This question is related to a previous question of mine. More specifically, it results from my attempts to understand the simplest incarnation of a phenomenon mentioned therein.

Put a grading on the elements of the coordinate algebra of $SU(2)$ by setting deg$(\hat{a})$=deg$(\hat{c})=1$, and deg$(\hat{b})$=deg$(\hat{d})=-1$; where $$ \hat{a}\left(\array{a & b \\\ c & d} \right) = a $$ and so on. We can identify the coordinate algebra of $\mathbb{CP}^1 = SU(2,\mathbb{C})/U(1)$, with the elements of degree $0$, $\Omega^{(0,1)}(\mathbb{CP}^1)$ with the elements of degree $2$. Does there exist an element $X$ of $\mathfrak {su}(2)$ (or its enveloping algebra) such that $$ X(f) = \overline{\partial}(f) \in \Omega^{(0,1)}(\mathbb{CP}^1), $$ for all coordinate functions $f$ on $\mathbb{CP}^1$, where $X$ is the canonical action of $\mathfrak {su}(2)$ on the coordinate algebra of $SU(2)$, that is, $$ X(f)(g) = \frac{d}{dt}_{t=0}(f(\exp(-tX)g)). $$ If so, can someone explain in a more general context why so? (Perhaps in the context of spaces of the form $G/T$, where $G$ is a compact Lie group, and $T$ is a maximal torus.)

EDIT: I've been doing some more reading, and it seems that the required element of $\mathfrak{su} (2)$ does exist. Moreover, there exists a formula for extending this to the general $\mathbb{CP}^n$ case, see page 32 of this paper by D'Andrea and Dabrowski. What I still don't see though is why.