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

Abstract

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) 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 GL n (F), when the fields are close enough compared to the depth of the representations.

Depth and the Local Langlands Correspondence

“…This was proved in [ABPS2,Lemma 4.4] under the additional assumption SL 2 (C) ⊂ ker φ. We will reduce to that special case.…”

“…It fits very well with the aforementioned work of Lusztig. In positive depth there is the result of Yu [Yu2,§7.10], who proved (1) for unramified tori. For GL n (F ), (1) was claimed in [Yu1,§2.3.6] and proved in [ABPS2,Proposition 4.5]. For GSp 4 (F ), (1) is proved in [Gan,§ 10].…”

On the Cohomology of Some Simple Shimura Varieties With Bad Reduction

“…Moreover, for any irreducible smooth representation π of G i (Q p ) with L-parameter ϕ π (cf. [2,4,23]), we have the following character identity We remark that in the quaternionic case with h as the characteristic function of C 0 p divided by the volume of C 0 p , the test function φ τ,h should have the same twisted orbital integrals as the function defined in [19] based on the study of the local structures of these varieties. When D is totally indefinite, the description of S D,C (F p r ) and the definition of φ τ,h can be done more directly by using the PEL moduli problems for these varieties (cf.…”

On the Local Langlands Correspondence for Split Classical Groups Over Local Function Fields

“…In [41], Kazhdan These results suggest that, if one understands the representation theory of Gal(F/F) for all local fields F of characteristic 0, then one can use it to understand the representation theory of Gal(F /F ) for a local field F of characteristic p, and similarly, with an understanding of the representation theory of G(F) for all local fields F of characteristic 0, one can study the representation theory of G(F ), for F of characteristic p. Such applications of this philosophy can be found in [8,9,29,49]. For the rest of the introduction, H will denote one of the split classical groups Sp 2n , SO 2n+1 , or SO 2n and N will denote the rank of the group Langlands dual to H. These groups can be realized as endoscopic groups of GL N , where GL N is the twisted space of GL N with respect to a nontrivial outer automorphism θ .…”