Abstract
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.
Highlights
Let F be a non-archimedean local field
The Langlands correspondence (LLC) for inner forms of SLn(F ) is derived from the above, in the sense that every L-packet for G consists of the irreducible constituents of ResGG ( φ(G))
Via the Langlands correspondence, the additional ones are associated with irreducible representations of non-split inner forms of GLn(F )
Summary
The image should consist of all such pairs, which satisfy an additional relevance condition on ρ This form of the LLC was proven for “GLn-generic” representations of G in [29], under the assumption that the underlying local field has characteristic zero. The hope is that one can turn (2) into a bijection by defining a suitable equivalence relation on the set of inner forms and taking the corresponding union of the sets Irr(H) on the left-hand side Such a statement was proven for unipotent representations of simple p-adic groups in [41]. Langlands parameters for GLn(F ) (respectively, SLn(F )) take values in the dual group, and they must be considered up to conjugation. It is worth noting that our group C(φ ) coincides with the group Sφ+ defined by Kaletha in [32] with Z taken to be equal to the centre of SLn(F )
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