I've been looking at an example in the non-commutative geometry literature and I'm having trouble figuring out what the classical motivation is. I'll just describe the classical case here: Recall that $\mathbb{CP}^2 = SU(\mathbb{C},3)/U(\mathbb{C},2)$. Recall also that since $T\_\mathbb{C}^\*(\mathbb{CP}^2)$ can be viewed as a vector bundle associated to the $SU(2)$-bundle $SU(\mathbb{C},3) \to \mathbb{CP}^2$, we can view $\Omega\_\mathbb{C}^1(\mathbb{CP}^2)$ as a subset of $\mathcal{O}(\mathbb{CP}^2) \oplus \mathcal{O}(\mathbb{CP}^2)$. Now in the example, the action of each of the Dolbeault operators $\partial,\overline{\partial}$ is given in terms of two mappings, $\partial\_1,\partial\_2$ and $\overline{\partial\_1},\overline{\partial\_2}$, such that 
$$
\partial(f) = (\partial\_1(f),\partial\_2(f)) \in \mathcal{O}(\mathbb{CP}^2) \oplus \mathcal{O}(\mathbb{CP}^2),
$$
and similarly for $\overline{\partial}$. The mappings ${\partial}\_i, \overline{\partial}\_i$ are constructed using an action of the Lie algebra $\mathfrak{su}(3)$ on $\mathcal{O}(SU(3))$ constructed using the canonical pairing between Lie algebras and coordinate algebras.


I have a feeling this is a complicated incarnation of a simple classical object. Does any of this ring any bells with anyone?


Please ask if you would like more details.


Edit: This question has been superseded by [this question](https://mathoverflow.net/questions/16880/dolbeault-operators-for-cp1-as-mathfraksu2-actions) and I am voting to close. I would ask others to do likewise.



Why can the Dolbeault Operators be Realised as Lie Algebra Actions
The Dolbeault operators are usually defined in terms of the de Rham operator and the complex structure (see e.g. Wells' book or Griffith and Harris). The example you outline generalizes to the situation $G\_{\mathbb{C}} / P = G / G\_0$, where $G$ is compact, $G\_{\mathbb{C}}$ is the complexification, $P$ is a parabolic subgroup, and $G\_0 = G \cap P$. The holomorphic tangent bundle is the homogeneous vector bundle $G\_{\mathbb{C}} \times\_P \mathfrak{g}\\_{\mathbb{C}} / \mathfrak{p}$, and the cotangent bundle is a homogeneous vector bundle in similar fashion.


In this case, the Dolbeault complex with coefficients in a homogeneous vector bundle $G\_{\mathbb{C}} \times\_P V$ translates to the Koszul complex for the relative
Lie algebra cohomology $H^\*(\mathfrak{g},\mathfrak{g}\\_0,V \otimes C^{\infty}(G))$. The Dolbeault operator $\overline{\partial}$ translates to the boundary operator for Lie algebra cohomology, which of course involves the action of $\mathfrak{g}$. I haven't worked out what
happens to $\partial$ in this situation, but most likely a similar expression in terms of the Lie algebra can be derived. The translation works by thinking about the smooth sections as elements of $C^{\infty}(G) \otimes V)^{\mathfrak{g}\\_0}$ (for holomorphic sections use $C^{\infty}(G) \otimes V)^{\mathfrak{p}}$). 


