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 G . We show that the index-conjugacy class of H H in G G is uniquely determined by its embedding of indices. For the cases where G G is absolutely simple of exceptional type and H H is maximal connected in G G , we classify all possibilities for the embedding of indices of H ⊂ G H \subset G . Finally, we establish some existence results. In particular, we consider which embeddings of indices exist when k k has cohomological dimension 1 (resp. k = R k=\mathbb {R} , k k is p \mathfrak {p} -adic).