Characters of the Norm-One Units of Local Division Algebras of Prime Degree

We give an explicit construction of all complex continuous irreducible characters of the group ${\rm SL}_1(D)$, where $D$ is a division algebra of prime degree $\ell$ over a local field of odd residual characteristic different than $\ell$. For $\ell$ odd, we show that all such characters of ${\rm SL}_1(D)$ are induced from linear characters of compact-open subgroups of ${\rm SL}_1(D)$. We also compute an explicit formula for the representation zeta function of ${\rm SL}_1(D)$.