What is the natural action of $U(\mathfrak{g})$ on $\mathbb{C}[G]$?

Question

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. What is the natural action of $U(\mathfrak{g})$ on $\mathbb{C}[G]$? It seems that the natural action comes from the following. We have a natural comultiplication $\mathbb{C}[G] \to \mathbb{C}[G] \otimes \mathbb{C}[G]$. If we choose a natural pairing $U(\mathfrak{g}) \otimes \mathbb{C}[G] \to \mathbb{C}$, then we have a map $$ U(\mathfrak{g}) \otimes \mathbb{C}[G] \to U(\mathfrak{g}) \otimes \mathbb{C}[G] \otimes \mathbb{C}[G] \to \mathbb{C} \otimes \mathbb{C}[G] = \mathbb{C}[G]. $$ I think that this map is an action. What is the natural pairing $U(\mathfrak{g}) \otimes \mathbb{C}[G] \to \mathbb{C}$? Are there some references which give explicit formula for the map $U(\mathfrak{g}) \otimes \mathbb{C}[G] \to \mathbb{C}$? Thank you very much.

Answer 1

This is making things too hard. This is the same as a $\mathfrak g$ action on $\mathbb C[G]$, which is given by differentiation by right invariant vector fields. Extending to $U(\mathfrak g)$ is given by taking the associated differential operator. The pairing you want is applying this operator and then evaluating at $e$.