Following on this question about how to characterise Spinors mathematically:
First, given a universal cover $\pi:G' \rightarrow G$ of a lie group $G$, is it correct to say we can always lift representations of $G$ to those of $G'$ essentially by pre-composition by $\pi$? (presumably, modulo questions about the exact smooth structure of the space $End (V)$ for a representation $V$ of $G$).
Secondly, there are representations of $G'$ that do not descend to $G$.
Is it fair to call these Spinor representations, since when $G$ is the connected component of the identity of $SO(p,q)$, its universal cover is the double cover $Spin(p,q)$?