TY - JOUR
T1 - The local period integrals and essential vectors
AU - Jo, Yeongseong
N1 - 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.
PY - 2023/1
Y1 - 2023/1
N2 - 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.
AB - 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.
KW - local Rankin–Selberg L-functions
KW - local period integrals
KW - newforms
KW - test vector problems
UR - http://www.scopus.com/inward/record.url?scp=85144100543&partnerID=8YFLogxK
U2 - 10.1002/mana.202100392
DO - 10.1002/mana.202100392
M3 - Article
AN - SCOPUS:85144100543
SN - 0025-584X
VL - 296
SP - 339
EP - 367
JO - Mathematische Nachrichten
JF - Mathematische Nachrichten
IS - 1
ER -