Elementary abelian p-subgroups of algebraic groups

Abstract

Let $\mathbb{K}$ be an algebraically closed field and let G be a finite-dimensional algebraic group over $\mathbb{K}$ which is nearly simple, i.e. the connected component of the identity G ⁰ is perfect, C _G(G ⁰)=Z(G ⁰) and G ⁰/Z(G ⁰) is simple. We classify maximal elementary abelian p-subgroups of G which consist of semisimple elements, i.e. for all primes p ≠ char $\mathbb{K}$.

Call a group quasisimple if it is perfect and is simple modulo the center. Call a subset of an algebraic group toral if it is in a torus; otherwise nontoral. For several quasisimple algebraic groups and p=2, we define complexity, and give local criteria for whether an elementary abelian 2-subgroup of G is toral.

For all primes, we analyze the nontoral examples, include a classification of all the maximal elementary abelian p-groups, many of the nonmaximal ones, discuss their normalizers and fusion (i.e. how conjugacy classes of the ambient algebraic group meet the subgroup). For some cases, we give a very detailed discussion, e.g. p=3 and G of type E ₆, E ₇ and E ₈. We explain how the presence of spin up and spin down elements influences the structure of projectively elementary abelian 2-groups in Spin(2n, ℂ). Examples of an elementary abelian group which is nontoral in one algebraic group but toral in a larger one are noted.

Two subsets of a maximal torus are conjugate in G iff they are conjugate in the normalizer of the torus; this observation, with our discussion of the nontoral cases, gives a detailed guide to the possibilities for the embedding of an elementary abelian p-group in G. To give an application of our methods, we study extraspecial p-groups in E ₈($\mathbb{K}$).