  Let K be a non-archimedean local field and let G be a connected reductive
K-group which splits over an unramified extension of K. We investigate
supercuspidal unipotent representations of the group G(K). We establish a
bijection between the set of irreducible G(K)-representations of this kind and
the set of cuspidal enhanced L-parameters for G(K), which are trivial on the
inertia subgroup of the Weil group of K. The bijection is characterized by a
few simple equivariance properties and a comparison of formal degrees of
representations with adjoint $\gamma$-factors of L-parameters.
  This can be regarded as a local Langlands correspondence for all
supercuspidal unipotent representations. We count the ensueing L-packets, in
terms of data from the affine Dynkin diagram of G. Finally, we prove that our
bijection satisfies the conjecture of Hiraga, Ichino and Ikeda about the formal
degrees of the representations.
