A geometric approach to complete reducibility

• 成果类型: Article
• 署名作者: Bate, M; Martin, B; Röhrle, G
• 署名单位: University of Birmingham; University of Canterbury
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-004-0425-9
• 发表日期: 2005
• 页码: 177-218
• 关键词: conjugacy classes; etale slices; lie-algebras; SUBGROUPS; semisimplicity; ELEMENTS

摘要

Let G be a connected reductive linear algebraic group. We use geometric methods to investigate G-completely reducible subgroups of G, giving new criteria for G-complete reducibility. We show that a subgroup of G is G-completely reducible if and only if it is strongly reductive in G; this allows us to use ideas of R.W. Richardson and Hilbert-Mumford-Kempf from geometric invariant theory. We deduce that a normal subgroup of a G-completely reducible subgroup of G is again G-completely reducible, thereby providing an affirmative answer to a question posed by J.-P. Serre, and conversely we prove that the normalizer of a G-completely reducible subgroup of G is again G-completely reducible. Some rationality questions and applications to the spherical building of G are considered. Many of our results extend to the case of non-connected G.