Effective local Langlands correspondence

Bushnell, Colin J.; Kim, Minhyong; Diamond, Fred; Kassaei, Payman L

doi:10.1017/cbo9781107446335.005

Cited by 14 publications

(59 citation statements)

References 36 publications

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“…To this end, one seeks to make the correspondence explicit/effective. For GL n , this has been the subject of a series of papers by Bushnell-Henniart [6][7][8]10]; for other groups, work has concentrated on regular depth zero irreducible cuspidal representations [13,24,26] and epipelagic irreducible cuspidal representations [9,18,25,27,46,47], with the most general work by Kaletha [28] on regular cuspidal representations.…”

Section: Introductionmentioning

confidence: 99%

On depth zero L‐packets for classical groups

Lust

Stevens

2020

Proc. London Math. Soc.

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By computing reducibility points of parabolically induced representations, we construct, to within at most two unramified quadratic characters, the Langlands parameter of an arbitrary depth zero irreducible cuspidal representation π of a classical group (which may be not-quasisplit) over a non-archimedean local field of odd residual characteristic. From this, we can explicitly describe all the irreducible cuspidal representations in the union of one, two, or four L-packets, containing π. These results generalize the work of DeBacker-Reeder (in the case of classical groups) from regular to arbitrary tame Langlands parameters.

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Section: Introductionmentioning

confidence: 99%

On depth zero L‐packets for classical groups

Lust

Stevens

2020

Proc. London Math. Soc.

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“…Endo-classes (endo-equivalence classes of ps-characters) have subsequently been extended to inner forms of general linear groups [5], and have proved fundamental in understanding fine properties of the local Langlands correspondence [10,11] and the Jacquet-Langlands correspondence [19,37], as well as in the study of Galois-distinguished cuspidal representations [1,33] and in Bernstein decompositions of the category of smooth representations over fields of positive characteristic [36].…”

Section: Introductionmentioning

confidence: 99%

Endo-parameters for p-adic classical groups

2020

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For a classical group over a non-archimedean local field of odd residual characteristic p, we prove that two cuspidal types, defined over an algebraically closed field $${\mathbf {C}}$$ C of characteristic different from p, intertwine if and only if they are conjugate. This completes work of the first and third authors who showed that every irreducible cuspidal $${\mathbf {C}}$$ C -representation of a classical group is compactly induced from a cuspidal type. We generalize Bushnell and Henniart’s notion of endo-equivalence to semisimple characters of general linear groups and to self-dual semisimple characters of classical groups, and introduce (self-dual) endo-parameters. We prove that these parametrize intertwining classes of (self-dual) semisimple characters and conjecture that they are in bijection with wild Langlands parameters, compatibly with the local Langlands correspondence.

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“…Let denote the set of endo-classes of simple characters over . (For the notion of endo-class, see [BH96] or the summary in any of [Bus14, BH03, BH13]. )…”

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confidence: 99%

“…More precisely, defines an element of the orbit space , namely the orbit of irreducible components of . The Langlands correspondence induces a canonical bijection ([BH03, 8.2 Theorem], [BH14b, 6.1]) by Results developed in [BH96, BH99, BH03, BH05a, BH05b, BH10] and particularly [BH14b] show that the map (A) is central to understanding of the Langlands correspondence.…”

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confidence: 99%

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Local Langlands correspondence and ramification for Carayol representations

Bushnell¹,

Henniart²

2019

Compositio Math.

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Let F be a non-Archimedean locally compact field of residual characteristic p with Weil group W F . Let σ be an irreducible smooth complex representation of W F , realized as the Langlands parameter of an irreducible cuspidal representation π of a general linear group over F . In an earlier paper, we showed that the ramification structure of σ is determined by the fine structure of the endo-class Θ of the simple character contained in π, in the sense of Bushnell-Kutzko. The connection is made via the Herbrand function Ψ Θ of Θ. In this paper, we concentrate on the fundamental Carayol case in which σ is totally wildly ramified with Swan exponent not divisible by p. We show that, for such σ, the associated Herbrand function satisfies a certain functional equation, and that this property essentially characterizes this class of representations. We calculate Ψ Θ explicitly, in terms of a classical Herbrand function arising naturally from the classification of simple characters. We describe exactly the class of functions arising as Herbrand functions Ψ Ξ , as Ξ varies over the set of totally wild endoclasses of Carayol type. In a separate argument, we derive a complete description of the restriction of σ to any ramification subgroup and hence a detailed interpretation of the Herbrand function. This gives concrete information concerning the Langlands correspondence.2000 Mathematics Subject Classification. 22E50, 11S37, 11S15.

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Effective local Langlands correspondence

Cited by 14 publications

References 36 publications

On depth zero L‐packets for classical groups

On depth zero L‐packets for classical groups

Endo-parameters for p-adic classical groups

Local Langlands correspondence and ramification for Carayol representations

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