My question is about the relation between representations of a Lie group and its quotient.
Let $G$ be a compact lie group and $H$ a central subgroup of $G$. What is the relation between representations of $G$ and those of $G/H$? Exactly, I want to know that, is any representation of $G/H$ correspond to a representation of $G$ which is trivial on $H$?
* All representations are in a fixed vector space.
* It can be supposed that $G$ is semisimple or simple, if it is necessary.