Fix a $\mathbb Z\_+^n$-graded Lie algebra ${\frak a}=\oplus\_{r \in\mathbb Z\_+^n}^{} {\frak a}[r]$ such that ${\frak g}:={\frak a}[0]$ is a finite-dimensional semisimple Lie algebra over the complex numbers and ${\frak a}[r]$ is a finite-dimensional $\frak g$-module for all $r\in\mathbb Z\_+^n$. 


We have a natural ideal given by ${\frak a\_+}=\oplus\_{r \in\mathbb Z\_+^n, r\ne 0}^{} {\frak a}[r]$. Denote by $U(\frak a\_+)$ its universal enveloping algebra. Notice that $U(\frak a\_+)$ inherits a $\mathbb Z\_+^n$-gradation from $\frak a$. 


How to describe ${U(\frak a\_+)}[k]$ as a ${\frak g}$-module using the PBW Theorem ?
(the graded pieces of $U({\frak a}\_+))$



PBW-Theorem and multigraded Lie algebras
As $\mathfrak{a}[0]$-modules we have: 
$$
U(\mathfrak{a}\_+)[k]=\bigoplus\_{\substack{l\geq1 \\ c\_1r\_1+\cdots+c\_lr\_l=k}}S^{c\_1}(\mathfrak{a}\_+[r\_1])\otimes\cdots\otimes S^{c\_l}(\mathfrak{a}\_+[r\_l])
$$


PS: I assume you work over a field of characteristic zero and I use the isomorphism of graded $\mathfrak a$-modules $S(\mathfrak a)\to U(\mathfrak a)$ given by the symmetrization map. 


