Homology of the Lie Algebra of Vector Fields on the Line with Coefficients in Symmetric Powers of its Adjoint Representation

We compute the homology of the Lie algebra $W_1$ of (polynomial) vector fields on the line with coefficients in symmetric powers of its adjoint representation. We also list the results obtained so far for the homology with coefficients in tensor powers and, in turn, use them for partially computing the homology of the Lie algebra of $W_1$-valued currents on the line.