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 those 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 ${}^2\mathrm D_{2n}$.)