The local Langlands correspondence for inner forms of SL _n

Let F be a non-archimedean local field. We establish the local Langlands correspondence for all inner forms of the group SL _n(F). It takes the form of a bijection between, on the one hand, conjugacy classes of Langlands parameters for SL _n(F) enhanced with an irreducible representation of an S-group and, on the other hand, the union of the spaces of irreducible admissible representations of all inner forms of SL _n(F) up to equivalence. An analogous result is shown in the archimedean case. For p-adic fields, this is based on the work of Hiraga and Saito. To settle the case where F has positive characteristic, we employ the method of close fields. We prove that this method is compatible with the local Langlands correspondence for inner forms of GL _n(F) , when the fields are close enough compared to the depth of the representations.