How does one construct a non-continuous representation $\rho:\operatorname{SL}\_2(\mathbf{R})\rightarrow G$ for some connected (finite dimensional) Lie group $G$?



Non-continuous representations of $\operatorname{SL}_2(\mathbf{R})$
Example can be found, for instance, in
[Boris Weisfeiler's](https://en.wikipedia.org/wiki/Boris_Weisfeiler) paper "Abstract homomorphisms of big subgroups of algebraic groups", pages 149-150, see


[Link](https://projecteuclid.org/ebooks/notre-dame-mathematical-lectures/Topics-in-the-Theory-of-Algebraic-Groups/chapter/Abstract-Homomorphisms-of-Big-Subgroups-of-Algebraic-Groups/ndml/1175197662)


His example of a discontinuous representation $\rho$ of $SO(n, {\mathbb R})$ to a semidirect product $H$ of $SO(n, {\mathbb R})$ with the abelian group ${\mathbb R}^N$ (the Lie algebra of $SO(n)$), works for $SL(2, {\mathbb R})$ as well. Actually, Weisfeiler's example is even more dramatic: The image of the compact group $SO(n)$ under $\rho$ is *dense* in the noncompact Lie group $H$. Weisfeiler's paper also lists many positive results on rigidity of abstract homomorphisms of Lie groups.


