Let $\mathfrak{g}$ be a simple complex Lie algebra and $U$ be its universal enveloping algebra. We have an action $\mathfrak{g} \to \mathfrak{gl}(U)$ by extending the adjoint action of $\mathfrak{g}$ on itself to higher degree terms by derivations. It is an immediate consequence of the PBW theorem and the commutation relations of elements of a Cartan subalgebra that $U$ decomposes as a direct sum of weight spaces, where the weights appearing consists of the full lattice generated by the roots.
Letting $U_n$ denote the PBW filtration on $U$, that is, the filtration by total degree, one also checks that each $U_n$ is invariant under $\mathfrak{g}$. Since it is finite dimensional, it is completely reducible. My question is:
Is there a description of the irreducible factors of $U_n$ for each $n$ in terms of the root system of $\mathfrak{g}$?
For $\mathfrak{sl}_2$ they are simple enough that can be written down by hand, though I haven’t tried to get any sort of formula. Already for rank 2 algebras the situation gets quite messy when trying to find a decomposition for $U_2$ and $U_3$. I’ve also tried searching it for I guess this is certainly a problem someone else has tackled before, but I couldn’t find anything yet, maybe I’m not using the right terms. Any help is appreciated!