Non continous representations of $SL_2(\mathbf{R})$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T01:33:46Z http://mathoverflow.net/feeds/question/91992 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91992/non-continous-representations-of-sl-2-mathbfr Non continous representations of $SL_2(\mathbf{R})$ Hugo Chapdelaine 2012-03-23T10:26:27Z 2012-03-23T13:33:55Z <p>Q: How does one construct a non continuous representation $\rho:SL_2(\mathbf{R})\rightarrow G$ for some connected (finite dimensional) Lie group $G$?</p> http://mathoverflow.net/questions/91992/non-continous-representations-of-sl-2-mathbfr/91995#91995 Answer by Marc Palm for Non continous representations of $SL_2(\mathbf{R})$ Marc Palm 2012-03-23T10:52:52Z 2012-03-23T10:57:57Z <p>A partial answer:</p> <p>No measurable constructions are possible. Any measurable group homomorphism between locally compact groups is automatically continuous, in fact $C^\infty$ for Lie groups. You can have a look at the answers in an old question of mine:</p> <p><a href="http://mathoverflow.net/questions/64116/are-measurable-automorphism-of-a-locally-compact-group-topological-automorphisms" rel="nofollow">http://mathoverflow.net/questions/64116/are-measurable-automorphism-of-a-locally-compact-group-topological-automorphisms</a></p> <p>If I would like to find something non continuous, I personally would start with finding some non measurable automorphism of the circle first.</p> http://mathoverflow.net/questions/91992/non-continous-representations-of-sl-2-mathbfr/92002#92002 Answer by Misha for Non continous representations of $SL_2(\mathbf{R})$ Misha 2012-03-23T11:45:07Z 2012-03-23T13:33:55Z <p>Example can be found, for instance, in <a href="http://en.wikipedia.org/wiki/Boris_Weisfeiler" rel="nofollow">Boris Weisfeiler's</a> paper "Abstract homomorphisms of big subgroups of algebraic groups", pages 149-150, see </p> <p><a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.ndml/1175197662" rel="nofollow">http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.ndml/1175197662</a></p> <p>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 <em>dense</em> in the noncompact Lie group $H$. Weisfeiler's paper also lists many positive results on rigidity of abstract homomorphisms of Lie groups. </p> http://mathoverflow.net/questions/91992/non-continous-representations-of-sl-2-mathbfr/92006#92006 Answer by paul garrett for Non continous representations of $SL_2(\mathbf{R})$ paul garrett 2012-03-23T12:34:29Z 2012-03-23T12:34:29Z <p>There are natural function spaces on Lie groups that are nevertheless not continuous (and, thus, are not representations in any usual, useful sense). For example, already on $G=\mathbb R$, the Frechet space $V$ of all continuous functions, and/or the Frechet space of bounded continuous functions, with the translation action of $G$, are not repn spaces, in the sense that $G\times V\rightarrow V$ is not continuous. The reason is the existence of not-uniformly-continuous continuous functions. For example, $f(x)=\sin(x^2)$.</p>