Problems arising (and some solutions) from Vogan's conception of the local Langlands correspondence

David Vogan's 1993 paper on the local Langlands correspondence showed that there are advantages to simultaneously considering representations of a p-adic group together with its pure inner forms. Combined with other insights, this opened the door to introducing ABV-packets for p-adic groups, which provide a vast generalization of A-packets, and also suggested questions about the structure of the category of perverse sheaves on a variety of certain Langlands parameters. In this talk we survey recent developments on these matters, including: ABV-packets for non-Arthur type Langlands parameters of GL(N); ABV-packets for unipotent representations of G(2); and comparing the category of perverse sheaves, above, with the Koszul dual of the corresponding generalized Steinberg representations. We will also discuss several problems not yet resolved that are on the horizon.