We work over $\mathbb{C}$. Let $\mathfrak{k}$ be a Lie algebra, and let $V,W$ be finite-dimensional $\mathfrak{k}$-modules. Consider the $\mathfrak{k}$-algebra $A:=S^\bullet V$. Write $\mathfrak{m}$ for the maximal ideal at $0\in V^*$, a $\mathfrak{k}$-module.
Let $M$ be a $\mathfrak{k}$-equivariant, free $A$-module with $M/\mathfrak{m}M\cong W$ as $\mathfrak{k}$-modules. Being $\mathfrak{k}$-equivariant means that $\mathfrak{k}$ acts on $M$ such that for $m\in M$, $a\in A$, and $x\in\mathfrak{k}$ we have $$ x(am)=x(a)m+ax(m). $$ Under what conditions on $V$ and $W$ can we conclude that $M$ is in fact globally trivial, i.e. that $M\cong W\otimes A$ as $\mathfrak{k}$-modules? I assume there must be a nice cohomology group that classifies the set of all such $M$ up to isomorphism, and that someone has studied it before; does anyone have a reference? Obviously if $\mathfrak{k}$ is reductive then it is true, but I'm working with nonreductive Lie algebras so this doesn't help me.
Thanks for any help!
