Rational cohomology of algebraic solvable groups (Q1821399)

The author proves, for algebraic solvable groups and trivial \({\mathbb{Q}}\) coefficients, an analogue of a theorem of G. Mostow and Van Est. Namely, let G be a solvable affine algebraic group over the rational numbers \({\mathbb{Q}}\) and let D be an arithmetic subgroup in G contained in the identity component \(G^ 0_{{\mathbb{R}}}\) of the real points of G. Suppose that D is cocompact in \(G^ 0_{{\mathbb{R}}}\) and the torus factor T in \(G=N\rtimes T\) is \({\mathbb{Q}}\) anisotropic and \({\mathbb{R}}\) split. Then there is an isomorphism of cohomology rings \(H^*({\mathfrak g}_{{\mathbb{Q}}}, {\mathbb{Q}})\overset \sim \rightarrow H^*(D, {\mathbb{Q}})\). This isomorphism is described explicitly.