Representation theory of SO(1, d-1) and O(1, d-1)

Question

Under a Lorentz transformation, a spinor living in \(d\) dimensions transforms as

\(\psi (x) \rightarrow \psi'(x') = e^{\frac{1}{2} \lambda^{\mu \nu} \Sigma_{\mu \nu}} \psi (x)\)

up to some numerical factors in the exponential from convention. This is only true if we're dealing with the orthochronous proper Lorentz transforms \(SO^+ (1,3)\), because the projective spinor representations mean we can just deal with the algebra, and the \(\Sigma_{\mu \nu}\) are representations of \(so (1, d-1)\). How does this change when we want to think about \(P\) and \(T\)? That is, how does one derive the action of elements of \(O (1, d-1)\) on spinors?

The only progress I've made towards understanding this is the coordinate-dependent interpretation put forth in standard texts on QFT, where \(P\)  and \(T\) act with some product of gamma matrices. I was looking for a slightly more general definition.

A related question: is the number of of disconnected components of \(O (1, d-1)\) the same for even and odd dimension?

Comments

For $d \geq 2$, $O(1,d-1)$ has always four connected components.