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 ${\mathrm{S}\mathrm{L}}_1(D)$, where $D$ is a division algebra of prime degree $\mathrm{\ell }$ over a local field of odd residual characteristic different than $\mathrm{\ell }$. For $\mathrm{\ell }$ odd, we show that all such characters of ${\rm SL}_1(D)$areinducedfromlinearcharactersofcompact-opensubgroupsof${\rm SL}_1(D)$. We also compute an explicit formula for the representation zeta function of ${\mathrm{S}\mathrm{L}}_1(D)$.