Nilpotent orbits over ground fields of good characteristic

Details Nilpotent orbits over ground fields of good characteristic Nilpotent orbits over ground fields of good characteristic

2002-09-13

arxiv.org/abs/math/0209151v2

Mathematische Annalen 329, 49--85 (2004)

Mathematics

Representation Theory

32 pages, AMSLaTeX. To appear: Math. Annalen. This version has a new title; it also contains various corrections of typographi

Scientific paper

10.1007/s00208-004-0510-9

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).

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