Hi, Let $G_1, G_2$ be topological groups with $G_1 \subset G_2$ is closed. Let $\rho:G_1 \to Aut(V)$ be a smooth irreducible representation. Can anyone tell me if there is a criterion/example/idea about when $\rho$ cannot be extended to $G_2$, i.e., there does not exist smooth representation $\rho':G_2 \to Aut(V)$ such that $\rho'_{G_1}=\rho$.

The above comments show that in general linear representations do not extend, so a better question is to ask when linear representations do extend! This leads rather quickly to Margulis' superrigidity and the subsequent programme generated by this result. 

