Hello,  

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. 



Representation of quotient group
Yes, if $H$ is closed (i.e. not some irrational-flow subgroup inside the closed subgroup $Z(G)$).


