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 SLn(F ). It takes the form of a bijection between, on the one hand, conjugacy classes of Langlands parameters for 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 SLn(F ) up to equivalence. 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 GLn(F ), when the fields are close enough compared to the depth of the representations. Contents 1. Introduction 1 2. Inner forms of GL n (F ) 5 3. Inner forms of SL n (F ) 9 4. Characterization of the LLC for some representations of GL n (F ) 14 5. The method of close fields 20 6. Close fields and Langlands parameters 26 References 30

The infinite base change lifting associated to an APF extension of a $p$-adic field

“…Moreover, from β we can canonically define an isomorphism of triples γ : Tr l (F 2 ) → Tr l (F 1 ). The following compatibility of β * with γ * via the local Langlands correspondence was proved by Aubert, Baum, Plymen and Solleveld in their preprint [2].…”

Depth and the Local Langlands Correspondence

“…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 -Adic Cohomology of Some -Adically Uniformized Shimura Varieties

“…We note that σ(JL(ρ)) is the Weil-Deligne representation associated to ρ ∈ Irr(D × ) by the local Langlands correspondence for the inner form D × of GL n , as expected naturally, cf. [2,20].…”

“…Combining lemma 3.1 with theorem 3.3, we can now prove the following theorem. Since the local reductive group G Qp is a product of inner forms of GL n (together with G m ), the local Langlands correspondence for it is known ( [2,20]). For any smooth irreducible representation π p of G(Q p ), let ϕ πp : W Qp −→ L (G Qp ) be the associated local Langlands parameter.…”

“…The local reductive group G Qp is a product of some inner forms of GL n (together with G m ), so the local Langlands correspondence has been known, cf. [2,20]. For any smooth irreducible representation π p of G(Q p ), let ϕ πp be the associated local Langlands parameter.…”