An Ennola duality for subgroups of groups of Lie type

TL;DR

This paper develops a theory of Ennola duality for subgroups of finite groups of Lie type, relating twisted and untwisted groups, and explores related conjectures about orthogonal representations and cyclotomic fields.

Contribution

It extends Ennola duality to subgroups of various Lie types, including exceptional types, and proposes conjectures about orthogonal forms and cyclotomic fields in representations.

Findings

Established Ennola duality for types other than A.

Proved results for exceptional types with known maximal subgroups.

Conjectured a relation between orthogonal form determinants and cyclotomic fields.

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