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The local Langlands correspondence for inner forms of \(\mathrm{SL}_{n}\). (English) Zbl 1394.22015
Summary: Let \(F\) be a non-archimedean local field. We establish the local Langlands correspondence for all inner forms of the group \(\mathrm{SL}_n (F)\). It takes the form of a bijection between, on the one hand, conjugacy classes of Langlands parameters for \(\mathrm{SL}_n (F)\) 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 \(\mathrm{SL}_n (F)\) 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 \(\mathrm{GL}_n (F)\), when the fields are close enough compared to the depth of the representations.

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
20G05 Representation theory for linear algebraic groups
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