Local theta correspondence of tempered representations and Langlands parameters

Atobe, Hiraku; Gan, Wee Teck

doi:10.1007/s00222-017-0730-8

Cited by 43 publications

(124 citation statements)

References 43 publications

Supporting

Mentioning

121

Contrasting

Order By: Relevance

“…The proof of the uniqueness is essentially same as we have done in (4.3). In this case, however, we cannot deduce from Proposition 5.4 in [2] that for π n+1 ∈ Π θ(φ 2 ) , Θ ψ,W ǫ ′ n ,V ǫ n+1 (π n+1 ) is irreducible and tempered. Instead, it follows from by our assumption and from Proposition 5.5 in [2].…”

Section: Pair Of Characters Of the Component Group Is Specified As Fomentioning

confidence: 90%

“…n ,V ǫ n+1 Θ ψ,V ǫ n+1 ,W ǫ ′ n(π) is nonzero by the above property (iii) and Theorem 4.1 in[2]. Note that Θ ψ,W ǫ ′ n ,V ǫ n+1 Θ ψ,V ǫ n+1 ,W ǫ ′ n (π) ⊠ Θ ψ,V ǫ n+1 ,W ǫ ′ n (π) is the maximal Θ ψ,V ǫ n+1 ,W ǫ ′ n (π)-isotypic quotient of ω ψ,V n+1 ,Wn and π⊠Θ ψ,V ǫ n+1 ,W ǫ ′ n (π) is a Θ ψ,V ǫ n+1 ,W ǫ ′ n (π)-isotypic quotient of ω ψ,V n+1 ,Wn .…”

mentioning

confidence: 80%

See 1 more Smart Citation

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

Haan

2016

Journal of Number Theory

View full text Add to dashboard Cite

The local Gan-Gross-Prasad conjecture of unitary groups, which is now settled by the works of Plessis, Gan and Ichino, says that for a pair of generic L-parameters of (U (n + 1), U (n)), there is a unique pair of representations in their associated Vogan L-packets which produces the Bessel model. In this paper, we examined the conjecture for a pair of L-parameters of U (n + 1), U (n) as fixing a special non-generic parameter of U (n + 1) and varing tempered L-parameters of U (n) and observed that there still exist a Gan-Gross-Prasad type formulae depending on the choice of L-parameter of U (n). 2 Tr E/F (δx)). So, it is reasonable to separate J ψ using the notations J ψ E and J ψ according to ε = +1 or ε = −1. However, when n is even, we writefor convenience and even when n is odd, we use the same notation J ψ for the canonical bijection.Hereafter, we fix an additive character of ψ : F × → C once and for all and when it comes to the LLC of unitary groups, we shall use a bijectionfor all L-parameter φ of U (V ± ) as above.Using these fixed bijections, all irreducible smooth representations of U(V ± ) can be labelled as π(φ, η) for some unique pair of L-parameter φ of U (V ± ) and η ∈ Irr(S φ ).Lastly, we briefly list some properties of the LLC for unitary group.

show abstract

Section: Pair Of Characters Of the Component Group Is Specified As Fomentioning

confidence: 90%

mentioning

confidence: 80%

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

Haan

2016

Journal of Number Theory

View full text Add to dashboard Cite

show abstract

“…We prove the proposition by induction on k. When k = 1 and l > 0, this is Corollary B.4. When k = 1 and l = 0 so that δ = 1, the irreducibility of µ −1 ⋊ π follows from Theorem B.1 (6).…”

Section: Local Main Theoremmentioning

confidence: 95%

A theory of Miyawaki liftings: the Hilbert–Siegel case

Atobe

2019

Math. Ann.

Self Cite

View full text Add to dashboard Cite

The Miyawaki liftings are defined by the pullbacks of Ikeda liftings. Recently, Ikeda and Yamana extended the theory of Ikeda liftings. In this paper, using their results, we establish a theory of Miyawaki liftings, both locally and globally. In the local theory, we describe the Miyawaki liftings for almost tempered unitary representations explicitly. In the global theory, we discuss the non-vanishing of the Miyawaki liftings using seesaw identities and the global Gan-Gross-Prasad conjecture. As an application of local Miyawaki liftings, we prove a new case of the local Gan-Gross-Prasad conjecture.

show abstract

“…We are grateful to professor Atsushi Ichino for his kind suggestion to pursue the non-tempered aspects of the GGP conjecture. The author expresses his deepest thanks to the referee for his many invaluable comments and bringing our attention to the work of H.Atobe and Gan [2] containing Theorem 4.1 and Proposition 5.4, both of which are crucial to prove our main theorem. This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP)(ASARC, NRF-2007-0056093).…”

Section: From This We See That For Having Hommentioning

confidence: 97%

“…Theorem 4.1. Let φ (1) , φ (2) be tempered L-parameters of U (V ± 1 ) and U (V ± 2 ) respectively and suppose that φ (2) does not contain χ −3 . Let…”

Section: Main Theoremmentioning

confidence: 99%

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

Haan

2017

Pacific J. Math.

View full text Add to dashboard Cite

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 ),

show abstract

Local theta correspondence of tempered representations and Langlands parameters

Cited by 43 publications

References 43 publications

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

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

A theory of Miyawaki liftings: the Hilbert–Siegel case

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

Contact Info

Product

Resources

About