Representation of Lie algebra of germs of smooth/holomorphic functions

$$\def\O{\mathcal{O}} \def\g{\mathfrak{g}}$$ Suppose $$G$$ is a real or complex Lie group, with Lie algebra $$\g$$. Write $$\O_{G,1}$$ (resp. $$\O_{\g,0}$$) for the ring of germs of smooth/holomorphic function on $$G$$ at $$1$$ (resp. on $$\g$$ at $$0$$). If we identify $$\g$$ with left-invariant vector fields on $$G$$, then we get a Lie algebra representation of $$\g$$ on $$\O_{G,1}$$. The exponential map $$\exp:\g \to G$$ is a local isomorphism, so induces an isomorphism of $$\O_{\g,0}$$ and $$\O_{G,1}$$, so we also get a Lie algebra representation of $$\g$$ on $$\O_{\g,0}$$.

Can the representation of $$\g$$ on $$\O_{\g,0}$$ be described in a way that is intrinsic to $$\g$$, without reference $$G$$?

• I think it might be more or less the dual of the adjoint action of $\mathfrak{g}$ on its universal enveloping algebra $U(\mathfrak{g})$. – Qiaochu Yuan Mar 10 at 3:21