  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.
