Simple supercuspidal L-packets of symplectic groups over dyadic fields

Henniart, Guy; Oi, Masao

doi:10.48550/arxiv.2207.12985

Cited by 3 publications

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“…Thus a corollary of the fourth statement in Theorem 4.4 is that each representation π x (λ, χ, ψ) lies in a singleton L-packet. This is in contrast with simple supercuspidals for simply connected G, where the size of the L-packet depends on the residue cardinality q (for an example see Theorem 1.1 and Remark 1.2 of [HO22]).…”

Section: Langlands Parameters For Simple Supercuspidal Representation...mentioning

confidence: 94%

“…Simple supercuspidals form a special case of epipelagic representations: they were first introduced in [GR10] for simply connected groups, and the construction of [RY14] can be seen as a generalization of this earlier construction. Simple supercuspials have inspired a wealth of research, for example [Kal13], [Adr16], [AK19], [Oi21], [HO22], but they have not been fully explored in the case of adjoint groups, and most work on them has been restricted to classical groups and GL n .…”

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confidence: 99%

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2024

Represent. Theory

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A paper of Reeder-Yu [J. Amer. Math. Soc. 27 (2014), pp. 437-477] gives a construction of epipelagic supercuspidal representations of p-adic groups. The input for this construction is a pair (λ, χ) where λ is a stable vector in a certain representation coming from a Moy-Prasad filtration, and χ is a character of the additive group of the residue field. We say two such pairs are equivalent if the resulting supercuspidal representations are isomorphic. In this paper we describe the equivalence classes of such pairs. As an application, we give a classification of the simple supercuspidal representations for split adjoint groups. Finally, under an assumption about unramified base change, we describe properties of the Langlands parameters associated to these simple supercuspidals, showing that they have trivial L-functions and minimal Swan conductors, and showing that each of these simple supercuspidals lies in a singleton L-packet. Contents

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Section: Langlands Parameters For Simple Supercuspidal Representation...mentioning

confidence: 94%

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confidence: 99%

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2024

Represent. Theory

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“…We next consider the case where p = 2. In this case, we determined explicitly the L-parameter of a simple supercuspidal representation of Sp 2n (F ) in [22]. However, we cannot deduce the formal degree conjecture for simple supercuspidal representations of Sp 2n (F ) from the description in loc.…”

Section: 3mentioning

confidence: 96%

On Swan exponents of symmetric and exterior square Galois representations

Henniart,

2024

Rad Hrvatske akademije znanosti i umjetnosti

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Let F be a local non-Archimedean field and E a finite Galois extension of F , with Galois group G. If ρ is a representation of G on a complex vector space V , we may compose it with any tensor operation R on V , and get another representation R • ρ. We study the relation between the Swan exponents Sw(ρ) and Sw(R•ρ), with a particular attention to the cases where R is symmetric square or exterior square. Indeed those cases intervene in the local Langlands correspondence for split classical groups over F , via the formal degree conjecture, and we present some applications of our work to the explicit description of the Langlands parameter of simple cuspidal representations. For irreducible ρ our main results determine Sw(Sym 2 ρ) and Sw(∧ 2 ρ) from Sw(ρ) when the residue characteristic p of F is odd, and bound them in terms of Sw(ρ) when p is 2. In that case where p is 2 we conjecture stronger bounds, for which we provide evidence.

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“…Based on this motivation, the second and fourth authors investigated the case of Sp 2n over a dyadic field in [HO22] by elaborating on the method of [Oi19a,Oi19b,Oi18]. (As mentioned above, the second author had already obtained the result for F = Q 2 in [Hen23] prior to [HO22].) Now let us state the main result of this paper:…”

Section: Introductionmentioning

confidence: 98%

“…Any odd p can be covered uniformly by this approach, but the dyadic case (i.e., p = 2) still remains. Based on this motivation, the second and fourth authors investigated the case of Sp 2n over a dyadic field in [HO22] by elaborating on the method of [Oi19a,Oi19b,Oi18]. (As mentioned above, the second author had already obtained the result for F = Q 2 in [Hen23] prior to [HO22].)…”

Section: Introductionmentioning

confidence: 99%

On The Langlands parameter of a simple supercuspidal representation: even orthogonal groups

Adrian

Kaplan

2021

Isr. J. Math.

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We consider the split special orthogonal group SO N defined over a p-adic field. We determine the structure of any L-packet of SO N containing a simple supercuspidal representation (in the sense of Gross-Reeder). We also determine its endoscopic lift to a general linear group. Combined with the explicit local Langlands correspondence for simple supercuspidal representations of general linear groups, this leads us to get an explicit description of the L-parameter as a representation of the Weil group of F . Our result is new when p = 2 and our method provides a new proof even when p = 2. Contents 1. Introduction 2. Simple supercuspidal representations 2.1. Self-dual simple supercuspidal representations of GL N 2.2. Simple supercuspidal representations of SO 2n+1 2.3. Simple supercuspidal representations of SO 2n 3. Local Langlands correspondence for SO N 3.1. Local Langlands correspondence for SO N 3.2. Result of Moeglin and Xu 3.3. Formal degree conjecture of Hiraga-Ichino-Ikeda 4. Analysis of symmetric and exterior square L-factors 5. Twisted gamma factor for simple supercuspidal representations of SO 2n 5.1. Simple supercuspidal representations of SO 2n 5.2. The twisted γ-factors 5.3. The computation of γ(s, π × τ, ψ) 5.4. L-parameter 6. L-packets and L-parameters for simple supercuspidals of SO N 6.1. Rough form of the L-parameters 6.2. From twisted γ-factor to Swan conductor 6.3. Utilization of the formal degree conjecture 6.4. Main Theorem Appendix A. Several remarks on simple supercuspidal representations A.1. Iwahori subgroups A.2. Another parametrization of simple supercuspidal representations A.3. Comparison of our approach with others Appendix B. On lifting from classical groups to GL N Appendix C. Unramified case of Arthur's classification theorem C.1. Classical groups as twisted endoscopy of GL N C.2. Fundamental lemma of Lemaire-Moeglin-Waldspurger C.3. Arthur's local classification theorem 47 C.4. Unramified representation in an A-packet 49 References 51, On the cuspidal support of discrete series for p-adic quasisplit Sp(N ) and SO(N ), Manuscripta Math. 154 (2017), no. 3-4, 441-502.

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Simple supercuspidal L-packets of symplectic groups over dyadic fields

Cited by 3 publications

References 0 publications

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On Swan exponents of symmetric and exterior square Galois representations

On The Langlands parameter of a simple supercuspidal representation: even orthogonal groups

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