The local Gan–Gross–Prasad conjecture for U(3)×U(2): The non-generic case

Haan, Jaeho

doi:10.1016/j.jnt.2016.01.007

Cited by 4 publications

(4 citation statements)

References 23 publications

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“…It is remarkable that for the supercuspidal L-parameter θ s (φ (1) , φ (2) ) with θ(φ (2) ) as above, the recipe, which is sugegsted in (FJ) 3 , does not occur from the theta lift from U (V ± 1 ) and U (V ± 2 ). This is quite similar with the Proposition 4.6 in [14], which concerns the non-generic aspect of Bessel case of the GGP conjecture.…”

Section: Sincesupporting

confidence: 86%

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On the Fourier–Jacobi model for some endoscopic Arthur packet of U(3) × U(3) : the nongeneric case

Haan

2017

Pacific J. Math.

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Abstract. For a generic L-parameter of U (n) × U (n), it is conjectured that there is a unique representation in their associated relevant Vogan L-packet which produces the unique Fourier-Jacobi model. We investigated this conjecture for some non-generic L-parameters of U (3) × U (3) and discovered that this conjecture is still true for some non-generic L-parameter and false for some non-generic L-parameter. In the case when it holds, we specified such representation under the local Langlands correspondence for unitary group. IntroductionThe local Gan-Gross-Prasad conjecture deals with certain restriction problems between p-adic groups. In this paper, we shall investigate it for some non-generic case not yet treated before.Let E/F be a quadratic extension of number fields and G = U (3) be the quasi-split unitary group of rank 3 relative to E/F . Then H = U (2) × U (1) is the unique elliptic endoscopic group for G. In [23], Rogawski has defined a certain enlarged class of L-packets, or A-packets, of G using endoscopic transfer of one-dimensional characters of H to G. In more detail, let ̺ = ⊗ v ̺ v be a one-dimensional automorphic character of H. The A-packet Π(̺) ≃ ⊗Π(̺ v ) is the transfer of ̺ with respect to functoriality for an embedding of L-groups ξ : L H → L G. Then for all places v of F , Π(̺ v ) contains a certain non-tempered representation π n (̺ v ) and it contains an additional supercuspidal representaton π s (̺ v ) precisely when v remains prime in E. Gelbart and Rogawski [12] showed that the representations in this A-packet arise in the Weil representation of G. Our goal is to study the branching rule of the representations in this A-packet.For the branching problem, there is a fascinating conjecture, the so-called Gan-Gross-Prasad (GGP) conjecture, which was first proposed by Gross and Prasad [7] for orthogonal group and later they, together with Gan, extended it to all classical group in [6]. Since our main theorem has to do with it, we shall give a brief review on the GGP conjecture, especially for unitary group. Let E/F be a quadratic extension of non-archimedean local fields of characteristic zero. Let V n+1 be a Hermitian space of dimension n + 1 over E and W n a skew-Hermitian space of dimension n over E. Let V n ⊂ V n+1 be a nondegenerate subspace of codimension 1, so that if we set G n = U(V n ) × U(V n+1 ) or U(W n ) × U(W n ) and H n = U(V n ) or U(W n ),

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Section: Sincesupporting

confidence: 86%

“…Now we shall consider the theta correspondence for U(V ǫ 1 ) × U(W ǫ ′ 3 ). The following is a compound of Theorem 3.4 and Theorem 3.5 in [14].…”

Section: If We Setmentioning

confidence: 95%

On the Fourier–Jacobi model for some endoscopic Arthur packet of U(3) × U(3) : the nongeneric case

Haan

2017

Pacific J. Math.

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“…Disregarding one-dimensional representations of , these A-parameters are of the form , with , one-dimensional characters that are conjugate dual of appropriate sign. This restriction problem has been studied in two papers [Haa16, Haa17] of Haan. His results are in accordance with the expectations of Conjecture 6.1.…”

Section: Low-rank Examplesmentioning

confidence: 99%

Branching laws for classical groups: the non-tempered case

Gan¹,

Gross²,

Prasad

2020

Compositio Math.

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This paper generalizes the Gan–Gross–Prasad (GGP) conjectures that were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the non-generic L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of Arthur parameters that governs the branching laws for ${{\rm GL}}_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in Gan et al. [Symplectic local root numbers, central critical L-values and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad. I, Astérisque 346 (2012), 1–109] including Bessel models and Fourier–Jacobi models.

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“…By Proposition 11.6 in [2], the non-vanishing of these two local zeta integral functions Z v,s=− 1 2 and Z v is equivalent to the non-vanishing of the local theta lifts Θ W1,v,V2,v (π 2,v ) and Θ W1,v,L1,v (µ v ) respectively. Since we assumed that theta lift [9] and it is the analog of the local Ichino-Ikeda conejctures for non-tempered case.…”

Section: The Rallis Inner Product Formulamentioning

confidence: 99%

The Bessel Period of U(3) and U(2) involving a non-tempered representation

Haan¹

2014

Preprint

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In [11], Neal Harris has given a refined Gross-Prasad conjecture for unitary group as an analogue of Ichino and Ikeda's paper [13] concerning special orthogonal groups. In his paper, he stated a conjecture under the assumption that the pair of given representations should be tempered. In this paper, we consider a specific pair involving a non-tempered representation of U (3) and suggest its refined period formula restriced to U (2).

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The local Gan–Gross–Prasad conjecture for U(3)×U(2): The non-generic case

Cited by 4 publications

References 23 publications

On the Fourier–Jacobi model for some endoscopic Arthur packet of U(3) × U(3) : the nongeneric case

On the Fourier–Jacobi model for some endoscopic Arthur packet of U(3) × U(3) : the nongeneric case

Branching laws for classical groups: the non-tempered case

The Bessel Period of U(3) and U(2) involving a non-tempered representation

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