Yang-Mills connections on compact complex tori

Let $G$ be a connected reductive complex affine algebraic group and $K\subset G$ a maximal compact subgroup. Let $M$ be a compact complex torus equipped with a flat K\"ahler structure and $(E_G ,\theta)$ a polystable Higgs $G$-bundle on $M$. Take any $C^\infty$ reduction of structure group $E_K \subset E_G$ to the subgroup $K$ that solves the Yang--Mills equation for $(E_G ,\theta)$. We prove that the principal $G$-bundle $E_G$ is polystable and the above reduction $E_K$ solves the Einstein--Hermitian equation for $E_G$. We also prove that for a semistable (respectively, polystable) Higgs $G$-bundle $(E_G , \theta)$ on a compact connected Calabi--Yau manifold, the underlying principal $G$-bundle $E_G$ is semistable (respectively, polystable).