Geometric structure in smooth dual and local Langlands conjecture

Aubert, Anne-Marie; Baum, Paul; Plymen, Roger; Solleveld, Maarten

doi:10.1007/s11537-014-1267-x

Cited by 20 publications

(51 citation statements)

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“…Together with the Jacquet-Langlands correspondence [7,19,43], this provides the LLC for inner forms G = GL m (D) of GL n (F ), see [4,29]. Recall that (GL m (D)) (GL n (F )) if GL m (D) is not split.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2016

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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 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.

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Section: Introductionmentioning

confidence: 99%

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

et al. 2016

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“…Whenever Conjecture 3 holds for s, one can apply [Sol2,§5.4]. This proves an earlier version of Conjecture 2 for Irr s (G) (formulated in terms of an extended quotient of the first kind, see [ABPS2]). To obtain Conjecture 2 completely more work is required, which has been carried out in the cases listed on page 19.…”

Section: Inner Forms Of Glmentioning

confidence: 59%

“…It is topologized via the Jacobson topology for the Hecke algebra of G, and in this way it is automatically rather close to an algebraic variety. We propose a generalization of our earlier conjectures [ABP,ABPS2], which make the structure of Irr(G) much more precise. To formulate these conjectures, we need extended quotients and the Bernstein decomposition.…”

Section: A Bijective Version Of the Llcmentioning

confidence: 83%

“…We note that in order to establish Conjecture 1 for (a collection of) groups, it suffices to prove it for tempered representations and bounded enhanced L-parameters. This follows from comparing the geometry of the spaces Irr(G) and e (G) [ABPS2], or from the Langlands classification for Irr(G) [Ren,§VII.4] and its counterpart for L-parameters [SiZi].…”

Section: A Bijective Version Of the Llcmentioning

confidence: 99%

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Conjectures about 𝑝-adic groups and their noncommutative geometry

Aubert¹,

Baum²,

Plymen

et al. 2017

Contemporary Mathematics

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Abstract. Let G be any reductive p-adic group. We discuss several conjectures, some of them new, that involve the representation theory and the geometry of G.At the heart of these conjectures are statements about the geometric structure of Bernstein components for G, both at the level of the space of irreducible representations and at the level of the associated Hecke algebras. We relate this to two well-known conjectures: the local Langlands correspondence and the BaumConnes conjecture for G. In particular, we present a strategy to reduce the local Langlands correspondence for irreducible G-representations to the local Langlands correspondence for supercuspidal representations of Levi subgroups. Contents

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“…Together with the Jacquet-Langlands correspondence this provides the LLC for essentially squareintegrable representations of inner forms G = GL m (D) of GL n (F ). It is extended to all irreducible G-representations via the Zelevinsky classification [Zel, DKV], see [HiSa,ABPS1]. For these groups every L-packet is a singleton and the LLC is a canonical bijective map…”

Section: Introductionmentioning

confidence: 99%