The local period integrals and essential vectors

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Abstract

By applying the formula for essential Whittaker functions established by Matringe and Miyauchi, we study five integral representations for irreducible admissible generic representations of GLn 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 (Formula presented.) for certain distinguished or unitary representations. The period integrals are also served as standard nonzero distinguished forms.

Original languageEnglish
Pages (from-to)339-367
Number of pages29
JournalMathematische Nachrichten
Volume296
Issue number1
DOIs
StatePublished - Jan 2023

Bibliographical note

Funding Information:
This project is inspired by the response to a question raised by James Cogdell as to whether the space of ‐invariant Shalika functionals in [ 31 , Lemma 3.2] is trivial or not. The author is indebted to J. Cogdell for drawing the author's attention to this problem. Yeongseong Jo would like to thank Muthu Krishnamurthy for kindly explaining their joint work [ 8 ], encouraging to write this paper, and giving many invaluable comments over the year. We are also grateful to Peter Humphries for many helpful suggestions on earlier versions of this paper. Finally, we express our sincere appreciation to the referee for a number of constructive comments, which significantly improved the exposition of literature in the paper. This research was supported by the Ewha Womans University Research Grant of 2022.

Publisher Copyright:
© 2022 The Authors. Mathematische Nachrichten published by Wiley-VCH GmbH.

Keywords

  • local Rankin–Selberg L-functions
  • local period integrals
  • newforms
  • test vector problems

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