Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

A weak kernel formula for Bessel functions


Author: Jingsong Chai
Journal: Trans. Amer. Math. Soc. 369 (2017), 7139-7167
MSC (2010): Primary 22E50; Secondary 11F70
DOI: https://doi.org/10.1090/tran/6884
Published electronically: April 24, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we prove a weak kernel formula of Bessel functions attached to irreducible generic representations of $ p$-adic $ GL(n)$. As an application, we show that the Bessel function defined by Bessel distribution coincides with the Bessel function defined via uniqueness of Whittaker models on the open Bruhat cell.


References [Enhancements On Off] (What's this?)

  • [1] Moshe Adrian, Baiying Liu, Shaun Stevens, and Peng Xu, On the Jacquet conjecture on the local converse problem for $ p$-adic $ \mathrm {GL}_N$, Represent. Theory 20 (2016), 1-13. MR 3452696, https://doi.org/10.1090/ert/476
  • [2] Ehud Moshe Baruch, On Bessel distributions for $ {\rm GL}_2$ over a $ p$-adic field, J. Number Theory 67 (1997), no. 2, 190-202. MR 1486498, https://doi.org/10.1006/jnth.1997.2173
  • [3] Ehud Moshe Baruch, On Bessel distributions for quasi-split groups, Trans. Amer. Math. Soc. 353 (2001), no. 7, 2601-2614 (electronic). MR 1828462, https://doi.org/10.1090/S0002-9947-01-02778-7
  • [4] Ehud Moshe Baruch, Bessel functions for GL(3) over a $ p$-adic field, Pacific J. Math. 211 (2003), no. 1, 1-33. MR 2016587, https://doi.org/10.2140/pjm.2003.211.1
  • [5] Ehud Moshe Baruch, Bessel distributions for $ \rm GL(3)$ over the $ p$-adics, Pacific J. Math. 217 (2004), no. 1, 11-27. MR 2105763, https://doi.org/10.2140/pjm.2004.217.11
  • [6] Ehud Moshe Baruch, Bessel functions for $ {\rm GL}(n)$ over a $ p$-adic field, Automorphic representations, $ L$-functions and applications: progress and prospects, Ohio State Univ. Math. Res. Inst. Publ., vol. 11, de Gruyter, Berlin, 2005, pp. 1-40. MR 2192818, https://doi.org/10.1515/9783110892703.1
  • [7] Ehud Moshe Baruch and Zhengyu Mao, Bessel identities in the Waldspurger correspondence over a $ p$-adic field, Amer. J. Math. 125 (2003), no. 2, 225-288. MR 1963685
  • [8] Ehud Moshe Baruch and Zhengyu Mao, Bessel identities in the Waldspurger correspondence over the real numbers, Israel J. Math. 145 (2005), 1-81. MR 2154720, https://doi.org/10.1007/BF02786684
  • [9] Ehud Moshe Baruch and Zhengyu Mao, Central value of automorphic $ L$-functions, Geom. Funct. Anal. 17 (2007), no. 2, 333-384. MR 2322488, https://doi.org/10.1007/s00039-007-0601-3
  • [10] Joseph N. Bernstein, $ P$-invariant distributions on $ {\rm GL}(N)$ and the classification of unitary representations of $ {\rm GL}(N)$ (non-Archimedean case), Lie group representations, II (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984, pp. 50-102. MR 748505, https://doi.org/10.1007/BFb0073145
  • [11] Jingsong Chai, Bessel functions and local converse conjecture of Jacquet, preprint, 2015.
  • [12] Jiang-Ping Jeff Chen, The $ n\times (n-2)$ local converse theorem for $ {\rm GL}(n)$ over a $ p$-adic field, J. Number Theory 120 (2006), no. 2, 193-205. MR 2257542, https://doi.org/10.1016/j.jnt.2005.12.001
  • [13] J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro, and F. Shahidi, Functoriality for the classical groups, Publ. Math. Inst. Hautes Études Sci. 99 (2004), 163-233. MR 2075885, https://doi.org/10.1007/s10240-004-0020-z
  • [14] J. W. Cogdell and I. I. Piatetski-Shapiro, Converse theorems for $ {\rm GL}_n$. II, J. Reine Angew. Math. 507 (1999), 165-188. MR 1670207, https://doi.org/10.1515/crll.1999.507.165
