Recall that Lannes’ \(T\)-functor is the left adjoint to the tensor product in unstable modules over the mod \(p\) Steenrod algebra with \(H\), the mod \(p\) cohomology of \(B \mathbb{Z}/p\). Understanding the action of \(T\) on the summands (in unstable modules) of \(H^*V\), the mod \(p\) group cohomology of an elementary abelian \(p\)-group \(V\), is of significant interest. These summands are in one-to-one correspondence with the projective \(\mathbb{F}_p[\mathrm{End} (V)]\)-modules; the summand corresponding to \(P\) is denoted \(L_P\).
The authors prove that \(T L_P = L_ P \oplus H \otimes L_{\delta (P)}\), for \(\delta (P)\) a projective \(\mathbb{F}_p[\mathrm{End} (V')]\)-module, where \(V' \oplus \mathbb{Z}/p \cong V\). Importantly, \(\delta\) can be understood as a functor. A striking property is that the reduced \(T\)-functor of \(L_P\) is always a free \(H\)-module in the category of unstable modules, namely \(H \otimes L_{\delta (P)}\).
The calculation of \(T H^* V\) is due to Lannes: \(T H^* V \cong \mathbb{F}_p^{V} \otimes H^* V\). By choice of a non-zero form in \(V^*\), the right hand side can be considered as an \(H\)-module, but this structure is not natural with respect to \(V\). The keystone of the paper is the elegant observation that, for the purposes of the calculation, the right hand side can be replaced by \(\mathbb{F}_p [V^*] \otimes H^* V\) and the latter has a natural \(H\)-module structure. The reduced \(T\)-functor then corresponds to restricting to the augmentation ideal \(\overline{\mathbb{F}_p [V^*]}\); this is free as an \(H\)-module.
The results of the paper follow directly from this, in particular the functoriality of \(\delta\). The underlying algebra is related to Harish-Chandra restriction for representations of \(\mathrm{Aut}(V)\); this follows from the identification \(\mathbb{F}_p \otimes_ H (\overline{\mathbb{F}_p [V^*]} \otimes H^* V) \cong \bigoplus_{\phi \in V^* \backslash 0} H^*(\ker \phi)\).