The irreducible smooth representations of p-adic reductive groups and the enhanced Langlands parameters of these latter can both be partitioned into series indexed by "cuspidal data". On the representation side, cuspidality refers to supercuspidal representations of Levi subgroups, while on the Galois side, it refers to "cuspidal unipotent pairs", as introduced by Lusztig, in certain subgroups of the Langlands dual groups.
In addition, on both sides, the elements in a given series are in bijection with the simple modules of a generalized affine Hecke algebra.
The cuspidal data on one side are expected to be in bijection with the cuspidal data on the other side. We will formulate conditions on this bijection that will guarantee the existence of a bijection between the simple modules of the attached generalized affine Hecke algebras. For the exceptional group of type G_2 and for all pure inner forms of quasi-split classical groups, the Hecke algebras are actually isomorphic.