The inductive blockwise Alperin weight condition for PSLn(q) and PSUn(q) with cyclic outer automorphism groups

In this paper, we prove that if (n,q) is such chosen that the outer automorphism groups of PSLn(q) and PSUn(q) are cyclic, PSLn(q) and PSUn(q) satisfy the inductive blockwise Alperin weight condition for any prime ℓ different from the defining characteristic. To do this, it suffices to study actions of automorphisms on the Brauer characters and weights of GLn(q) and GUn(q) for ℓ, and to show that the blockwise bijection between Brauer characters and weights given in [1], [3], [4] and [5] is equivariant.