The local Langlands correspondence for inner forms of SL$$_{n}$$n

By Anne-Marie Aubert, P. Baum, R. Plymen, M. Solleveld. 2013

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

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/07490d0680c6bdcf9e60dc48d5d0b040ec77fbdf