Cohomology of solvable lie algebras and solvmanifolds

The cohomology Hλω* (G/Γ,ℂ) of the de Rham complex Λ*(G/Γ) ⊗ ℂ of a compact solvmanifold G/Γ with deformed differential dλω = d + λω, where ω is a closed 1 -form, is studied. Such cohomologies naturally arise in Morse-Novikov theory. It is shown that, for any completely solvable Lie group G containing a cocompact lattice Γ ⊂ G, the cohomology Hλω*(G/Γ, ℂ) is isomorphic to the cohomology Hλω*(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{g}$$ \end{document}) of the tangent Lie algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{g}$$ \end{document} of the group G with coefficients in the one-dimensional representation ρλω : \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{g}$$ \end{document} → \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{K}$$ \end{document} defined by ρλω(ξ) = λω(ξ). Moreover, the cohomology Hλω*(G/Γ,ℂ) is nontrivial if and only if -λ[ω] belongs to a finite subset \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\tilde \Omega _\mathfrak{g} $$ \end{document} of H1(G/Γ,ℂ) defined in terms of the Lie algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{g}$$ \end{document}.