Let $G$ be a locally compact Hausdorff group. Assume that $\theta:G\to U\_d$ is a group homomorphism where $U\_d$ is a finite dimensional unitary group. Consider a action of $G$ on $U\_d$ by $g.u:=\theta(g)u,$ $u\in U\_d.$ Consider $U=\overline{\theta(G)}.$ Is it true that $G$ acts ergodically on $U$ where $U$ is equipped with the invariant probability measure of $U\_d$?



Ergodic action on unitary group
Since $U\_d$ is connected, every subgroup proper of it had infinite index, so unless $U=U\_d$ it must be a set of measure zero there, and the literal answer is "no".


On the other hand, the action of $G$ on $U$ is ergodic for the Haar measure of $U$. Indeed, let $f\in L^1(U)$ be $G$-invariant (e.g. the characteristic function of a $G$-invariant set). Then by the continuity of the action of $U$ on $L^1(U)$, $f$ is $U$-invariant, hence constant a.e.


