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## Characteristic classes of flags of foliations and lie algebra cohomology

Transformation Groups. 2016. Vol. 21. No. 2. P. 479-518.

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.