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

Abstract

Let k be any field and let G be a connected reductive algebraic k-group. Associated to G is an invariant first studied by Satake [23,24] and Tits [31] that is called the index of G (a Dynkin diagram along with some additional combinatorial information). Tits [31] 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, Satake [23] and Tits [31] classified all possibilities for the index of G. 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 ⊂ 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 ⊂ G. Finally, we establish some existence results. In particular, we consider which embeddings of indices exist when k has cohomological dimension 1 (resp. k = R, k is p-adic).