Characteristic classes of flags of foliations and Lie algebra cohomology

TL;DR

This paper proves a conjecture on the Lie algebra cohomology of formal vector fields, computes cohomology related to flags of foliations, and introduces a new approach that simplifies complex invariant theory computations.

Contribution

It provides a new proof of the Lie algebra cohomology conjecture and computes all symmetric powers simultaneously using spectral sequences.

Findings

Confirmed the conjecture by Feigin, Fuchs, and Gelfand.

Computed cohomology of vector fields preserving a flag.

Simplified the computation process by avoiding complex invariant theory.

Abstract

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.