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Let X be an F-rational nilpotent element in the Lie algebra of a connectedand reductive group G defined over the ground field F. Suppose that the Liealgebra has a non-degenerate invariant bilinear form. We show that theunipotent radical of the centralizer of X is F-split. This property has severalconsequences. When F is complete with respect to a discrete valuation witheither finite or algebraically closed residue field, we deduce a uniform proofthat G(F) has finitely many nilpotent orbits in Lie(G)(F). When the residuefield is finite, we obtain a proof that nilpotent orbital integrals converge.Under some further (fairly mild) assumptions on G, we prove convergence forarbitrary orbital integrals on the Lie algebra and on the group. Theconvergence of orbital integrals in the case where F has characteristic 0 wasobtained by Deligne and Ranga Rao (1972).Comment: 32 pages, AMSLaTeX. To appear: Math. Annalen. This version has a new  title; it also contains various corrections of typographic errors and such.  More significantly, it contains "cleaner" statements of convergence for  unipotent (as opposed to nilpotent) orbital integrals; see sections 8.5-8.

Nilpotent orbits over ground fields of good characteristic