Local root numbers and spectrum of the local
descents for orthogonal groups : p-adic case

Jiang, Dihua; Zhang, Lei

doi:10.2140/ant.2018.12.1489

Cited by 8 publications

(13 citation statements)

References 33 publications

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“…Assume that ϕ π = ⊗ ν ϕ πν and φ s = ⊗ ν φ τν ⊗σν ,s = ⊗ ν φ s,ν are factorizable vectors, which yields the factorization in (4. 45), and that the pair (σ, π) has a nonzero Bessel period. Then for the real part of s large, it can be written as an Euler product:…”

Section: First We Havementioning

confidence: 99%

Arthur parameters and cuspidal automorphic modules of classical groups

Jiang

Zhang

2020

Ann. of Math. (2)

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The endoscopic classification via the stable trace formula comparison provides certain character relations between irreducible cuspidal automorphic representations of classical groups and their global Arthur parameters, which are certain automorphic representations of general linear groups. It is a question of J. Arthur and W. Schmid that asks: How to construct concrete modules for irreducible cuspidal automorphic representations of classical groups in term of their global Arthur parameters? In this paper, we formulate a general construction of concrete modules, using Bessel periods, for cuspidal automorphic representations of classical groups with generic global Arthur parameters. Then we establish the theory for orthogonal and unitary groups, based on certain well expected conjectures. Among the consequences of the theory in this paper is that the global Gan-Gross-Prasad conjecture for those classical groups is proved in full generality in one direction and with a global assumption in the other direction.

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Section: First We Havementioning

confidence: 99%

Arthur parameters and cuspidal automorphic modules of classical groups

Jiang

Zhang

2020

Ann. of Math. (2)

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“…In the p-adic case, the local Gan-Gross-Prasad conjecture has been resolved by J.-L. W2,W3,MW] for orthogonal groups, by R. Beuzart-Plessis [BP1,BP2] and W. T. Gan and A. Ichino [GI] for unitary groups, and by H. Atobe [Ato] for symplectic-metaplectic groups. On the other hand, D. Jiang and L. Zhang [JZ1] study the local descents for p-adic orthogonal groups, whose results can be viewed as a refinement of the local Gan-Gross-Prasad conjecture, and the descent method has important applications towards the global problem (see [JZ2]).…”

Section: Introductionmentioning

confidence: 99%

“…Roughly speaking, for fixed G(V ) and its representation π, the descent problem seeks the smallest member G(W ) among a Witt tower which has an irreducible representation π ′ satisfying m(π, π ′ ) = 0, and all such π ′ give the first descent of π. To give the precise notion of descent, let us sketch the definition of the data (H, ν) following [GGP1] and [JZ1].…”

Section: Introductionmentioning

confidence: 99%

Descents of unipotent representations of finite unitary groups

Liu

Wang

2020

Trans. Amer. Math. Soc.

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Inspired by the Gan-Gross-Prasad conjecture and the descent problem for classical groups, in this paper we study the descents of unipotent cuspidal representations of orthogonal and symplectic groups over finite fields.Date: August 29, 2019. 2010 Mathematics Subject Classification. Primary 20C33; Secondary 22E50.Let G(V ) be the identity component of the automorphism group of V and G(W ) ⊂ G(V ) the subgroup which acts as identity on W ⊥ . Let π and π ′ be irreducible representations of G(V ) and G(W ) respectively. The Gan-Gross-Prasad conjecture is concerned with the multiplicity Theorem 15.1], and will be explained in details shortly. According to whether dimV − dimW is odd or even, the above-Hom space is called the Bessel model or Fourier-Jacobi model. In the case of finite unitary groups, W. T. Gan, B. H. Gross and D. Prasad ([GGP2, Proposition 5.3]) showed that if π and π ′ are both cuspidal, then m(π, π ′ ) ≤ 1.We should mention that our formulation of multiplicities differs slightly from that in the Gan-Gross-Prasad conjecture [GGP1], up to taking the contragradient of π ′ . This is more suitable for the purpose of descents (c.f. [LW2]), which will be clear from the discussion below. On the other hand, in this paper we will restrict our attention to unipotent cuspidal representations of SO n (F q ) and Sp 2n (F q ), which are self-dual (see Proposition 5.5), and thus for π and π ′ unipotent cuspidal the above Hom-space coincides with Hom H(Fq) (π ⊗ π ′ , ν).Roughly speaking, for fixed G(V ) and its representation π, the descent problem seeks the smallest member G(W ) among a Witt tower which has an irreducible representation π ′ satisfying m(π, π ′ ) = 0, and all such π ′ give the first descent of π. To give the precise notion of descent, let us sketch the definition of the data (H, ν) following [GGP1] and [JZ1].We first consider the Bessel model. Let V n be an n-dimensional space over F q with a nondegenerate symmetric bilinear form (, ), which defines the special orthogonal group SO(V n ). We will consider various pairs of Hermitian spaces V n ⊃ V n−2ℓ−1 and the following partitions of n, (1.1) p ℓ = [2ℓ + 1, 1 n−2ℓ−1 ], 0 ≤ ℓ < n/2.Assume that V n has a decomposition V n = X + V n−2ℓ + X ∨

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“…In the p-adic case, the local Gan-Gross-Prasad conjecture has been resolved by J.-L. W3,W4,MW] for orthogonal groups, by R. Beuzart-Plessis [BP1,BP2] and W. T. Gan and A. Ichino [GI] for unitary groups, and by H. Atobe [Ato] for symplectic-metaplectic groups. On the other hand, D. Jiang and L. Zhang [JZ1] study the local descents for p-adic orthogonal groups, whose results can be viewed as a refinement of the local Gan-Gross-Prasad conjecture, and the descent method has important applications towards the global problem (see [JZ2]).…”

Section: Introductionmentioning

confidence: 99%