  Based upon the general theory, developed by the author, on the
parametrization of the irreducible representations of the hyper special compact
groups corresponding to the regular adjoint orbit, supercuspidal
representations of $SL_n(F)$ are explicitly constructed for which the formal
degree conjecture and the root number conjecture are verified with respect to
certain $L$-parameter defined, by means of Kaletha, that is, the local
Langlands correspondence of tori and the Langlands-Schelstad procedure, by the
data parametrizing the irreducible representations of the hyper special compact
subgroup $SL_n(O_F)$.
