We have a smooth space $X$ with an action from $G\_1 \times G\_2$ on it; we also have a differential operator $P \in \mathscr{D}(X)$. If $P$ takes $G\_1 \times G\_2$-invariant functions to $G\_1 \times G\_2$-invariant functions and also is $G\_1$-invariant, is it necessarily true that $P$ takes $G\_2$-invariant functions to $G\_2$-invariant functions?


Would this be true if we require that $G\_1$ be a connected semisimple Lie group?



Differential operators and commuting actions
Let $X$ be ${\mathbb R}\times{\mathbb R}$ and let $G\_1\cong G\_2$ be the group of reals, $G\_1$ acting by translation on the first argument, $G\_2$ on the second.
Let $f$ be any smooth function on $\mathbb R$ and let $D=f(y)\frac{\partial}{\partial x}$, where the coordinates on $X$ are $(x,y)$. Then $D$ ist $G\_1$-invariant. Maps $G\_1\times G\_2$-invariants to zero, hence to $G\_1\times G\_2$-invariants, but does not map all $G\_2$-invariant functions to $G\_2$ invariants.


