Approximate Hermitian–Yang–Mills structures on semistable principal Higgs bundles

We generalize the Hitchin–Kobayashi correspondence between semistability and the existence of approximate Hermitian–Yang–Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {E}}$$\end{document} on a compact Kähler manifold, with structure group a connected linear algebraic reductive group G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document}, is semistable if and only if it admits an approximate Hermitian–Yang–Mills structure.