Equivalences between blocks of p-local Mackey algebras

Let G be a finite group and (K,O,k) be a p-modular system. Let R=O or k. There is a bijection between the blocks of the group algebra and the blocks of the so-called p-local Mackey algebra μR1(G). Let b be a block of RG with abelian defect group D. Let b′ be its Brauer correspondant in NG(D). It is conjectured by Broué that the blocks RGb and RNG(D)b′ are derived equivalent. Here we look at equivalences between the corresponding blocks of p-local Mackey algebras. We prove that an analogue of the Broué's conjecture is true for the p-local Mackey algebras in the following cases: for the principal blocks of p-nilpotent groups and for blocks with defect 1.