Let $G$ be a Lie Group and $H$ a closed subgroup such that $G/H$ (the set of right cosets) is a complex manifold manifold. Now $\Omega^1(G/H)$, the space of complex one forms, is a $H$-equivariant bundle (with respect to left multiplication of $H$). What I want to know is: Are the Dolbeault maps $\partial, \overline{\partial}$, $H$-equivariant, that is, is it true that $$ \partial(f)(h.\overline{g}) = h.(\partial(f)(\overline{g})), $$ and $$ \overline{\partial}(f)(h.\overline{g}) = h.(\overline{\partial}(f)(\overline{g})), $$ for all $f \in \mathbb{C}[G/H]$, $h \in H$, $g \in G$.

The example I have in mind $SU(n+1)/U(n)$.