Non-unitary representations of nilpotent groups, I: Cohomologies, extensions and neutral cocycles

Let λ be a finite-dimensional representation of a connected nilpotent group G and U be a unitary representation of G. We investigate the structure of the extensions of λ by U   and, correspondingly, the group H1(λ,U)H1(λ,U) of 1-cohomologies. A spectral criterion of triviality of H1(λ,U)H1(λ,U) is proved and systematically used in the study of various types of decomposition of the extensions. We consider a special type of (λ,U)(λ,U)-cocycles – neutral cocycles, which play a crucial role in the theory of J  -unitary representations of groups on Pontryagin ΠkΠk-spaces.