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 $\rho'|\_{G\_1}=\rho$. 



Extending smooth irreducible representations
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' super-rigidity and the subsequent programme generated by this result.


