The local Langlands correspondence for inner forms of $S{L}_n$

Abstract

Let F be a non-archimedean local field. We establish the local Langlands correspondence for all inner forms of the group $S{L}_n(F)$. It takes the form of a bijection between, on the one hand, conjugacy classes of Langlands parameters for $S{L}_n(F)$ 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 $S{L}_n(F)$. 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 $G{L}_n(F)$, when the fields are close enough compared to the depth of the representations.

In the second version Theorem 5.3 was restricted to n' = n-1 and the proof was modified accordingly. Also references to the work of Ganapathy were added. It turned out that the proof of Theorem 4.4 in versions 1 and 2 was incorrect and beyond repair, so we removed this result in version 3. Consequently Theorems 4.4 and 6.1 (from v1 and v2) only remain valid with worse lower bounds

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