Nilpotent Elements and Reductive Subgroups Over a Local Field
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Nilpotent Elements and Reductive Subgroups Over a Local Field George J. McNinch1 Received: 14 October 2019 / Accepted: 18 September 2020 / © Springer Nature B.V. 2020
Abstract Let K be a local field – i.e. the field of fractions of a complete DVR A whose residue field k has characteristic p > 0 – and let G be a connected, absolutely simple algebraic K-group G which splits over an unramified extension of K. We study the rational nilpotent orbits of G– i.e. the orbits of G(K) in the nilpotent elements of Lie(G)(K) – under the assumption p > 2h − 2 where h is the Coxeter number of G. A reductive group M over K is unramified if there is a reductive model M over A for which M = MK . Our main result shows for any nilpotent element X1 ∈ Lie(G) that there is an unramified, reductive K-subgroup M which contains a maximal torus of G and for which X1 ∈ Lie(M) is geometrically distinguished. The proof uses a variation on a result of DeBacker relating the nilpotent orbits of G with the nilpotent orbits of the reductive quotient of the special fiber for the various parahoric group schemes associated with G. Keywords Reductive groups · Local fields · Nilpotent orbits Mathematics Subject Classification (2010) 20G15 · 20G25 · 20G07
1 Introduction Let G be a connected and reductive algebraic group G over a field F. If Falg is an algebraic (or even just separable) closure, there is a central isogeny S ×G1 ×· · ·×Gd → GFalg where S is a torus and where each Gi is quasisimple over Falg . If Ri denotes the (irreducible) root system associated with Gi , then up to re-ordering, the list of root systems R1 , . . . , Rd is independent of any choices made. We write h for the supremum of the Coxeter number of the Ri , and throughout this paper, we refer to h as “the Coxeter number of G”.
Presented by: Michel Brion George J. McNinch
[email protected] 1
Department of Mathematics, Tufts University, 503 Boston Avenue, Medford, MA, 02155, USA
G.J. McNinch
1.1 Nilpotent Elements and their Orbits In this paper, we consider the rational orbits of G in its adjoint action on Lie(G) – i.e. the orbits of the group of F-points G(F) on the elements of Lie(G) = Lie(G)(F) 1 . More precisely, we study the rational orbits of nilpotent elements. An element X ∈ Lie(G) is nilpotent if dρ(X) is a nilpotent endomorphism of V for every algebraic representation (ρ, V ) of G. From another perspective, X is nilpotent provided that the derivation of the coordinate algebra F[G] determined by X act locally nilpotently on F[G] – see [38, Sections 2.4 and 4.4]; since E [G] = F[G] ⊗F E , this makes clear that X ∈ Lie(G) is nilpotent if and only if X is nilpotent in Lie(GE ) for some (any) field extension E of F. When the characteristic of F is bad for G, the adjoint orbits have some pathological properties – for example, in that case the scheme theoretic centralizer of a nilpotent element may fail to be smooth (i.e. reduced) over F. To avoid such problems, we suppose G to be a standard reductive group. Among other nice properties, one knows when G is standard
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