Parametrization of supercuspidal representations of depth zero for some simple adjoint groups

We construct a surjective map from the set of conjugacy classes of depth-zero cuspidal enhanced L-parameters to that of isomorphism classes of depth-zero supercuspidal representations for simple adjoint groups, and check the bijectivity in various cases. We also prove that the Hiraga--Ichino--Ikeda conjecture on the formal degree of essentially square-integrable irreducible representations holds for this parametrization if it is bijective.