Papers
Topics
Authors
Recent
Search
2000 character limit reached

Formal Degrees and Parabolic Induction: the Maximal Generic Case

Published 26 Feb 2024 in math.NT and math.RT | (2402.16456v2)

Abstract: We study the compatibility of the formal degree conjecture and the parabolic induction process in the simplest nontrivial case for quasi-split pp-adic groups. For a generic discrete series π\pi induced from an irreducible supercuspidal σ\sigma of a maximal Levi subgroup, we compute the quotient d(π)/d(σ)d(\pi)/d(\sigma) of formal degrees under some assumptions. As an application, we verify the conjecture for discrete series of split G2\mathrm{G}_2 supported on maximal Levi subgroups.

Authors (1)
Definition Search Book Streamline Icon: https://streamlinehq.com
References (28)
  1. Raphaël Beuzart-Plessis “Plancherel formula for GLn⁢(F)\GLn⁢(E)\subscriptGL𝑛𝐹subscriptGL𝑛𝐸{\rm GL}_{n}(F)\backslash{\rm GL}_{n}(E)roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_F ) \ roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_E ) and applications to the Ichino-Ikeda and formal degree conjectures for unitary groups” In Invent. Math. 225.1, 2021, pp. 159–297
  2. A. Borel “Automorphic L𝐿Litalic_L-functions” In Automorphic forms, representations and L𝐿Litalic_L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2 XXXIII, Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 1979, pp. 27–61
  3. Gaëtan Chenevier and al “Automorphic Forms, Geometry and Arithmetic (hybrid meeting)” Mathematisches Forschungsinstitut Oberwolfach, 2021. URL: https://math.umd.edu/~yhzhu/OWR_2021_39.pdf
  4. Brian Conrad “Reductive group schemes” In Autour des schémas en groupes. Vol. I 42/43, Panor. Synthèses Soc. Math. France, Paris, 2014, pp. 93–444
  5. Benedict H. Gross and Wee Teck Gan “Haar measure and the Artin conductor” In Trans. Amer. Math. Soc. 351.4, 1999, pp. 1691–1704
  6. Wee Teck Gan and Atsushi Ichino “Formal degrees and local theta correspondence” In Invent. Math. 195.3, 2014, pp. 509–672
  7. Benedict H. Gross and Mark Reeder “Arithmetic invariants of discrete Langlands parameters” In Duke Math. J. 154.3, 2010, pp. 431–508
  8. Benedict H. Gross “On the motive of a reductive group” In Invent. Math. 130.2, 1997, pp. 287–313
  9. Wee Teck Gan and Gordan Savin “Howe duality and dichotomy for exceptional theta correspondences” In Invent. Math. 232.1, 2023, pp. 1–78
  10. Wee Teck Gan and Gordan Savin “The Local Langlands Conjecture for G2subscriptG2\mathrm{G}_{2}roman_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT” In Forum Math. Pi 11, 2023, pp. Paper No. e28\bibrangessep42
  11. Volker Heiermann “Décomposition spectrale et représentations spéciales d’un groupe réductif p𝑝pitalic_p-adique” In J. Inst. Math. Jussieu 3.3, 2004, pp. 327–395
  12. Volker Heiermann “Orbites unipotentes et pôles d’ordre maximal de la fonction μ𝜇\muitalic_μ de Harish-Chandra” In Canad. J. Math. 58.6, 2006, pp. 1203–1228
  13. Volker Heiermann “Unipotent orbits and local L𝐿Litalic_L-functions” In J. Reine Angew. Math. 596, 2006, pp. 103–114
  14. Kaoru Hiraga, Atsushi Ichino and Tamotsu Ikeda “Correction to: “Formal degrees and adjoint γ𝛾\gammaitalic_γ-factors” [J. Amer. Math. Soc. 21 (2008), no. 1, 283–304; MR2350057]” In J. Amer. Math. Soc. 21.4, 2008, pp. 1211–1213
  15. Kaoru Hiraga, Atsushi Ichino and Tamotsu Ikeda “Formal degrees and adjoint γ𝛾\gammaitalic_γ-factors” In J. Amer. Math. Soc. 21.1, 2008, pp. 283–304
  16. Atsushi Ichino, Erez Lapid and Zhengyu Mao “On the formal degrees of square-integrable representations of odd special orthogonal and metaplectic groups” In Duke Math. J. 166.7, 2017, pp. 1301–1348
  17. “Bruhat-Tits theory—a new approach” 44, New Mathematical Monographs Cambridge University Press, Cambridge, 2023, pp. xxx+718
  18. George Lusztig “Some examples of square integrable representations of semisimple p𝑝pitalic_p-adic groups” In Trans. Amer. Math. Soc. 277.2, 1983, pp. 623–653
  19. Kazuki Morimoto “On a certain local identity for Lapid-Mao’s conjecture and formal degree conjecture: even unitary group case” In J. Inst. Math. Jussieu 21.4, 2022, pp. 1107–1161
  20. Goran Muić “The unitary dual of p𝑝pitalic_p-adic G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT” In Duke Math. J. 90.3, 1997, pp. 465–493
  21. “Spectral decomposition and Eisenstein series” Une paraphrase de l’Écriture [A paraphrase of Scripture]. 113, Cambridge Tracts in Mathematics Cambridge University Press, Cambridge, 1995
  22. Kazuma Ohara “On the formal degree conjecture for non-singular supercuspidal representations” In Int. Math. Res. Not. IMRN, 2023, pp. 10997–11034
  23. David Schwein “Formal Degree of Regular Supercuspidals” To appear on Journal of the European Mathematical Society. https://arxiv.org/abs/2101.00658
  24. Freydoon Shahidi “On the Ramanujan conjecture and finiteness of poles for certain L𝐿Litalic_L-functions” In Ann. of Math. (2) 127.3, 1988, pp. 547–584
  25. Freydoon Shahidi “Third symmetric power L𝐿Litalic_L-functions for GL⁢(2)GL2{\rm GL}(2)roman_GL ( 2 )” In Compositio Math. 70.3, 1989, pp. 245–273
  26. Freydoon Shahidi “A proof of Langlands’ conjecture on Plancherel measures; complementary series for p𝑝pitalic_p-adic groups” In Ann. of Math. (2) 132.2, 1990, pp. 273–330
  27. Allan J. Silberger “Special representations of reductive p𝑝pitalic_p-adic groups are not integrable” In Ann. of Math. (2) 111.3, 1980, pp. 571–587
  28. J.-L. Waldspurger “La formule de Plancherel pour les groupes p𝑝pitalic_p-adiques (d’après Harish-Chandra)” In J. Inst. Math. Jussieu 2.2, 2003, pp. 235–333

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

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

Generate Now

Tweets

Sign up for free to view the 1 tweet with 3 likes about this paper.

Sign Up for Free