Let $G$ is a semisimple algebraic group over characteristic 0. I'm curious about the finite set of differential operators $D_i$ on algebra $\Bbbk [V]$ ($V$ is a representation of $G$), this differential operators are satisfying the following conditions. For every irreducible $G$-representation $W$ (denote it highest weight as $\omega$), there exists an isomorphism of $W$ with the set of $f\in\Bbbk[V]$, s.t $D_i f=n_i(\omega) f$ (where $n_i(\omega)$ are the functions from the set of all highest weights to $\Bbbk$).

Example: $xp_x+yp_y=np$ - Euler equation on $k[x,y]$ defines an irreducible representation of $SL_2$.

So I want to know if there exist: $V$, $D_i$ and $n_i$.