The Local Langlands Conjectures for Non-quasi-split Groups

Kaletha, Tasho

doi:10.1007/978-3-319-41424-9_6

Cited by 25 publications

(35 citation statements)

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“…Let 𝜆 𝑧 be the character corresponding to the class of z under the Tate-Nakayama isomorphism Proof. Both of these statements follow from [Kal16a,Conjecture G]. For the independence of Whittaker datum, one can prove that the validity of this conjecture implies that if 𝔴 is replaced by another choice 𝔴 , then there is an explicitly constructed character (𝔴, 𝔴 ) of 𝜋 0 (𝑆 𝜙 /𝑍 ( 𝐺) Γ ) whose inflation to 𝜋 0 (𝑆 + 𝜙 ) satisfies 𝜏 𝑧,𝔴, 𝜎 = 𝜏 𝑧,𝔴 , 𝜎 ⊗ (𝔴, 𝔴 ) for any 𝜎 ∈ Π 𝜙 (𝐺) ∪Π 𝜙 (𝐺 𝑏 ).…”

Section: Construction Of 𝜹 𝝅𝝆 In the General Casementioning

confidence: 89%

“…We now drop the assumption that G is a B-inner form of 𝐺 In order to make this precise, we will need the material of [Kal16b] and [Kal18], some of which is summarized in [Kal16a]. First, we will need the cohomology set 𝐻 1 (𝑢 → 𝑊, 𝑍 → 𝐺 * ) defined in [Kal16b,§3] for any finite central subgroup 𝑍 ⊂ 𝐺 * defined over F. As in [Kal18, §3.2], it will be convenient to package these sets for varying Z into the single set…”

Section: Construction Of 𝜹 𝝅𝝆 In the General Casementioning

confidence: 99%

“…Our generalized Kottwitz conjecture is conditional on the refined local Langlands correspondence for supercuspidal L -parameters in the formulation of [Kal16a, Conjecture G]. In particular, it relies crucially on the endoscopic character identities satisfied by L -packets.…”

Section: Introductionmentioning

confidence: 99%

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On the Kottwitz conjecture for local shtuka spaces

Kaletha

Hansen

Weinstein³

2022

Forum of Mathematics, Pi

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Kottwitz’s conjecture describes the contribution of a supercuspidal representation to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze’s more general spaces of local shtukas. Using a new Lefschetz–Verdier trace formula for v-stacks, we prove the extended conjecture, disregarding the action of the Weil group, and modulo a virtual representation whose character vanishes on the locus of elliptic elements. As an application, we show that, for an irreducible smooth representation of an inner form of $\operatorname {\mathrm {GL}}_n$ , the L-parameter constructed by Fargues–Scholze agrees with the usual semisimplified parameter arising from local Langlands.

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Section: Construction Of 𝜹 𝝅𝝆 In the General Casementioning

confidence: 89%

Section: Construction Of 𝜹 𝝅𝝆 In the General Casementioning

confidence: 99%

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On the Kottwitz conjecture for local shtuka spaces

Kaletha

Hansen

Weinstein³

2022

Forum of Mathematics, Pi

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“…from section 2.1. We now wish to put this local Langlands correspondence for the inner form in the context of the refined local Langlands correspondence of Kaletha [Kal16]. We consider the Kottwitz set B(G) [Kot85; RR96] and let b ∈ B(G) be the basic element whose associated σ-centralizer J b = J.…”

Section: Theorem 22 [Gt14]mentioning

confidence: 99%

“…For the purposes of applying the weak form of the Kottwitz Conjecture proven in Hansen-Kaletha-Weinstein [HKW21], we formulate this correspondence in terms of the refined local Langlands of Kaletha [Kal16] with respect to the fixed choice of Whittaker datum m. Now, given a parameter φ that is either mixed supercuspidal or supercuspidal, we have by Theorem 2.1 (2) a correspondence between the L-packet Π φ (G) and the set of irreducible characters A ∨ φ . This in turn gives rise to an irreducible character of the group S φ via the composition:…”

