Papers
Topics
Authors
Recent
AI Research Assistant
AI Research Assistant
Well-researched responses based on relevant abstracts and paper content.
Custom Instructions Pro
Preferences or requirements that you'd like Emergent Mind to consider when generating responses.
Gemini 2.5 Flash
Gemini 2.5 Flash 89 tok/s
Gemini 2.5 Pro 43 tok/s Pro
GPT-5 Medium 24 tok/s Pro
GPT-5 High 24 tok/s Pro
GPT-4o 112 tok/s Pro
Kimi K2 199 tok/s Pro
GPT OSS 120B 449 tok/s Pro
Claude Sonnet 4 37 tok/s Pro
2000 character limit reached

Local converse theorems and Langlands parameters (2409.20240v4)

Published 30 Sep 2024 in math.RT and math.NT

Abstract: Let $F$ be a non Archimedean local field, and $G$ be the $F$-points of a connected quasi-split reductive group defined over $F$. In this note we propose a converse theorem statement for generic Langlands parameters of $G$ when the Langlands dual group of $G$ is acceptable. We then prove it when $G$ is $F$-split. Then we consider a variant which we prove for $G=\mathrm{G}_2(F)$ and all quasi-split classical groups. When $F$ has characteristic zero and assuming the validity of the Gross-Prasad and Rallis conjecture, this latter variant translates via the generic local Langlands correspondence of Jantzen and Liu, into the usual local converse theorems for classical groups expressed in terms of Shahidi's gamma factors.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (47)
  1. Local theta correspondence of tempered representations and Langlands parameters. Invent. Math., 210(2):341–415, 2017.
  2. Simple supercuspidal l-packets of split special orthogonal groups over dyadic fields. preprint, https://arxiv.org/abs/2305.09076, 2023.
  3. Representation theory of disconnected reductive groups. Doc. Math., 25:2149–2177, 2020.
  4. James Arthur. The endoscopic classification of representations, volume 61 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2013. Orthogonal and symplectic groups.
  5. A local converse theorem for Archimedean GL⁢(n)GL𝑛{\rm GL}(n)roman_GL ( italic_n ). Forum Math., 35(3):793–808, 2023.
  6. Hiraku Atobe. On the uniqueness of generic representations in an L𝐿Litalic_L-packet. Int. Math. Res. Not. IMRN, (23):7051–7068, 2017.
  7. A. Borel. Automorphic l-functions. In Automorphic forms, representations and L𝐿Litalic_L-functions, Part 2, Proc. Sympos. Pure Math., XXXIII, pages 27–61. Amer. Math. Soc., Providence, R.I., 1979.
  8. N. Bourbaki. Éléments de mathématique. Algèbre. Chapitre 8. Modules et anneaux semi-simples. Springer, Berlin, revised edition, 2012.
  9. G. Chenevier and W. Gan. Spin⁢(7)Spin7\textrm{Spin}(7)Spin ( 7 ) is unacceptable. Peking Mathematical Journal, 2024.
  10. Jingsong Chai. Bessel functions and local converse conjecture of Jacquet. J. Eur. Math. Soc. (JEMS), 21(6):1703–1728, 2019.
  11. Converse theorems for GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. Inst. Hautes Études Sci. Publ. Math., (79):157–214, 1994.
  12. Local langlands in families: the banal case. preprint, arXiv:2406.09283, 2024.
  13. The Gross-Prasad conjecture and local theta correspondence. Invent. Math., 206(3):705–799, 2016.
  14. On the decomposition of a representation of SOnsubscriptSO𝑛{\rm SO}_{n}roman_SO start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT when restricted to SOn−1subscriptSO𝑛1{\rm SO}_{n-1}roman_SO start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT. Canad. J. Math., 44(5):974–1002, 1992.
  15. Arithmetic invariants of discrete Langlands parameters. Duke Math. J., 154(3):431–508, 2010.
  16. Robert L. Griess, Jr. Basic conjugacy theorems for G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Invent. Math., 121(2):257–277, 1995.
  17. Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence. Compos. Math., 148(6):1655–1694, 2012.
  18. The local Langlands conjecture for G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Forum Math. Pi, 11:Paper No. e28, 42, 2023.
  19. G. Henniart. Une caractérisation de la correspondance de langlands locale pour GL⁢(n)GL𝑛\mathrm{GL}(n)roman_GL ( italic_n ). Bulletin de la Société Mathématique de France, 130(4):587–602, 2002.
