Simple wild L-packets

In a recent paper, Gross and Reeder study arithmetic properties of discrete Langlands parameters for semi-simple p-adic groups and conjecture that a special class of these -- the simple wild parameters -- should correspond to L-packets consisting of simple supercuspidal representations. We provide a construction of this correspondence and show that the simple wild L-packets satisfy many expected properties. In particular, they admit a description in terms of the Langlands dual group and contain a unique generic element for a fixed Whittaker datum. Moreover, we prove their stability on an open subset of the regular semi-simple elements, and show that they satisfy a natural compatibility with respect to unramified base-change.