I am quite sure that this is written somewhere in the literature but I could not find a reference.
Let $\mathfrak{gl}_n$ be the Lie algebra of (complex) $n\times n$ matrices and let $E_{ij}$ be the standard basis of $\mathfrak{gl}_n$ consisting of matrix units. In particular, they satisfy the following Lie algebra structure is given by the following commutation relations: $$ [E_{ij},E_{kl}]=\delta_{jk}E_{il}-\delta_{li}E_{kj},~\text{for all}~i,j,k,l\in\{1,2,\ldots,n\}, $$ where $\delta_{\alpha\beta}$ is the Kronecker delta.
Now, consider the coordinate ring $\mathbb{C}[\mathfrak{gl}_n]\simeq\mathfrak{gl}_n^*$. This ring is just a polynomial ring in $n^2$ variables $(x_{ij})_{i,j=1}^{n}$, which correspond to the $(i,j)$-entry of a matrix in $\mathfrak{gl}_n$.
Next, it is known that the universal enveloping algebra $U(\mathfrak{gl}_n)$ has a faithful representation in the space of the differential operators in variables $x_{ij}$ with polynomial coefficients. Namely, the corresponding map $L$ sends $E_{ij}$ to $$ L(E_{ij})=\sum_{k=1}^{n}x_{ik}\frac{\partial}{\partial x_{jk}}. $$ In particular, this gives a representation of $\mathfrak{gl}_n$ (or $U(\mathfrak{gl}_n)$) on the space $\mathbb{C}[\mathfrak{gl}_n]$.
Question. Is it known how does the representation $(L,\mathbb{C}[\mathfrak{gl}_n])$ decompose into the irreducible representations of $\mathfrak{gl}_n$?
I am interested in the explicit description of these subrepresentations inside $\mathbb{C}[\mathfrak{gl}_n]$, but this might be not too realistic. It would be nice to have at least the description of all occuring highest weights.
Some general ideas: one can check that the described action of $\mathfrak{gl}_n$ preserves many gradings. To be more precise, for all $i,j,k\in\{1,2,\ldots,n\}$ the operator $L(E_{ij})$ preserves the total degree in variables that appear in $k$-th row (or column). This already gives some non-trivial decomposition of $\mathbb{C}[\mathfrak{gl}_n]$.