  We study the de Rham 1-cohomology H^1_{DR}(M,G) of a smooth manifold M with
values in a Lie group G. By definition, this is the quotient of the set of flat
connections in the trivial principle bundle $M\times G$ by the so-called gauge
equivalence. We consider the case when M is a compact K\"ahler manifold and G
is a solvable complex linear algebraic group of a special class which contains
the Borel subgroups of all complex classical groups and, in particular, the
group $T_n(\Bbb C)$ of all triangular matrices. In this case, we get a
description of the set H^1_{DR}(M,G) in terms of the 1-cohomology of M with
values in the (abelian) sheaves of flat sections of certain flat Lie algebra
bundles with fibre $\frak g$ (the Lie algebra of G) or, equivalently, in terms
of the harmonic forms on M representing this cohomology.
