A geometric approach to complete reducibility
A geometric approach to complete reducibility
复制标题
完全还原性的几何方法
DOI：
--
复制
发表时间：
2004
期刊：
影响因子：
--
通讯作者：
G. Röhrle
中科院分区：
文献类型：
--
作者：
M. Bate;B. Martin;G. Röhrle
设G是一个连通的约化线性代数群.利用几何方法研究了G的G-完全可约子群，给出了G-完全可约的新判据。我们证明了G的子群是G-完全可约的当且仅当它在G中是强可约的，这使得我们可以利用R. W. Richardson和Hilbert-Mumford-Kempf的几何不变理论。我们推出G的G-完全可约子群的正规子群也是G-完全可约的，从而对J. P. Serre的方法，反过来证明了G的G-完全可约子群的正规化子也是G-完全可约的。讨论了G的球面构造的一些合理性问题及其应用。我们的许多结果推广到了非连通G的情形。
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.