A NOTE ON THE LANGLANDS CLASSIFICATION AND IRREDUCIBILITY OF INDUCED REPRESENTATIONS OF p-ADIC GROUPS

Konno, Takuya

doi:10.2206/kyushujm.57.383

Cited by 37 publications

(25 citation statements)

References 9 publications

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“…By Lemma 2.9 ν ′ − ν ∈ā * ,− 0 \ {0}, which by [Kon,Claim 3.5.1] implies ν > ν ′ . Since cc G (J(λ)) 2 = cc G (J(λ ′ )) 2 , we conclude that cc M (σ) 2 < cc M ′ (σ ′ ) 2 .…”

Section: )mentioning

confidence: 82%

Periodic cyclic homology of reductive $p$-adic groups

Solleveld

2009

J. Noncommut. Geom.

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Let G be a reductive p-adic group, H(G) its Hecke algebra and S(G) its Schwartz algebra. We will show that these algebras have the same periodic cyclic homology. This provides an alternative proof of the Baum-Connes conjecture for G, modulo torsion. As preparation for our main theorem we prove two results that have independent interest. Firstly, a general comparison theorem for the periodic cyclic homology of finite type algebras and certain Fréchet completions thereof. Secondly, a refined form of the Langlands classification for G, which clarifies the relation between the smooth spectrum and the tempered spectrum. Acknowledgements.This paper is partly based on the author's PhD-thesis, which was written at the Universiteit van Amsterdam under the supervision of Eric Opdam. The author is grateful for the support and advice that professor Opdam has given him during his PhD-research. He would also like to thank Ralf Meyer, Christian Voigt and the referee for their comments, which lead to substantial clarifications of some proofs.

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Section: )mentioning

confidence: 82%

Periodic cyclic homology of reductive $p$-adic groups

Solleveld

2009

J. Noncommut. Geom.

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“…It is known from [29, Theorem B.2.d] and [11] that for inner forms of GL n (F ) normalized parabolic induction sends irreducible square-integrable (modulo centre) representations to irreducible tempered representations. Together with the Langlands classification [43] this implies that there exists a natural bijection between Irr(G) and the right hand side of (19). It sends (M, σ) to the Langlands quotient L I G M (σ) , where the parabolic induction goes via a parabolic subgroup with Levi factor M, with respect to which the central character of σ is "positive".…”

Section: The General Linear Group and Its Inner Formsmentioning

confidence: 95%

Geometric structure in smooth dual and local Langlands conjecture

et al. 2014

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We prove that a strengthened form of the local Langlands conjecture is valid throughout the principal series of any connected split reductive p-adic group. The method of proof is to establish the presence of a very simple geometric structure, in both the smooth dual and the Langlands parameters. We prove that this geometric structure is present, in the same way, for the general linear group, including all of its inner forms. With these results as evidence, we give a detailed formulation of a general geometric structure conjecture.

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“…Furthermore, the absolute value of the character of ω 1 on Z(M 1 ) is regular in the sense that no root of (G, Z(M 1 )) annihilates it. Hence, there exists a unique parabolic subgroup P 2 of G with Levi factor M 1 , such that (P 2 , ω 1 ) satisfies the hypothesis of the Langlands classification [34,37]. That result says that I G P 2 (ω 1 ) has a unique irreducible quotient L(P 2 , ω 1 ) and that every irreducible G-representation can be obtained in this way, from data that are unique up to G-conjugation.…”

Section: Then Recmentioning

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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A NOTE ON THE LANGLANDS CLASSIFICATION AND IRREDUCIBILITY OF INDUCED REPRESENTATIONS OF p-ADIC GROUPS

Abstract: In this paper, we present a proof of the Langlands classification of the irreducible admissible representations of reductive p-adic groups. Then we deduce certain irreducibility results for parabolically induced modules from discrete series representations.

Cited by 37 publications

References 9 publications

Periodic cyclic homology of reductive $p$-adic groups

Periodic cyclic homology of reductive $p$-adic groups

Geometric structure in smooth dual and local Langlands conjecture

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

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