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.
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What do you mean by Casimir operators? Are they just elements of the centre of the universal enveloping 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 ${\mathbb C}$).
You might find the comments on the following question helpful:
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$\begingroup$ I edited my question to make it more precise, answering your initial questions. The problem is indeed that the Jacobi Lie algebra is not semisimple; so that the standard stuff does not apply. I don't see how the embedding into sp(2n+2) helps; it seems to me that its Casimirs are unlike to generate those of the Jacobi Lie algebra. $\endgroup$ Apr 26, 2017 at 7:37
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$\begingroup$ 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. $\endgroup$ Apr 26, 2017 at 9:43