# Casimir operators for Jacobi Lie algebra

I am looking for a complete description (if possible with proof) of the Casimir operators (i.e., generators and relations for the center of the universal enveloping algebra) for the Jacobi Lie algebra over the complex numbers, the semidirect product of the symplectic Lie algebra with a Heisenberg algebra.

I think that in general, when you drop the assumption that your Lie algebra is semisimple, then describing its centre becomes tricky. It may help to observe that your Lie algebra $\mathfrak{sp}_{2n}\ltimes{\mathfrak h}_n$ is the derived subalgebra of a maximal parabolic subalgebra of $\mathfrak{sp}_{2n+2}$ (at least over ${\mathbb C}$).
• Well, by the Duflo isomorphism, $Z(U({\mathfrak g}))$ is isomorphic to $S({\mathfrak g})^{\mathfrak g}$ for any finite-dimensional Lie algebra. Now if ${\mathfrak g}=\mathfrak{sp}_{2(n+1)}$ and our Lie algebra is ${\mathfrak g}_0$, then obviously $S(\mathfrak{g})^{\mathfrak g}\subset S(\mathfrak{g})^{{\mathfrak g}_0}$, and further we have a ${\mathfrak g}_0$-stable decomposition ${\mathfrak g}={\mathfrak g}_0+I$, so I think we have a map from $S(\mathfrak{g})^{\mathfrak g}$ to $S(\mathfrak{g}_0)^{{\mathfrak g}_0}$. It's a start though it certainly isn't surjective. – Paul Levy Apr 26 '17 at 9:43