Loss of regularity for left invariant vector field

Question

Consider $G$ a compact Lie group. Then there is a natural Laplacian operator and Fourier theory in $G$ so we can define Sobolev spaces $H^s(G)$ for each $s\in\mathbb{R}$. It is easy to prove that every left invariant vector field in $G$ can be seen as a first order differential operator and so is bounded from $H^s(G)$ to $H^{s-1}(G)$, for each $s$. I was wondering, is it possible for there to exist some group $G$ and left invariant vector field $X$ such that $X$ is bounded from $H^{s}(G)$ to $H^{s-\epsilon}(G)$, for some $0<\epsilon<1$? Sounds like it shouldn't be possible, but I can't prove it, nor think of an example...