  Let $F$ be a non-archimedean local field. We show that any representation of
a maximal compact subgroup of $\mathbf{SL}_N(F)$ which is typical for an
essentially tame supercuspidal representation must be induced from a
Bushnell--Kutzko maximal simple type. From this, we explicitly count and
describe the conjugacy classes of such typical representations, and give an
explicit description of an inertial Langlands correspondence for essentially
tame irreducible $N$-dimensional projective representations of the Weil group
of $F$.
