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Test vectors for local periods

  • U. K. Anandavardhanan ORCID logo EMAIL logo and Nadir Matringe
Published/Copyright: December 25, 2016
Forum Mathematicum
From the journal Forum Mathematicum Volume 29 Issue 6

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 [14] 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.

Keywords: Distinguished representation; local period; test vector; Asai L-function
MSC 2010: 22E50; 11F33; 11F70; 11F85

Communicated by Freydoon Shahidi

Funding source: Agence Nationale de la Recherche

Award Identifier / Grant number: ANR-13-BS01-0012 FERPLAY

Funding statement: The second author was supported by the grant ANR-13-BS01-0012 FERPLAY.

Acknowledgements

The authors would like to thank Dipendra Prasad for asking the question about an explicit test vector for the invariant linear form for (Gn⁢(E),Gn⁢(F)). They would also like to thank Omer Offen for useful conversations on the theme of this paper. This paper began when the authors were visiting CIRM Luminy as part of the Chaire Jean-Morlet 2016 Programme and they thank Dipendra Prasad and Volker Heiermann for the invitation to participate in the programme. Thanks are due to the anonymous referee for a careful reading of the manuscript and several useful suggestions.

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Received: 2016-8-3
Revised: 2016-11-17
Published Online: 2016-12-25
Published in Print: 2017-11-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. The role of the Rogers–Shephard inequality in the characterization of the difference body
  3. Test vectors for local periods
  4. The local Langlands conjecture for p-adic GSpin4, GSpin6, and their inner forms
  5. A family of irretractable square-free solutions of the Yang–Baxter equation
  6. On Sylow permutable subgroups of finite groups
  7. Using Steinberg algebras to study decomposability of Leavitt path algebras
  8. The ℓ∞-semi-norm on uniformly finite homology
  9. Minimal potential results for Schrödinger equations with Neumann boundary conditions
  10. On a unique continuation for Aharonov–Bohm magnetic Schrödinger equation
  11. Sup-norm bounds for Eisenstein series
  12. Permutohedral structures on E2-operads
  13. On the complete integrability of the periodic quantum Toda lattice
  14. Fans, decision problems and generators of free abelian ℓ-groups
  15. Standard cocycles: Variations on themes of C. Kassel’s and R. Wilson’s
  16. Crossed modules as maps between connected components of topological groups
  17. A local converse theorem for U(2,2)
  18. Probabilistic trace and Poisson summation formulae on locally compact abelian groups
https://doi.org/10.1515/forum-2016-0169
Keywords for this article
Distinguished representation; local period; test vector; Asai L-function

Articles in the same Issue

  1. Frontmatter
  2. The role of the Rogers–Shephard inequality in the characterization of the difference body
  3. Test vectors for local periods
  4. The local Langlands conjecture for p-adic GSpin4, GSpin6, and their inner forms
  5. A family of irretractable square-free solutions of the Yang–Baxter equation
  6. On Sylow permutable subgroups of finite groups
  7. Using Steinberg algebras to study decomposability of Leavitt path algebras
  8. The ℓ∞-semi-norm on uniformly finite homology
  9. Minimal potential results for Schrödinger equations with Neumann boundary conditions
  10. On a unique continuation for Aharonov–Bohm magnetic Schrödinger equation
  11. Sup-norm bounds for Eisenstein series
  12. Permutohedral structures on E2-operads
  13. On the complete integrability of the periodic quantum Toda lattice
  14. Fans, decision problems and generators of free abelian ℓ-groups
  15. Standard cocycles: Variations on themes of C. Kassel’s and R. Wilson’s
  16. Crossed modules as maps between connected components of topological groups
  17. A local converse theorem for U(2,2)
  18. Probabilistic trace and Poisson summation formulae on locally compact abelian groups
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