Descent of splendid Rickard equivalences in GLn(q)

Abstract

Let n be a positive integer and q a prime power. We prove that a refined version of Broué's abelian defect group conjecture holds for unipotent ℓ-blocks of GLn(q), where ℓ∤q. We also give a sufficient condition on general ℓ-blocks of GLn(q) to satisfy the refined abelian defect group conjecture.