Cohomology of some nilpotent subalgebras of the Virasoro and Kac-Moody Lie algebras

The homology of the Lie algebra of algebraic vector fields in the complex line with trivial 3-jet at the point 0 with the coefficients in irreducible highest weight representations of the Virasoro Lie algebra is calculated. The same is done for vector fields with trivial 1-jets at two distinguished points. The class of quasi- finite representations of the Virasoro Lie algebra naturally arises which is the substitute for the class of finite-dimensional representations. The similar results for Kac-Moody Lie algebras are given as well as some conjectures and announcements.