Let $G$ be a compact real Lie group which is also a complex manifold (note that I do not want $G$ to be a complex Lie group). Let $\Omega^{p , q}(G)$ denote the space of smooth $(p , q)$-forms on $G$, and let $\Omega_L^{p , q}(G)$ (resp. $\Omega^{p , q}_I(G)$) denote the space of left-invariant (resp. bi-invariant) $(p , q)$-forms.
Suppose $G$ has a bi-invariant hermitian metric. Then one can define the Hodge-$*$ operator (See Griffiths and Harris, page 82), and therefore the adjoint operators $\partial^*: \Omega^{p , q}(G) \to \Omega^{p - 1 , q}(G)$ and $\overline{\partial}^*: \Omega^{p , q}(G) \to \Omega^{p , q - 1}(G)$, and the Laplacians:
$\Delta_\partial = \partial^\ \partial^* + \partial^* \partial^\ : \Omega^{p , q}(G) \to \Omega^{p , q}(G)$
and
$\Delta_\overline{\partial} = \overline{\partial}^\ \overline{\partial}^* + \overline{\partial}^* \overline{\partial}^\ : \Omega^{p , q}(G) \to \Omega^{p , q}(G)$.
Let $\mathcal{H}_\partial^{p , q}(G)$ and $\mathcal{H}_\overline{\partial}^{p , q}(G)$ denote the kernels of these operators, respectively. Forms that belong to theses spaces are called Harmonic.
What is the relationship, if any, between the spaces $\Omega^{p , q}_L(G)$, $\Omega^{p , q}_I(G)$ , $\mathcal{H}_\partial^{p , q}(G)$, and $\mathcal{H}_\overline{\partial}^{p , q}(G)$ (all of which are contained in $\Omega^{p , q}(G)$)? In particular, is it true that all Harmonic forms are left-invariant?
I don't want to assume anything about $G$ being Kähler, if I can help it.