Non-basic rigid packets for discrete L-parameters

We formulate a new version of the local Langlands correspondence for discrete L-parameters which involves (Weyl orbits of) packets of representations of all twisted Levi subgroups of a connected reductive group G through which a given parameter factors and prove that this version of the correspondence is true if one assumes the pre-existing local Langlands conjectures. Twisted Levi subgroups are crucial objects in the study of supercuspidal representations; this work is a step towards deepening the relationship between the representation theory of p-adic groups and the Langlands correspondence. This talk will also serve as a rough introduction to the refined local Langlands correspondence (and its generalizations). This is joint work with David Schwein.