I encountered the following definition:- "The $Spin^{\mathbb{C}}(p,q)$ group is $$Spin^{\mathbb{C}}(p,q):=Spin(p,q)\times U(1)\bigg/(a,e^{i\theta})\sim(-a,-e^{i\theta})$$ and I also found the following statement:
There exists a well defined double cover $$r^{\mathbb{C}}:Spin^{\mathbb{C}}(p,q)\rightarrow SO(p,q)\times U(1)$$ $$[(a,e^{i\theta})]\mapsto(r(a),(e^{i\theta})^2)$$ where, given $a\in\Gamma(p,q)$(the space of all invertible element of $Cl(p,q)=\mathbb{R}^{n}$),I used $$r(a):x\mapsto r(a)(x):=\alpha(a)xa^{-1}\equiv -axa^{-1}$$
This was presented as a Corollary, so there's no proof. It seems that the statement should be trivial, can someone explain it? I think the problem lies in the fact that I've still not very clear how a "double cover" is defined.
It means the index of the kernel of the group homomorphism has index $2$ in its group.
That means that each element of the image has two things in the group that map to it. Modulo the kernel these two things are the same thing.
So each element of the image is "covered" by two things from the domain.