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Working paper

Lie algebra cohomology representing characteristic classes of flags of foliations

This article contains the computation of the Lie algebra cohomology of the infinite-dimensional Lie algebra of formal vector fields with coefficients in symmetric powers of the coadjoint representation. At the same time we compute the cohomology of the Lie algebra of formal vector fields that preserve a given flag at the origin. The resulting cohomology are known to be responsible for the characteristic classes of the flags of foliations and are well used in the local Riemann-Roch theorem by Feigin and Tsygan and later on by Felder,Shoikhet and collaborators.      We use the degeneration theorems of appropriate Hochschild-Serre spectral sequences and provide the method which allows us to avoid one of the most complicated computation in the invariant theory which was done by Gelfand, Feigin and Fuchs in order to cover the case of first symmetric power. The method we use gives a uniform and beautiful answer for all symmetric powers at the same time.