The local Langlands correspondence for inner forms of SL_n

This is a beamer presentation of MIMS eprint 2013.38 of the same title (joint authors are A-M. Aubert, P. Baum, R. Plymen, M. Solleveld). 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 $SL_n (F)$. An analogous result is shown in the archimedean case. 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