Lattices invariant under the affine general linear group (Q1886826)

Let \(V\) be the additive group of the field of \(p^{mt}\) elements. Then the affine general linear group \(G = \text{AGL}_{m}(p^{t})\) acts (doubly transitively) on \(V\) and hence on the group ring \(\Lambda = {\mathbb Z}[V]\), fixing a natural scalar product on it. The main result of the paper classifies all sublattices \(\Lambda'\) of \(\Lambda\) which are invariant under \(G\) (and contain \(p^{k}\Lambda\) for some \(k \geq 0\)). The cases \(m = 1\) and \(t = 1\) were handled in previous papers of the author. In the case \(p^{t} = 2\), since \(G\) can be embedded in the automorphism group of the Barnes-Wall lattice \(\text{BW}_{2^{m}}\), one should expect to have \(\text{BW}_{2^{m}}\) realized as one of the invariant sublattices, and one such explicit construction is presented by the author.