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confidence: 99%

Compatibility of the Fargues-Scholze and Gan-Takeda Local Langlands

Hamann¹

2021

Preprint

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Given a prime p, a finite extension L/Q p , a connected p-adic reductive group G/L, and a smooth irreducible representation π of G(L), Fargues-Scholze [FS21] recently attached a semisimple Weil parameter to such π, giving a general candidate for the local Langlands correspondence. It is natural to ask whether this construction is compatible with known instances of the correspondence after semisimplification. For G = GL n and its inner forms, Fargues-Scholze and Hansen-Kaletha-Weinstein [HKW21] show that the correspondence is compatible with the correspondence of Harris-Taylor/Henniart [Hen00; HT01]. We verify a similar compatibility for G = GSp 4 and its unique non-split inner form G = GU 2 (D), where D is the quaternion division algebra over L, assuming that L/Q p is unramified and p > 2. In this case, the local Langlands correspondence has been constructed by Gan-Takeda and Gan-Tantono [GT11; GT14]. Analogous to the case of GL n and its inner forms, this compatibility is proven by describing the Weil group action on the cohomology of a local Shimura variety associated to GSp 4 , using basic uniformization of abelian type Shimura varieties due to Shen [She17], combined with various global results of Kret-Shin [KS16] and Sorensen [Sor10] on Galois representations in the cohomology of global Shimura varieties associated to inner forms of GSp 4 over a totally real field. After showing the parameters are the same, we apply some ideas from the geometry of the Fargues-Scholze construction explored recently by Hansen [Han20], to give a more precise description of the cohomology of this local Shimura variety, verifying a strong form of the Kottwitz conjecture in the process. LINUS HAMANN Applications 48 References 51 AcknowledgementsIt is a pleasure to thank my advisor David Hansen for giving me this project and for numerous suggestions and ideas related to it, as well as Eric Chen, Arthur-César Le-Bras, Yoichi Mieda, Jack Sempliner, and Zhiyu Zhang for some nice conversations related to this work. Special thanks also go to Peter Scholze for sharing with me the draft of [FS21], pointing out some errors, and generously initiating me in these ideas during my masters thesis, Sug-Woo Shin for pointing out some errors in section 5, and the MPIM Bonn for their hospitality during part of the completion of this project.

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Local Langlands and Springer Correspondences

Aubert

2019

Progress in Mathematics

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This notes are the written version of a course given by the author at the workshop "Representation theory of p-adic groups" that was held at IISER Pune, India, in July 2017 (to appear in Representations of p-adic groups: Contributions from Pune, Progress in Math., Birkhäuser). They give notably an overview of results obtained jointly with Ahmed Moussaoui and Maarten Solleveld on the local Langlands correspondence, focusing on the links of the latter with both the generalized Springer correspondence and the geometric conjecture, the so-called ABPS Conjecture, introduced in collaboration with Paul Baum, Roger Plymen and Maarten Solleveld. Langlands parameters correspond by the LLC to the irreducible supercuspidal representations of G. The validity of this conjecture is proved for representations with unipotent reduction of the group G of the F-rational points of any connected reductive algebraic group which splits over an unramified extension of F in [FOS, Theorem 2] (when G is simple of adjoint type it is a special case of [Lus4], [Lus5]), for the Deligne-Lusztig depth-zero supercuspidal representations (as a consequence of [DeRe]), and also for general linear groups and split classical p-adic groups (any representation) (see [Mou1]), for inner forms of linear groups and of special linear groups, and for quasi-split unitary p-adic groups (any representation) (see [AMS1, § 6]).

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The Local Langlands Conjectures for Non-quasi-split Groups

Cited by 25 publications

References 42 publications

On the Kottwitz conjecture for local shtuka spaces

On the Kottwitz conjecture for local shtuka spaces

Compatibility of the Fargues-Scholze and Gan-Takeda Local Langlands

Local Langlands and Springer Correspondences

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