Spinor reps in $\mathbb{R}^{1,3}\times{}B$ space-times

Question

I am considering spinors in a space-time which is $\mathbb{R}^{1,3}\times{}B$ being $B$ a compact manifold of $D$ dimensions. 

I know that in ordinary 4 dimensional space-time spinors are representations of $O(1,3)$. Now, in my case, are spinors expected to be reps of $O(1,3+D)$?

Does the compactness of $B$ impose some restrictions to this? I have the feeling that we have to expect spinors to be reps of $O(1,3)\times{}O(D)$ since I don't feel that making a boost in the compact space is allowed but I am not sure.

Any clarification on the reps spinors are in in the mentioned space-time will be greatly appreciated.