Spaces, Actions, Representations and Curvature

Abstract

The present chapter contains background material. We fix conventions on the structure of the Lie group SO(1, n) and its Lie-algebra. Starting from the light-cone model, we derive formulas for its actions on Sⁿ−1 and ℝⁿ⁻¹ ∪ {∞} using stereographic projection. These actions induce two models of the spherical principal series representations. We define these in a form which emphasizes the con-formal nature of the geometric actions with respect to canonical metrics in each model. Moreover, we prove the equivariance of the corresponding Poisson transformations. These intertwine the principal series representations with eigenspace representations for the Laplacian on hyperbolic space. Finally, we derive the standard transformation laws for the Riemannian curvature, Ricci curvature and scalar curvature with respect to conformal changes of the metric.