Test vectors for finite periods and base change

Anandavardhanan, U. K.; Matringe, Nadir

doi:10.1016/j.aim.2019.106915

Cited by 5 publications

(3 citation statements)

References 25 publications

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“…Conversely, one can check that any nondegenerate character of UpEq that is trivial on UpFq can be obtained in this way. In view of Lemma 6.1, one can check easily that µ is actually a ψ-Whittaker linear form (see [AM,Remark 7]). Proposition 6.3.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

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Distinguished representations, Shintani base change and a finite field analogue of a conjecture of Prasad

Yang

2020

Advances in Mathematics

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Let E{F be a quadratic extension of fields, and G a connected quasi-split reductive group over F. Let G op be the opposition group obtained by twisting G by the duality involution considered by Prasad. Assume that the field F is finite. Let π be an irreducible generic representation of GpEq. When π is a Shintani base change lift of some representation of G op pFq, we give an explicit nonzero GpFq-invariant vector in terms of the Whittaker vector of π. This shows particularly that π is GpFq-distinguished.When the field F is p-adic, the paper also proves that the duality involution takes an irreducible admissible generic representation of GpFq to its contragredient. As a special case of this result, all generic representations of G 2 , F 4 or E 8 are self-dual.

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Section: Proof Of Theorem 11mentioning

confidence: 99%

“…It turns out that it is easier to evaluate µpB π,ψ q. The arguments in [AM,Lemma 3.4] works verbatim here. We include the proof here for the sake of completeness.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

Distinguished representations, Shintani base change and a finite field analogue of a conjecture of Prasad

Yang

2020

Advances in Mathematics

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“…We may mention here in passing that somewhat surprisingly even the finite field analogue of this characterization of distinction in a generic L-packet turned out to be non-trivial and is settled only fairly recently [AM18]. However, if π is a cuspidal representation, then a uniform proof of this characterization for both p-adic and finite fields can be given in an elementary manner [AP18, Proposition 4.2 & Remark 4].…”

Section: Introductionmentioning

confidence: 99%

Some questions on global distinction for $\mathrm{SL}(n)$

Anandavardhanan¹,

Matringe²

2019

Preprint

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Let E/F be a quadratic extension of number fields and let π be an SL n (A F )distinguished cuspidal automorphic representation of SL n (A E ). Using an unfolding argument, we prove that an element of the L-packet of π is distinguished if and only if it is ψ-generic for a non-degenerate characterwhere N n is the group of unipotent upper triangular matrices of SL n . We then use this result to analyze the non-vanishing of the period integral on different realizations of a distinguished cuspidal automorphic representation of SL n (A E ) with multiplicity > 1, and show that in general some canonical copies of a distinguished representation inside different L-packets can have vanishing period. We also construct examples of everywhere locally distinguished representations of SL n (A E ) the L-packets of which do not contain any distinguished representation.

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Finite period vectors and Gauss sums

Yeongseong Jo

2024

Finite Fields and Their Applications

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Test vectors for finite periods and base change

Abstract: Let E/F be a quadratic extension of finite fields. By a result of Gow, an irreducible representation π of G

Cited by 5 publications

References 25 publications

Distinguished representations, Shintani base change and a finite field analogue of a conjecture of Prasad

Distinguished representations, Shintani base change and a finite field analogue of a conjecture of Prasad

Some questions on global distinction for $\mathrm{SL}(n)$

Finite period vectors and Gauss sums

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