Operators and measures

Abstract

This chapter deals with operators and measures. The Casimir operator Ξ on PSO(1, d), i.e., the second-order differential operator associated with the Killing form, is the fundamental operator of the theory. It induces the Laplace operator D on the affine group A d , and the hyperbolic Laplacian ∆. After proving some fundamental properties of Haar measures on groups, we determine the Haar measure of PSO(1, d). The chapter ends with a presentation of harmonic, Liouville and volume measures. These can all be derived from the Haar measure, and their analytical expressions are derived in this way.