transforms that simplify a representation when dealing with representation theory. These transforms are related to going over from points of the homogeneous space X to new geometrical objects in this space. The role of integral geometry in representation theory lies just in this transition from the space of certain geometrical objects to the space of others. In most homogeneous spaces, there exist geometrical objects such that this kind of transition introduces a significant simplification into the representation.