We give a definition of differentiable cohomology of a Lie group G (possibly
infinite-dimensional) with coefficients in any abelian Lie group. This
differentiable cohomology maps both to the cohomology of the group made
discrete and to Lie algebra cohomology. We show that the secondary
characteristic classes of Beilinson lead to differentiable cohomology classes
with coefficients in C*. These may be viewed as an enrichment of the
Chern-Simons differential forms.
By transgression, classes in differentiable cohomology of a Lie group G lead
to differentiable cohomology classes for gauge groups Map(M,G). These classes
generalize the central extensions of loop groups. We also discuss holomorphic
cohomology of complex Lie groups as the natural place to construct secondary
classes.
We present several conjectures relating the above cohomology classes to the
differential forms of Bott-Shulman-Stasheff.