Highest weight vectors for the adjoint action of GL_n on polynomials

Let G=GL_n be the general linear group over an algebraically closed field k and let g=gl_n be its Lie algebra. Let U be the subgroup of G which consists of the upper unitriangular matrices. Let k[g] be the algebra of regular functions on $\g$. For 2(n-1)-1 weights we give explicit bases for the k[g]^G-module k[g]^U_\lambda of highest weight vectors of weight \lambda. For 5 of those weights we show that this basis is algebraically independent over the invariants k[g]^G and generates the k[g]^G-algebra $\bigoplus_{r\ge0}k[\g]^U_{r\lambda}$. Finally we formulate a question which asks whether in characteristic zero k[g]^G-module generators of k[g]^U_\lambda can be obtained by applying one explicit highest weight vector of weight \lambda in the tensor algebra T(g) to varying tuples of fundamental invariants.