  Let X be an F-rational nilpotent element in the Lie algebra of a connected
and reductive group G defined over the ground field F. Suppose that the Lie
algebra has a non-degenerate invariant bilinear form. We show that the
unipotent radical of the centralizer of X is F-split. This property has several
consequences. When F is complete with respect to a discrete valuation with
either finite or algebraically closed residue field, we deduce a uniform proof
that G(F) has finitely many nilpotent orbits in Lie(G)(F). When the residue
field is finite, we obtain a proof that nilpotent orbital integrals converge.
Under some further (fairly mild) assumptions on G, we prove convergence for
arbitrary orbital integrals on the Lie algebra and on the group. The
convergence of orbital integrals in the case where F has characteristic 0 was
obtained by Deligne and Ranga Rao (1972).
