$\def\O{\mathcal{O}} \def\g{\mathfrak{g}}$ 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 $G$?

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

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