I'm trying to compute the Kuranishi space of complex compact manifold $\mathcal{M}$ with $\textit{SL}_2(\mathbb{C})$ as universal cover. In order to do that, I use the correspondance between a subspace of harmonic $(0,1)$-differential forms with value in $T^{1,0}\textit{SL}_2(\mathbb{C})$ which descends on the manifold $\mathcal{M}$.
I have some difficulties to write explicitly the right invariance of $\omega \in A^{0,1}(T^{1,0}\textit{SL}_2(\mathbb{C}))$ by an element $g\in \textit{SL}_2(\mathbb{C})$.
Consider the left-invariant standard basis $\langle e_i\rangle$ of $\mathfrak{sl}_2(\mathbb{C})$ (with $[e_1,e_2]=e_3,\ [e_1,e_3]=-2e_1$ and $[e_2,e_3]=-2e_3$) and the complexified dual basis $\langle e^1,e^2,e^3, \overline{e^1},\overline{e^2},\overline{e^3} \rangle$. We can write $\omega$ as $\displaystyle\sum_{i,j=1}^3 (f_i^j(x)\ \overline{e^j}) \otimes e_i$ with $f_i^j$ $\mathcal{C}^\infty$ functions on $\textit{SL}_2(\mathbb{C})$. Now take $g\in \textit{SL}_2(\mathbb{C})$ and consider $L_g$ the left translation by $g$. As $e_i$ is left invariant we have $(L_g)^*\omega=\omega \ \iff \ f_i^j(gx)=f_i^j(x)$ for all $x\in \textit{SL}_2(\mathbb{C})$. Now, what is the condition to have $(R_g)^*\omega=\omega$ ?