from the comparison between the results for Lie groups in [@kn:homolie; @kn:mostow] and the situation of algebraic groups in [@kn:hmbb].

For example, in the first mentioned papers and in a rather laborious way, the authors prove the following result.

[@kn:mostow Theorem 4.1] Let $H
\subseteq G$ be a closed normal inclusion in the category of (real or complex) analytic groups and denote as $N$ the radical of the commutator subgroup $G'$ of $G$. Assume that $\rho$ is a finite dimensional representation of $H$ and that $\rho'$ is the semisimple representation associated to $\rho$. Then, $\rho$ can be extended to $G$ (with a finite dimensional extension) if and only if the following three conditions hold:

1.  $\rho'$ is trivial in $H\cap N$;

2.  The representation $\sigma$ of $HN$ defined by $\sigma(xu)=\rho'(x)$ for $x \in H\,,\, u \in N$ is continuous when $HN$ is endowed with the topology induced by $G$;

3.  Call $G_{f}$ the intersection of all the kernels of all the finite dimensional representations of $G$. Then $\rho$ is trivial in $G_f \cap H$.

The above theorem is the main result of [@kn:mostow], whereas in the first paper [@kn:homolie], a particular case is proved with additional topological conditions. It is 