@QiaochuYuan: Thanks a lot for the extra comments. So if I have understood correctly, for a Lie group $G$, will the irreps of $G$ coincide with the irreps of $\frak{g}$ when $\pi(G)$ is trivial?

So the example I am thinking about is $U_2$ and $SU_2 \times U_1$. Does $SU_2 \times U_1$ have more irreducible representations than $U_2$?

Great, I didn't hope there would be a such nice answer! Thanks a lot!

I am thinking of geometric examples, for instance the direct sum of algebraic sections of line bundles over the $2$-sphere. Here it all works if I understand it well. I was wanting to know if injectivity is a geometric result, or a general algebra result.