For an algebraic variety $V$, denote its ring of regular functions by ${\cal O}(V)$. The Kahler differentials of $V$ are the quotient of the kernel $M$ of the multiplication map $$ m: {\cal O}(V) \otimes {\cal O}(V)\to {\cal O}(V) $$ by the ideal $M^2$.
What can be said about the maximal proper submodules of $M$?
Is there any sense/specific-case in which the submodule $M^2$ is maximal?
I am particularly interested in the homogeneous variety case, specifically the flag variety case. For example, is $M^2$ a maximal right $G$-invariant proper submodule when $V$ a a $G$-homogeneous variety.