Triangular de Rham cohomology of compact Kahler manifolds

The de Rham $H^1_{DR}(M,G)$ of a smooth manifold $M$ with values in a group Lie $G$ is studied. By definition, this is the quotient of the set of flat connections in the trivial principal bundle $M\times G$ by the so-called gauge equivalence. The case under consideration is the one when $M$ is a compact Kahler manifold and $G$ is a soluble complex linear algebraic group in a special class containing the Borel subgroups of all complex classical groups and, in particular, the group of all triangular matrices. In this case a description of the set $H^1_{DR}(M,G)$ in terms of the cohomology of $M$ with values in the (Abelian) sheaves of flat sections of certain flat Lie algebra bundles with fibre $\mathfrak g$ (the tangent Lie algebra of $G$) or, equivalently, in terms of the harmonic forms on $M$ representing this cohomology is obtained.