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 SL1(D), where D is a division algebra of prime degree ℓ over a local field of odd residual characteristic different from ℓ. For ℓ odd, we show that all such characters of SL1(D) are induced from linear characters of compact-open subgroups of SL1(D). We also compute an explicit formula for the representation zeta function of SL1(D).