Optimal SL(2)-homomorphisms

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dc.creatorMcNinch, George Joseph
dc.date2003-09-23
dc.date2005-02-21
dc.date.accessioned2026-07-07T06:24:42Z
dc.date.available2026-07-07T06:24:42Z
dc.descriptionLet G be a semisimple group over an algebraically closed field of very good characteristic for G. In the context of geometric invariant theory, G. Kempf has associated optimal cocharacters of G to an unstable vector in a linear G-representation. If the nilpotent element X in Lie(G) lies in the image of the differential of a homomorphism SL(2) --> G, we say that homomorphism is optimal for X, or simply optimal, provided that its restriction to a suitable torus of SL(2) is optimal for X in Kempf's sense. We show here that any two SL(2)-homomorphisms which are optimal for X are conjugate under the connected centralizer of X. This implies, for example, that there is a unique conjugacy class of principal homomorphisms for G. We show that the image of an optimal SL(2)-homomorphism is a completely reducible subgroup of G; this is a notion defined recently by J-P. Serre. Finally, if G is defined over the (arbitrary) subfield K of k, and if X in Lie(G)(K) is a K-rational nilpotent element whose p-th power is 0, we show that there is an optimal homomorphism for X which is defined over K.
dc.descriptionAMS-LaTeX, 26 pages. To appear in Comment. Math. Helv. The most substantial modification found in the revision is a proof of the G(K)-conjugacy of any 2 optimal SL(2)-homomorphisms for X in Lie(G)(K) which are defined over K; see Prop/Def 21 and Theorem 47
dc.identifierhttps://arxiv.org/abs/math/0309385
dc.identifierhttp://arxiv.org/abs/math/0309385
dc.identifierComment. Math. Helv. 80 (2005), 391 -- 426.
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/96638
dc.subjectRepresentation Theory
dc.subjectAlgebraic Geometry
dc.subject20G15
dc.titleOptimal SL(2)-homomorphisms
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