Reference request: indefinite orthogonal groups $O(p,q)$, spin groups $\mathrm{Spin}(p,q)$, and projective orthogonal groups $PO(p,q)$

Question

The indefinite orthogonal group $O(p,q)$ is the orthogonal group preserving the standard scalar product of signature $(p,q)$ on $\mathbb{R}^{p+q}$.

Are there any good references that discuss the properties of these indefinite orthogonal groups and related groups such as indefinite spin groups and projective orthogonal groups? Some things that I'm looking for include:

• the representation theory of $O(p,q)$ and its orbits in the standard representation
• the spin groups $\mathrm{Spin}(p,q)$ and their representation theory
• the universal cover of the identity component $O_0(p,q)$
• the projective groups $PGO(p,q)$ and $PSO(p,q)$, and their representation theory

Answer 1

References or addressing some of your points are given in answers to this question.