Gamma Factors, Root Numbers, and Distinction

Matringe, Nadir; Offen, Omer

doi:10.4153/cjm-2017-011-6

Cited by 7 publications

(8 citation statements)

References 22 publications

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“…Therefore, by Proposition 4.4, and by observing that the epsilon factor is trivial since π is distinguished with respect to G n (F) [MO16,Theorem 3.6], we get…”

Section: Theorem 63 Let π Be An Irreducible Generic Unitary Represementioning

confidence: 90%

“…G n 2 (F)), then ǫ(1/2, π 1 × π 2 , ψ) = 1 where ψ is a character of E which is trivial on F. This result was established for cuspidal representations in Youngbin Ok's PhD thesis [Ok97], where a cuspidal relative converse theorem for the pair (G n (E), G n (F)) was also proved. Both these results from [Ok97] are generalized in [MO16], to which we refer for more explanations about this topic. In fact, the aforementioned result of [MO16] is [Ana08, Conjecture 5.1].…”

Section: Lemma 43 Let E/f Be Ramified If π Is An Irreducible Admismentioning

confidence: 99%

“…Both these results from [Ok97] are generalized in [MO16], to which we refer for more explanations about this topic. In fact, the aforementioned result of [MO16] is [Ana08, Conjecture 5.1].…”

Section: Lemma 43 Let E/f Be Ramified If π Is An Irreducible Admismentioning

confidence: 99%

“…To conclude Theorem 1.2, in addition to a fine knowledge of the properties of the essential vector, we also need to appeal to the main result of [MO16], which relates distinction for the pair (GL n (E), GL n (F)) to the behavior of epsilon factor for pairs. Both Theorem 1.1 and Theorem 1.2 are proved in Section 6, under the assumption that π is a unitary representation.…”

Section: Introductionmentioning

confidence: 99%

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Test vectors for local periods

Anandavardhanan

Matringe

2016

Forum Mathematicum

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Abstract. Let E/F be a quadratic extension of non-archimedean local fields of characteristic zero. An irreducible admissible representation π of GL(n, E) is said to be distinguished with respect to GL(n, F) if it admits a non-trivial linear form that is invariant under the action of GL(n, F). It is known that there is exactly one such invariant linear form up to multiplication by scalars, and an explicit linear form is given by integrating Whittaker functions over the F-points of the mirabolic subgroup when π is unitary and generic. In this paper, we prove that the essential vector of [JPSS81] is a test vector for this standard distinguishing linear form and that the value of this form at the essential vector is a local L-value. As an application we determine the value of a certain proportionality constant between two explicit distinguishing linear forms. We then extend all our results to the non-unitary generic case.

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“…Therefore, by Proposition 4.4, and by observing that the epsilon factor is trivial since π is distinguished with respect to G n (F) [MO16,Theorem 3.6], we get…”

Section: Theorem 63 Let π Be An Irreducible Generic Unitary Represementioning

confidence: 90%

Section: Lemma 43 Let E/f Be Ramified If π Is An Irreducible Admismentioning

confidence: 99%

Section: Lemma 43 Let E/f Be Ramified If π Is An Irreducible Admismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Test vectors for local periods

Anandavardhanan

Matringe

2016

Forum Mathematicum

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“…In fact, appealing to [3, Theorem 6.3] is enough in the cuspidal case, since a distinguished cuspidal representation of G is always unitary (as its central character is). Now the only ingredient in the proof of [3,Theorem 6.3] which uses this rectriction on the characteristic of F is that the Godement-Jacquet epsilon factor ǫ(1/2, π, ψ) is equal to 1, for which [3] refers to [38], but the cuspidal case of this result is already in [41] and this reference does not assume the characteristic of F to be 0.…”

Section: Test Vectorsmentioning

confidence: 99%

Galois self-dual cuspidal types and Asai local factors

Anandavardhanan

Kurinczuk

Matringe

et al. 2018

Preprint

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Let F/F o be a quadratic extension of non-archimedean locally compact fields of odd residual characteristic and σ be its non-trivial automorphism. We show that any σ-self-dual cuspidal representation of GL n (F) contains a σ-self-dual Bushnell-Kutzko type. Using such a type, we construct an explicit test vector for Flicker's local Asai L-function of a GL n (F o )-distinguished cuspidal representation and compute the associated Asai root number. Finally, by using global methods, we compare this root number to Langlands-Shahidi's local Asai root number, and more generally we compare the corresponding epsilon factors for any cuspidal representation. NotationLet F/F o be a quadratic extension of locally compact non-archimedean fields of residual characteristic p = 2. Write σ for the non-trivial F o -automorphism of F. For any finite extension E of F o , we denote by O E its ring of integers, by p E the unique maximal ideal of O E and by k E its residue field. We abbreviate O F to O and O Fo to O o , and define similarly p, p o , k, k o . The involution σ induces a k o -automorphism of k, still denoted σ.

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The local period integrals and essential vectors

2022

Mathematische Nachrichten

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By applying the formula for essential Whittaker functions established by Matringe and Miyauchi, we study five integral representations for irreducible admissible generic representations of GL 𝑛 over 𝑝-adic fields. In each case, we show that the integrals achieve local formal 𝐿-functions defined by Langlands parameters, when the test vector is associated to the new form. We give the relation between local periods involving essential Whittaker functions and special values of formal 𝐿-factors at 𝑠 = 1 for certain distinguished or unitary representations. The period integrals are also served as standard nonzero distinguished forms.

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Gamma Factors, Root Numbers, and Distinction

Cited by 7 publications

References 22 publications

Test vectors for local periods

Test vectors for local periods

Galois self-dual cuspidal types and Asai local factors

The local period integrals and essential vectors

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