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 J 1 in dimension 56 given. (This tallies with Ennola duality for representations, where type D 2 n has no Ennola duality with 2 D 2 n .)