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

Let k be any field and let G be a connected reductive algebraic k-group. Associated to 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 (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-isogeny class of G is uniquely determined by its index and the k-isogeny class of its anisotropic kernel G_a. For the cases where G is absolutely simple, all possibilities for the index of 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 be a connected reductive k-subgroup of maximal rank in G. We introduce an invariant of the G(k)-conjugacy class of H in G called the embedding of indices of H \subset G. This consists of the index of H and the index of G along with an embedding map that satisfies certain compatibility conditions. We introduce an equivalence relation called index-conjugacy on the set of k-subgroups of G, and observe that the G(k)-conjugacy class of H in G is determined by its index-conjugacy class and the G(k)-conjugacy class of H_a in G. We show that the index-conjugacy class of H in G is uniquely determined by its embedding of indices. For the cases where G is absolutely simple of exceptional type and H is maximal connected in G, we classify all possibilities for the embedding of indices of H \subset G. Finally, we establish some existence results. In particular, we consider which embeddings of indices exist when k has cohomological dimension 1 (resp. k=\mathbb {R}, k is \mathfrak {p}-adic).