  20. Guy Henniart. Caractérisation de la correspondance de Langlands locale par les facteurs ϵitalic-ϵ\epsilonitalic_ϵ de paires. Invent. Math., 113(2):339–350, 1993.
  21. Guy Henniart. Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique. Inventiones mathematicae, 139(2):439–455, 2000.
  22. preprint, arXiv:2301.13847, 2023.
  23. A. Hazeltine and B. Liu. preprint, arXiv:2301.13847, 2023.
  24. The Geometry and Cohomology of Some Simple Shimura Varieties.(AM-151), volume 151. Princeton university press, 2001.
  25. Disconnected reductive groups: classification and representations. preprint, arXiv:2409.06375v1, 2024.
  26. Dihua Jiang. On local γ𝛾\gammaitalic_γ-factors. In Arithmetic geometry and number theory, volume 1 of Ser. Number Theory Appl., pages 1–28. World Sci. Publ., Hackensack, NJ, 2006.
  27. H. Jacquet and R. P. Langlands. Automorphic forms on GL⁢(2)GL2{\rm GL}(2)roman_GL ( 2 ), volume Vol. 114 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1970.
  28. On the local converse theorem for p𝑝pitalic_p-adic GLnsubscriptGL𝑛{\rm GL}_{n}roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. Amer. J. Math., 140(5):1399–1422, 2018.
  29. C. Jantzen and B. Liu. The generic dual of p-adic groups and applications. preprint, arXiv:2404.07111, 2024.
  30. Y. Jo. preprint, arXiv:2205.09004, 2024.
  31. Rankin-Selberg convolutions. Amer. J. Math., 105(2):367–464, 1983.
  32. Automorphic forms on GL⁢(3)GL3{\rm GL}(3)roman_GL ( 3 ). I. Ann. of Math. (2), 109(1):169–212, 1979.
  33. The local converse theorem for so(2n+1) and applications. Ann. of Math. (2), 157(3):743–806, 2003.
  34. Michael Larsen. On the conjugacy of element-conjugate homomorphisms. Israel J. Math., 88(1-3):253–277, 1994.
  35. Michael Larsen. On the conjugacy of element-conjugate homomorphisms. II. Quart. J. Math. Oxford Ser. (2), 47(185):73–85, 1996.
  36. Chung Pang Mok. Endoscopic classification of representations of quasi-split unitary groups. Mem. Amer. Math. Soc., 235(1108):vi+248, 2015.
  37. Kazuki Morimoto. On the irreducibility of global descents for even unitary groups and its applications. Trans. Amer. Math. Soc., 370(9):6245–6295, 2018.
  38. Dipendra Prasad. A relative langlands correspondence and geometry of parameter spaces. https://sites.google.com/view/dipendra-prasad/publications, 2020.
  39. Robert Steinberg. Regular elements of semisimple algebraic groups. Inst. Hautes Études Sci. Publ. Math., (25):49–80, 1965.
  40. Langlands classification for L𝐿Litalic_L-parameters. J. Algebra, 511:299–357, 2018.
  41. J. Tate. Number theoretic background. In Automorphic forms, representations and L𝐿Litalic_L-functions, Part 2, Proc. Sympos. Pure Math., XXXIII, pages 3–26. Amer. Math. Soc., Providence, R.I., 1979.
  42. M. Weidner. Pseudocharacters of homomorphisms into classical groups. Transform. Groups, 25(4):1345–1370, 2020.
  43. Jun Yu. Acceptable compact Lie groups. Peking Math. J., 5(2):427–446, 2022.
  44. P. Yan and Q. Zhang. preprint, arxiv:2302.06256, to appear in Proceedings of the AMS, 2023.
  45. A. V. Zelevinsky. Induced representations of reductive p-adic groups. II. on irreducible representations of GL⁢(n)GL𝑛\mathrm{GL}(n)roman_GL ( italic_n ). Annales Scientifiques de l’École Normale Supérieure, 13(2):165–210, 1980.
  46. Qing Zhang. A local converse theorem for Sp2⁢rsubscriptSp2𝑟{\rm Sp}_{2r}roman_Sp start_POSTSUBSCRIPT 2 italic_r end_POSTSUBSCRIPT. Math. Ann., 372(1-2):451–488, 2018.
  47. Qing Zhang. A local converse theorem for u(2r+1). Trans. Amer. Math. Soc., 371(8):5631–5654, 2019.
Citations (1)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
Lightbulb On Streamline Icon: https://streamlinehq.com

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Authors (1)

  1. Nadir Matringe
List To Do Tasks Checklist Streamline Icon: https://streamlinehq.com

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets

This paper has been mentioned in 2 posts and received 0 likes.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up to View