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 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).