Orbits and invariants of the unitriangular group

Hilbert’s fourteenth problem asks whether the algebra of invariants for an action of a linear algebraic group is finitely generated.
This is true for reductive groups and the problem is open for unipotent groups. We discuss the case of the adjoint action of a maximal unipotent subgroup U in GL_n(K) on the nilradical m of any parabolic subalgebra, where K is an algebraically closed field of zero characteristic. This action is extended to a representation in the algebra K[m]. I will show that the algebra of invariants K[m]^U is finitely generated. Besides, a set of algebraically independent invariants generating the field K(m)^U will be presented.