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.