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 ${ \rm SL}_1(D)$.
Categories: math.RT