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