An Ennola duality for subgroups of groups of Lie type

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 GU_d(q) correspond to subgroups of GL_d(−q), where −q is interpreted modulo |H|. Analogous results for types other than A are established, including for those exceptional types where the maximal subgroups are known, although the result for type 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 α = √det(M). If the representation has twice odd dimension, we conjecture that α 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 J₁ in dimension 56 given. (This tallies with Ennola duality for representations, where type D_2n has no Ennola duality with ²D_2n.)