It is expected that the local Langlands correspondence gives a parameterization of smooth irreducible representations of a connected reductive group over a local field by its enhanced L-parameters.
In this talk, I will present the construction of such a map for depth-zero supercuspidal representations when the group is adjoint and simple. I will also explain that this map satisfies, if it is bijective, the Hiraga—Ichino—Ikeda conjecture on the formal degree of essentially square-integrable irreducible representations.