Germs of left invariant differential operators on a group

Question

Are there germs at the identity of linear differential operators on a group which are not germs at the identity of left invariant differential operators? I feel like the answer is no but the statement could be adjusted...

Answer 1

I presume “group” means “Lie group”.

Invariant differential operators on the Lie group $\def\R{{\bf R}}\R$ have the form $∑_{k≥0}a_k {∂^k\over ∂x^k}$, where $a_k∈\R$.

Thus, any linear differential differential operator whose coefficients have a nonconstant germ at 0 will have a noninvariant germ at 0. For example, take $x{∂\over ∂x}$.