Maximal connected 𝑘-subgroups of maximal rank in connected reductive algebraic 𝑘-groups

Abstract

Let k k be any field and let G G be a connected reductive algebraic k k -group. Associated to G G is an invariant first studied in the 1960s by Satake [Ann. of Math. (2) 71 (1960), 77–110] and Tits [Théorie des Groupes Algébriques (Bruxelles, 1962), Librairie Universitaire, Louvain; Gauthier- Villars, Paris, 1962], [Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62] that is called the index of G G (a Dynkin diagram along with some additional combinatorial information). Tits [Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62] showed that the k k -isogeny class of G G is uniquely determined by its index and the k k -isogeny class of its anisotropic kernel G a G_a . For the cases where G G is absolutely simple, all possibilities for the index of G G have been classified in by Tits [Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62]. Let H H be a connected reductive k k -subgroup of maximal rank in G G . We introduce an invariant of the G ( k ) G(k) -conjugacy class of H H in G G called the embedding of indices of H ⊂ G H \subset G . This consists of the index of H H and the index of G G along with an embedding map that satisfies certain compatibility conditions. We introduce an equivalence relation called index-conjugacy on the set of k k -subgroups of G G , and observe that the G ( k ) G(k) -conjugacy class of H H in G G is determined by its index-conjugacy class and the G ( k ) G(k) -conjugacy class of H a H_a in G

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连通约化代数中最大秩的最大连通𝑘-subgroups𝑘-groups

设k k是任意域，G G是一个连通的约化代数k k群。与G相关的G是20世纪60年代由Satake [Ann]首次研究的不变量。的数学。(2) 71(1960)， 77-110)和Tits [thsamorie des Groupes algbriques(布鲁塞尔，1962)，鲁汶大学图书馆;[代数群与不连续子群]，《数学学报》，2002。纯粹数学。，博尔德，科罗拉多州，1965)，美国。数学。Soc。， Providence, r.i.， 1966, pp. 33-62]，被称为G G的索引(带有一些附加组合信息的Dynkin图)。代数群与不连续子群。纯粹数学。，博尔德，科罗拉多州，1965)，美国。数学。Soc。(Providence, r.i.， 1966, pp. 33-62)证明了G G的k k等同性类是由它的指数和它的各向异性核G a G_a的k k等同性类唯一决定的。对于G G绝对简单的情况，G G索引的所有可能性都被Tits[代数群和不连续子群](Proc. Sympos)分类。纯粹数学。，博尔德，科罗拉多州，1965)，美国。数学。Soc。[j]，《普罗维登斯，罗德岛，1966》，第33-62页。设H H是G G中最大秩的连通约化k k子群。我们在G G中引入H H的G(k) G(k)共轭类的一个不变量，称为H∧G H \子集G的指标嵌入。它由H H索引和G G索引以及满足一定兼容条件的嵌入图组成。在G G的k k -子群集合上引入了一个称为指标共轭的等价关系，并观察到G G中H H的G(k) G(k) -共轭类是由G

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