Universal enveloping algebra as algebra of differential operators

Question

Let $G$ be Lie group and $g$ be its Lie algebra. Is it true (and if not generally, then under which circumstances) that the the algebra of its differential operators is isomorphic to the universal envelopping algebra $U(g)$ of $g$? If yes, can you give me an isomorphism between these two algebras?

Answer 1

$U(g)$ is isomorphic to the algebra $D_{l}(G)$ of left invariant differential operators on the Lie group $G$. Its center $Z(U(g))$ is the algebra of bi-invariant differential operators.

To prove the first statement, one defines the map $U(g)\to D_l(G)$ by extending the inclusion $g\to D_l(G)$ (remember $g$ is the space of left invariant vector fields on $G$, which are precisely the left invariant differential operators of order one), after an easy check that this works. The surjectivity is more involved, of course. IIRC it is done in detail in Helgason's book on symmetric spaces.

The algebra of differential operators on $G$, on the other hand, is considerably larger. Even the algebra of differential operators on $G$ with polynomial coefficients (that is, regular functions---I am assuming $G$ is an algebraic group here) has Gelfand-Kirillov dimension twice that of $U(g)$.