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

Highest weight vectors for the adjoint action of GL_n on polynomials