The de Sitter group and its presence at the late-time boundary

TL;DR

This paper introduces the representation theory of the de Sitter group SO(d+1,1), focusing on its properties and applications at the late-time boundary, including inner products and tensor product constructions.

Contribution

It provides an accessible review of key intertwining maps and demonstrates their practical use in boundary representations of de Sitter space.

Findings

Explicit examples of complementary series inner product at the late-time boundary

Construction of tensor product representations from principal series irreducibles

Clarification of the role of intertwining maps in de Sitter representation theory

Abstract

Our main goal here is to provide an introduction on some of the well established properties of the representation theory of SO(d+1,1), for those considering to think on physical problems set in de Sitter space in terms of these representations. With this purpose we review two intertwining maps, the map G that is used in constructing a well defined inner product for the complementary series representations and the map Q that is involved in constructing composite representations. We give explicit examples from the late-time boundary of de Sitter on the practical use of the complementary series inner product and in building a tensor product representation from unitary principal series irreducible representations.