Diagonal p-permutation functors

Let k be an algebraically closed field of positive characteristic p, and let F be an algebraically closed field of characteristic 0. We consider the F-linear category FppkΔ of finite groups, in which the set of morphisms from G to H is the F-linear extension FTΔ(H,G) of the Grothendieck group TΔ(H,G) of p-permutation (kH,kG)-bimodules with (twisted) diagonal vertices. The F-linear functors from FppkΔ to F-Mod are called diagonal p-permutation functors. They form an abelian category FppkΔ. We study in particular the functor FTΔ sending a finite group G to the Grothendieck group FT(G) of p-permutation kG-modules, and show that FTΔ is a semisimple object of FppkΔ, equal to the direct sum of specific simple functors parametrized by isomorphism classes of pairs (P,s) of a finite p-group P and a generator s of a p′-subgroup acting faithfully on P. This leads to a precise description of the evaluations of these simple functors. In particular, we show that the simple functor indexed by the trivial pair (1,1) is isomorphic to the functor sending a finite group G to FK0(kG), where K0(kG) is the Grothendieck group of projective kG-modules.