Decomposition of universal enveloping algebra of the general linear Lie algebra

Hi all I have a question, let $\mathfrak g = \mathfrak{gl}(n)$ be the general linear Lie algebra, and $U$ the enveloping algebra of $\mathfrak g$. Now we regard $U$ as a $\mathfrak g$-module $U^{\text{ad}}$ through the adjoint action of $\mathfrak g$, that is, $x\cdot u = xu-ux$, for all $x\in \mathfrak g$ and $u\in U$.

$\bf My ~Question$: What is the $\mathfrak g$-decomposition of $U^{\text{ad}}$ into direct sum of irreducible $\mathfrak g$-modules? Thanks!

• Why do you expect it to decompose into irreducible modules? – Tobias Kildetoft Jun 4 '18 at 18:03
• $U^{\text{ad}}$ is locally $\mathfrak g$-finite I think and so it is completely reducible. – TLSu Jun 4 '18 at 20:00

For any reductive Lie algebra $\mathfrak{g}$, the enveloping algebra $U=U(\mathfrak{g})$ is isomorphic to $\mathrm{Sym}(\mathfrak{g})$ as a $\mathfrak{g}$-module. This fact is essentially the PBW theorem: the $\mathrm{ad}(\mathfrak{g})$-stable filtration $$U^{\leq d}=\mathbf{C} \{ x_1 \cdots x_e \ | \ x_i \in \mathfrak{g}, \ e \leq d \}$$ has associated graded algebra $\mathrm{Sym}(\mathfrak{g})$, and by complete reducibility $$U^{\leq d} \cong \bigoplus_{e \leq d} U^{\leq e}/U^{\leq e-1}$$ as $\mathfrak{g}$-modules.