In a recent paper, Gross and Reeder study the arithmetic properties of discrete Langlands parameters for semi-simple -adic groups, and they conjecture that a special class of these – the simple wild parameters – should correspond to -packets consisting of simple supercuspidal representations. We provide a construction of this correspondence, and show that the simple wild -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.

The normal residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients. We show that any S-arithmetic subgroup of a higher rank Chevalley group G has normal residual finiteness growth ndim (G).

For a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic 0, Langlands and Shelstad constructed a cohomological invariant called the splitting invariant, which is an important component of their endoscopic transfer factors. We study this invariant in the case of a split real group and prove a decomposition theorem which expresses this invariant for a general torus as a product of the corresponding invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants between different tori in the given real group.