  We prove the conjecture by Feigin, Fuchs and Gelfand describing the Lie
algebra cohomology of formal vector fields on an $n$-dimensional space with
coefficients in symmetric powers of the coadjoint representation. We also
compute the cohomology of the Lie algebra of formal vector fields that preserve
a given flag at the origin. The latter encodes characteristic classes of flags
of foliations and was used in the formulation of the local Riemann-Roch Theorem
by Feigin and Tsygan.
  Feigin, Fuchs and Gelfand described the first symmetric power and to do this
they had to make use of a fearsomely complicated computation in invariant
theory. By the application of degeneration theorems of appropriate
Hochschild-Serre spectral sequences we avoid the need to use the methods of
FFG, and moreover we are able to describe all the symmetric powers at once.
