On the conjugacy classes in maximal unipotent subgroups of simple algebraic groups
On the conjugacy classes in maximal unipotent subgroups of simple algebraic groups
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关于简单代数群的最大单能子群的共轭类
DOI：
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发表时间：
2006
期刊：
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通讯作者：
Simon M. Goodwin
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文献类型：
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作者：
Simon M. Goodwin
设G是特征p ≥ 0的代数闭域k上的单代数群.假设p为零或对G有利。设B是G的Borel子群，记U为B的幂幺根，记u为U的李代数。利用相对Springer同构分析了U在u中的伴随轨道。特别是，我们表明，伴随轨道的U在u包含一个独特的所谓的最小代表。在p > 0的情况下，假设G被定义并在p个元素的有限域Fp上分裂。设q是p的幂，G（q）是G的有限Fq-有理点群.设F是Frobenius态射，使得G（q）= GF.设B是F-稳定的，使得U也是F-稳定的，U（q）是G（q）的Sylow p-子群.我们证明了U（q）的共轭类与U在u中的F-稳定伴随轨道相对应。这使我们能够推导出关于U（q）的共轭类的结果。
Let G be a simple algebraic group over the algebraically closed field k of characteristic p ≥ 0. Assume p is zero or good for G. Let B be a Borel subgroup of G; we write U for the unipotent radical of B and u for the Lie algebra of U. Using relative Springer isomorphisms} we analyze the adjoint orbits of U in u. In particular, we show that an adjoint orbit of U in u contains a unique so-called minimal representative. In case p > 0, assume G is defined and split over the finite field of p elements Fp. Let q be a power of p and let G(q) be the finite group of Fq-rational points of G. Let F be the Frobenius morphism such that G(q) = GF. Assume B is F-stable, so that U is also F-stable and U(q) is a Sylow p-subgroup of G(q). We show that the conjugacy classes of U(q) are in correspondence with the F-stable adjoint orbits of U in u. This allows us to deduce results about the conjugacy classes of U(q).