  In this paper characters of the normaliser of $d$-split Levi subgroups in
$\mathrm {SL}_n(q)$ and $\mathrm {SU}_n(q)$ are parametrized with a particular
focus on the Clifford theory between the Levi subgroup and its normalizer.These
results are applied to verify the Alperin-McKay conjecture for primes $\ell$
with $\ell\nmid 6(q^2-1)$ and the Alperin weight conjecture for $\ell$-blocks
of those quasi-simple groups with abelian defect. The inductive Alperin-McKay
condition and inductive Alperin weight condition by the second author are
verified for certain blocks of $\mathrm {SL}_n(q)$ and $\mathrm {SU}_n(q)$.