  • [15] J. W. Cogdell, I. I. Piatetski-Shapiro, and F. Shahidi, Stability of $ \gamma $-factors for quasi-split groups, J. Inst. Math. Jussieu 7 (2008), no. 1, 27-66. MR 2398146, https://doi.org/10.1017/S1474748007000163
  • [16] J. W. Cogdell, F. Shahidi, and Tung-Lin Tsai, Local Langlands correspondence for $ GL_n$ and the exterior and symmetric square $ \epsilon $-factors, available at http://arxiv.org/abs/1412.1448.
  • [17] Brooke Feigon, Erez Lapid, and Omer Offen, On representations distinguished by unitary groups, Publ. Math. Inst. Hautes Études Sci. 115 (2012), 185-323. MR 2930996
  • [18] I. M. Gelfand and D. A. Kajdan, Representations of the group $ {\rm GL}(n,K)$ where $ K$ is a local field, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 95-118. MR 0404534
  • [19] Guy Henniart, Caractérisation de la correspondance de Langlands locale par les facteurs $ \epsilon $ de paires, Invent. Math. 113 (1993), no. 2, 339-350 (French, with English and French summaries). MR 1228128, https://doi.org/10.1007/BF01244309
  • [20] Roger Howe, Classification of irreducible representations of $ GL_2(F)$ ($ F$ a local field), unpublished preprint, 1978.
  • [21] Dihua Jiang, On local $ \gamma $-factors, Arithmetic geometry and number theory, Ser. Number Theory Appl., vol. 1, World Sci. Publ., Hackensack, NJ, 2006, pp. 1-28. MR 2258071, https://doi.org/10.1142/9789812773531_0001
  • [22] Dihua Jiang and Chufeng Nien, On the local Langlands conjecture and related problems over p-adic local fields, Proceedings to the 6th International Congress of Chinese Mathematicians, Taipei, 2013.
  • [23] Dihua Jiang, Chufeng Nien, and Shaun Stevens, Towards the Jacquet conjecture on the local converse problem for $ p$-adic $ {\rm GL}_n$, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 4, 991-1007. MR 3349305, https://doi.org/10.4171/JEMS/524
  • [24] Dihua Jiang and David Soudry, The local converse theorem for $ {\rm SO}(2n+1)$ and applications, Ann. of Math. (2) 157 (2003), no. 3, 743-806. MR 1983781, https://doi.org/10.4007/annals.2003.157.743
  • [25] Erez Lapid and Zhengyu Mao, Stability of certain oscillatory integrals, Int. Math. Res. Not. IMRN 3 (2013), 525-547. MR 3021791
  • [26] Chufeng Nien, A proof of the finite field analogue of Jacquet's conjecture, Amer. J. Math. 136 (2014), no. 3, 653-674. MR 3214273, https://doi.org/10.1353/ajm.2014.0020
  • [27] Freydoon Shahidi, Local coefficients as Mellin transforms of Bessel functions: towards a general stability, Int. Math. Res. Not. 39 (2002), 2075-2119. MR 1926651, https://doi.org/10.1155/S1073792802204171
  • [28] J. A. Shalika, The multiplicity one theorem for $ {\rm GL}_{n}$, Ann. of Math. (2) 100 (1974), 171-193. MR 0348047
  • [29] David Soudry, The $ L$ and $ \gamma $ factors for generic representations of $ {\rm GSp}(4,\,k)\times {\rm GL}(2,\,k)$ over a local non-Archimedean field $ k$, Duke Math. J. 51 (1984), no. 2, 355-394. MR 747870, https://doi.org/10.1215/S0012-7094-84-05117-2

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 22E50, 11F70

Retrieve articles in all journals with MSC (2010): 22E50, 11F70


Additional Information

Jingsong Chai
Affiliation: College of Mathematics and Econometrics, Hunan University, Changsha, 410082, People’s Republic of China
Email: jingsongchai@hotmail.com

DOI: https://doi.org/10.1090/tran/6884
Keywords: Bessel functions, Bessel distributions, weak kernel formula
Received by editor(s): August 3, 2015
Received by editor(s) in revised form: November 16, 2015, and December 3, 2015
Published electronically: April 24, 2017
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society