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In [1], I. N. Bernshtein and B. I. Rozenfeld introduced a new language which expresses differential geometric problems in terms of infinite-dimensional Lie algebras and their homogeneous spaces. In particular, their method is naturally applied to the formulation of characteristic classes of foliations. Let ctn be the Lie algebra of all formal vector fields on R. Then their results show that each element of the (continuous) cohomologies of closed transitive infinite Lie subalgebras of an can be regarded as an invariant of foliations with respect to the corresponding structures such as the complex structure. Many partial results, concerning the cohomologies of the subalgebras of an such as the Lie algebra of (formal) Hamiltonian vector fields, have appeared since I. M. Gel'fand and D. B. Fuks calculated the cohomology of &„ by the epoch-making method. But it seems that the difficulty in calculating such cohomologies hinders discovering new invariants in this approach. In this paper, although we do not obtain any new invariants, we reveal a new relation between the classical characteristic classes (e. g. Chern classes) and the secondary classes of foliations (e. g. the Godbillon-Vey class), by using the information we obtained about the cohomologies of the Lie algebras of formal G-invariant vector fields an 8. Here G is a finite dimensional Lie group with its Lie algebra Q, and the Lie algebra of formal G-invariant vector fields on

Continuous cohomologies of Lie algebras of formal {$G$}-invariant vector fields and obstructions to lifting foliations