On the orbit types of the action of the gauge group on connections modulo pointed gauge transformations

For a principal SU(n)-bundle over a compact manifold of dimension 2,3,4, we determine the orbit types of the action of the gauge group on the space of connections modulo pointed local gauge transformations. We find that they are given by Howe subgroups of SU(n) for which a certain characteristic equation is solvable. Depending on the base manifold, this equation leads to a linear, bilinear, or quadratic Diophantine equation.