  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 polynomial
functions on g and let k[g]^G be the algebra of invariants under the
conjugation action of G. For all weights chi in Z^n with chi_2<=0 or
chi_{n-1}>=0 we give explicit bases for the k[g]^G-module k[g]^U_chi of highest
weight vectors of weight chi. This extends earlier results to a much bigger
class of weights. To express our semi-invariants in terms of matrix powers we
prove certain Cayley-Hamilton type identities.
