Suppose that $G$ is a connected Lie Group with Lie algebra $\mathfrak{g}$ and $\Gamma$ is a cocompact lattice, and you choose a left-invariant metric on $G/\Gamma$ and let $\Delta$ be the Laplacian. Let $\lambda$ be a generalised eigenvalue for the Laplacian and let $n$ be a positive integer sufficiently large that $(\Delta-\lambda I)^{n}$ kills all of the generalised eigenspace.
Would there be any way of showing that $U(\mathfrak{g})/(\Delta-\lambda I)^{n}U(\mathfrak{g})$ is Artinian, or non-Artinian?
