I have a vague question regarding the Spin representations.
Is there a "quick" way of seeing that $Spin(2n)$ has exactly two irreducible representations which do not factor through $SO(2n)$, and one for the odd case?
Of course, there is the classification of representations and one can write out all the representations for both the groups and see that this is indeed the case. But I was wondering if without doing all that just by using the fact that $Spin(2n)$ double covers $SO(2n)$ can one see this?
Even more vaguely, is there some kind of a Galois correspondence between the irreducible representations of compact Lie groups and $Spin(2n+1)$ is not a Galois cover of $SO(2n+1)$?
Thank you.