Triangular de Rham cohomology of compact Kahler manifolds

The de Rham 1-cohomology H-DR(1) (M, G) of a smooth manifold M with values in a Lie group G is studied. By definition, this is the quotient of the set of flat connections in the trivial principal bundle M x G by the so-called gauge equivalence. The case under consideration is the one when M is a compact Kahler manifold and G 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 HDR(M; G) in terms of the l-cohomology of M with values in the (Abelian) sheaves;of flat sections of certain flat Lie algebra;bundles with fibre B (the tangent Lie algebra of G) or, equivalently, in terms of the harmonic forms on M representing this cohomology is obtained.