Self-tilting complexes yield unstable modules (Q2782659)

The author continues his investigation on the action of self-derived equivalences of the group algebra on the group cohomology. Here \(G\) is a finite group, \(R\) a field of characteristic \(p\), and \(\text{TrPic}_R(RG)\) is the group (previously introduced by R.~Rouquier and the author) of standard self-equivalences of the derived module category of the group algebra \(RG\). In another paper, the author has shown that the subgroup \(HD_R(G)\) of \(\text{TrPic}_R(RG)\) consisting of self-equivalences fixing the trivial \(RG\)-module acts on the cohomology ring \(H^*(RG)\). In the current paper, Rickard's splendid equivalences are also considered. It is shown that the action of \(HD_R(G)\) commutes with the action of the Steenrod algebra on \(H^*(G,R)\) for any prime \(p\). Moreover, there are two functors between the principal block of the group algebra \(\mathbb{F}_pHD_{\mathbb{F}_p}(\mathbb{F}_pG)\) and the category of unstable modules over the mod \(p\) Steenrod algebra. Properties of these functors, especially their relationship with Lannes' \(T\)-functor are studied.