TITLE: Why does the universal cover of $SL_{2}(\mathbb{R})$ have no finite-dimensional representations?
QUESTION [12 upvotes]: Why does the universal cover of $SL_{2}(\mathbb{R})$ have no finite-dimensional representations?

REPLY [5 votes]: This comes a bit late but I think it answers shortly the question:
Theorem: Any connected linear semisimple Lie group has finite center.
(This is mainly because of Schur's Lemma)
But $Z(\widetilde{SL_2(\mathbb{R})})$ is isomorphic to $\mathbb{Z}$ the fundamental group of $SL_2(\mathbb{R})$.