Let $n$ be a positive integer and $q$ a prime power. We prove that a refined version of Brou\'{ e}'s abelian defect group conjecture holds for unipotent $\ell$-blocks of ${ \rm GL}_n(q)$, where $\ell\nmid q$. We also give a sufficient condition on general $\ell$-blocks of ${ \rm GL}_n(q)$ to satisfy the refined abelian defect group conjecture. We explain by an example that this sufficient condition does not hold in general.
Categories: math.RT math.GR