Maximal subgroups of maximal rank in the classical algebraic groups

Let $k$ be an arbitrary field. We classify the maximal reductive subgroups of maximal rank in any classical simple algebraic $k$-group in terms of combinatorial data associated to their indices. This result complements [S, 2022], which does the same for the exceptional groups. We determine which of these subgroups may be realised over a finite field, the real numbers, or over a $\mathfrak{p}$-adic field. We also look at the asymptotics of the number of such subgroups as the rank grows large.

💡 Research Summary

This paper completes the classification of maximal reductive subgroups of maximal rank in classical simple algebraic groups over an arbitrary field k, using a refined equivalence relation called index‑conjugacy. The authors first recall the notion of the Tits index I(G), a combinatorial invariant consisting of a Dynkin diagram together with a Galois action on the simple roots. For a maximal‑rank reductive subgroup H⊂G they define the embedding of indices (I(G), I(H), θ), where θ is a linear map between the character lattices induced by a suitable conjugation. Two subgroups are index‑conjugate precisely when they share the same embedding of indices; this relation lies strictly between G(k̄)‑conjugacy and G(k)‑conjugacy.

The core of the work is a systematic enumeration of all possible embeddings of indices for each classical type Aₙ, Bₙ, Cₙ, D₄, and Dₙ (n>4). This is achieved by analysing almost‑primitive subsystems of the corresponding root systems, a theory originally developed in the context of Tits’ work and refined in the companion paper