On the conjugacy classes in maximal unipotent subgroups of simple algebraic groups

Let G be a simple algebraic group over the algebraically closed field k of characteristic p >= 0. Assume p is zero or good for G. Let B be a Borel subgroup of G; we write U for the unipotent radical of B and u for the Lie algebra of U. Using relative Springer isomorphisms we analyze the adjoint orbits of U in u. In particular, we show that an adjoint orbit of U in u contains a unique so-called minimal representative. In case p > 0, assume G is defined and split over the finite field of p elements F-p. Let q be a power of p and let G(q) be the finite group of F-q-rational points of G. Let F be the Frobenius morphism such that G(q) = G(F). Assume B is F-stable, so that U is also F-stable and U(q) is a Sylow p-subgroup of G(q). We show that the conjugacy classes of U(q) axe in correspondence with the F-stable adjoint orbits of U in u. This allows us to deduce results about the conjugacy classes of U(q).