Twisted endoscopy and the generic packet conjecture

Konno, Takuya

doi:10.1007/bf02773167

Cited by 10 publications

(4 citation statements)

References 14 publications

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“…Before these results of Arthur, Mok and Kaletha, there is a result of Konno [16]. Under the assumption that the residual characteristic of F is not two, Konno [16,Theorem 4.1] proved a twisted analogue of a result of Moeglin-Waldspurger [21], which states that the 'leading' coefficients in the (twisted) character expansion of an irreducible representation π of G(F ) at the identity element give the dimensions of certain spaces of degenerate Whittaker models.…”

Section: Introductionmentioning

confidence: 95%

“…Under the assumption that the residual characteristic of F is not two, Konno [16,Theorem 4.1] proved a twisted analogue of a result of Moeglin-Waldspurger [21], which states that the 'leading' coefficients in the (twisted) character expansion of an irreducible representation π of G(F ) at the identity element give the dimensions of certain spaces of degenerate Whittaker models. By comparing the twisted character expansion of GL n with the ordinary one of a classical group G, Konno concluded that for each tempered L-parameter φ of G, Π φ has a w ′ -generic representation for some Whittaker datum w ′ for G ( [16,Thorem 8.4]). As is mentioned in the remark after Proposition 8.3.2 in [1], one may seem to be able to prove the uniqueness (Desideratum 1.1) by the same method.…”

Section: Introductionmentioning

confidence: 99%

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On the uniqueness of generic representations in an $L$-packet

Atobe¹

2015

Preprint

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Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

On the uniqueness of generic representations in an $L$-packet

Atobe¹

2015

Preprint

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“…We note that reverse direction of the above theorem follows from [Art13, Proposition 8.3.2] or [Kon02] and [Art13, Theorem 2.2.1]. The uniqueness of a generic representation with respect to the Whittaker datum (B, χ) in Π φ follows from the works of [JS03,JS12,Liu11] or again [Kon02] and [Art13, Theorem 2.2.1].…”

mentioning

confidence: 93%

On the intersection of local Arthur packets for classical groups

Hazeltine¹,

Liu²,

Lo³

2022

Preprint

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In this paper, for symplectic and split odd special orthogonal groups, following Atobe's refinement on Moeglin's construction of local Arthur packets, we give new algorithms, comparing to those given by Atobe ([Ato22]), to determine whether a given representation is of Arthur type, and determine all local Arthur packets containing a representation of Arthur type. Our algorithms depend on certain operators on construction data, each of which preserves representations, forming a proper subset of those used by Atobe. In particular, for any representation π of Arthur type, we give a precise formula to compute the set {local Arthur parameter ψ | the local Arthur packet Π ψ contains π}.Our main results have four applications as follows. (1). We give a classification of all the local Arthur packets which contain tempered representations, and give the precise number of the tempered representations in any local Arthur packet and describe their L-data. We also give a classification of all the local Arthur packets which have non-trivial intersection with a given tempered local Arthur packet. (2). We prove the enhanced Shahidi's conjecture, which says that local Arthur packets are tempered if and only if they have generic members. In particular, this implies that any irreducible tempered generic representation lives in exactly one local Arthur packet. (3). We give a complete description of the construction data for representations in the L-packet of any given local Arthur packet. (4). For each representation of Arthur type, we specify a unique construction data satisfying the property that if this representation lies in the L-packet of a local Arthur packet then this construction data provides exactly the corresponding local Arthur parameter. This result suggests an answer to the mysterious question of determining which local Arthur parameter is "the" parameter for a representation if it lies in several local Arthur packets.

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“…It has been conjectured that, when k is local, every tempered L-packet contains a unique generic representation with respect to a fixed generic character of U(k), where U is the unipotent radical of a Borel subgroup of G over k [56]. Many cases of this conjecture are now proved [17,20,28,34,56,59,60]. In particular, for every fixed generic character of U(k), the corresponding generic tempered representations are in one-one correspondence with tempered L-packets.…”

Section: Introductionmentioning

confidence: 99%

Arthur packets and the Ramanujan conjecture

Shahidi¹

2011

Kyoto J. Math.

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The purpose of this paper is to show that under a part of generalized Arthur's A-packet conjecture, locally generic cuspidal automorphic representations of a quasisplit group over a number field are of Ramanujan type, i.e., are tempered at almost all places. The A-packet conjecture allows one to reduce the problem to a special case of a general local question about the components of the corresponding Langlands L-packet which is then answered here in its generality.

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Twisted endoscopy and the generic packet conjecture

Cited by 10 publications

References 14 publications

On the uniqueness of generic representations in an $L$-packet

On the uniqueness of generic representations in an $L$-packet

On the intersection of local Arthur packets for classical groups

Arthur packets and the Ramanujan conjecture

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