An Ennola duality for subgroups of groups of Lie type

Abstract

We develop a theory of Ennola duality for subgroups of finite groups of Lie type, relating subgroups of twisted and untwisted groups of the same type. Roughly speaking, one finds that subgroups $H$ of $\mathrm{GU}_d(q)$ correspond to subgroups of $\mathrm{GL}_d(-q)$, where $-q$ is interpreted modulo $|H|$. Analogous results for types other than $\mathrm A$ are established, including for exceptional types where the maximal subgroups are known, although the result for type $\mathrm D$ is still conjectural. Let $M$ denote the Gram matrix of a non-zero orthogonal form for a real, irreducible representation of a finite group, and consider $\alpha=\sqrt{\det(M)}$. If the representation has twice odd dimension, we conjecture that $\alpha$ lies in some cyclotomic field. This does not hold for representations of dimension a multiple of $4$, with a specific example of the Janko group $\mathrm J_1$ in dimension $56$ given. (This tallies with Ennola duality for representations, where type $\mathrm D_{2n}$ has no Ennola duality with ${}^2\mathrm D_{2n}$.)