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.

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    $\begingroup$ This is more a question about the first (or second?) isomorphism theorem than anything else, and hence more appropriate for math.stackexchange. $\endgroup$ – Allen Knutson Jan 17 '13 at 0:25

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

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