Let G =GLn be the general
linear group over an algebraically closed field k, and let g=gln be its Lie algebra.
Let U be the subgroup of G that consists of the upper unitriangular matrices. Let
k[g] be the algebra of polynomial functions on g, and let k[g]G be the algebra of
invariants under the conjugation action of G. For certain special weights, we give
explicit bases for the k[g]G-module k[g]λU of highest-weight vectors of weight λ.
For five of these special weights, we show that this basis is algebraically
independent over k[g]G and generates the k[g]G-algebra ⊕r≥0k[g]rλU. Finally, we
formulate the question whether in characteristic zero, k[g]G-module generators
of k[g]λU can be obtained by applying one explicit highest-weight vector
of weight λ in the tensor algebra T(g) to varying tuples of fundamental
invariants.
