Formulas for tamely ramified supercuspidal characters of 𝐺𝐿₃

Let F F denote a p p -adic local field of residual characteristic p ≠ 3 p\ne 3 . This article gives formulas, valid on the regular elliptic set, for the irreducible supercuspidal characters of GL 3 ⁡ ( F ) \operatorname {GL}_3(F) which correspond to characters of a ramified Cartan subgroup. In the case in which F F does not contain cube roots of unity, i.e., the case in which ramified cubic extensions of degree 3 3 over F F cannot be Galois, base change results concerning “simple types" due to Bushnell and Henniart (1996) are used in the proofs.