Formal degrees and adjoint 𝛾-factors

Hiraga, Kaoru; Ichino, Atsushi; Ikeda, Tamotsu

doi:10.1090/S0894-0347-07-00567-X

Formal degrees and adjoint $\gamma$ -factors

Authors: Kaoru Hiraga, Atsushi Ichino and Tamotsu Ikeda
Journal: J. Amer. Math. Soc. 21 (2008), 283-304
MSC (2000): Primary 22E50
DOI: https://doi.org/10.1090/S0894-0347-07-00567-X
Published electronically: June 5, 2007
Erratum: J. Amer. Math. Soc. 21 (2008), 1211-1213.
MathSciNet review: 2350057
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give a conjectural formula for the formal degree of a discrete series representation in terms of the adjoint $\gamma$ -factor. Our conjecture is supported by various examples and is compatible with the Weyl dimension formula. Using twisted endoscopy, we also verify the conjecture for a stable discrete series representation of $\operatorname{U}(3)$ over a non-archimedean local field of characteristic zero.

1. James Arthur, The local behaviour of weighted orbital integrals, Duke Math. J. 56 (1988), no. 2, 223–293. MR 932848, https://doi.org/10.1215/S0012-7094-88-05612-8
2. James Arthur, Unipotent automorphic representations: conjectures, Astérisque 171-172 (1989), 13–71. Orbites unipotentes et représentations, II. MR 1021499
3. Anne-Marie Aubert and Roger Plymen, Plancherel measure for 𝐺𝐿(𝑛,𝐹) and 𝐺𝐿(𝑚,𝐷): explicit formulas and Bernstein decomposition, J. Number Theory 112 (2005), no. 1, 26–66. MR 2131140, https://doi.org/10.1016/j.jnt.2005.01.005
4. Armand Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math. 35 (1976), 233–259. MR 0444849, https://doi.org/10.1007/BF01390139
5. Laurent Clozel, Characters of nonconnected, reductive 𝑝-adic groups, Canad. J. Math. 39 (1987), no. 1, 149–167. MR 889110, https://doi.org/10.4153/CJM-1987-008-3
6. Laurent Clozel, The fundamental lemma for stable base change, Duke Math. J. 61 (1990), no. 1, 255–302. MR 1068388, https://doi.org/10.1215/S0012-7094-90-06112-5
7. Laurent Clozel, Invariant harmonic analysis on the Schwartz space of a reductive 𝑝-adic group, Harmonic analysis on reductive groups (Brunswick, ME, 1989) Progr. Math., vol. 101, Birkhäuser Boston, Boston, MA, 1991, pp. 101–121. MR 1168480, https://doi.org/10.1007/978-1-4612-0455-8_6
8. S. DeBacker and M. Reeder, Depth-zero supercuspidal -packets and their stability, preprint.
9. P. Deligne, D. Kazhdan, and M.-F. Vignéras, Représentations des algèbres centrales simples 𝑝-adiques, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 33–117 (French). MR 771672
10. David Goldberg, Some results on reducibility for unitary groups and local Asai 𝐿-functions, J. Reine Angew. Math. 448 (1994), 65–95. MR 1266747, https://doi.org/10.1515/crll.1994.448.65
11. Benedict H. Gross, On the motive of a reductive group, Invent. Math. 130 (1997), no. 2, 287–313. MR 1474159, https://doi.org/10.1007/s002220050186
12. Benedict H. Gross, On the motive of 𝐺 and the principal homomorphism 𝑆𝐿₂→̂𝐺, Asian J. Math. 1 (1997), no. 1, 208–213. MR 1480995, https://doi.org/10.4310/AJM.1997.v1.n1.a8
13. Harish-Chandra, Harmonic analysis on real reductive groups. I. The theory of the constant term, J. Functional Analysis 19 (1975), 104–204. MR 0399356
14. Harish-Chandra, Harmonic analysis on real reductive groups. III. The Maass-Selberg relations and the Plancherel formula, Ann. of Math. (2) 104 (1976), no. 1, 117–201. MR 0439994, https://doi.org/10.2307/1971058
15. Harish-Chandra, Admissible invariant distributions on reductive 𝑝-adic groups, University Lecture Series, vol. 16, American Mathematical Society, Providence, RI, 1999. Preface and notes by Stephen DeBacker and Paul J. Sally, Jr. MR 1702257
16. Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001. With an appendix by Vladimir G. Berkovich. MR 1876802
17. Guy Henniart, Une preuve simple des conjectures de Langlands pour 𝐺𝐿(𝑛) sur un corps 𝑝-adique, Invent. Math. 139 (2000), no. 2, 439–455 (French, with English summary). MR 1738446, https://doi.org/10.1007/s002220050012
18. -, Correspondance de Langlands et fonctions des carrés extérieur et symétrique, preprint.
19. K. Hiraga and H. Saito, On -packets for inner forms of $\mathrm{SL}_n$ , preprint.
20. A. Ichino and T. Ikeda, On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture, preprint.
21. Takuya Konno, Twisted endoscopy and the generic packet conjecture, Israel J. Math. 129 (2002), 253–289. MR 1910946, https://doi.org/10.1007/BF02773167
22. Robert E. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), no. 3, 611–650. MR 757954, https://doi.org/10.1215/S0012-7094-84-05129-9
23. Robert E. Kottwitz, Stable trace formula: elliptic singular terms, Math. Ann. 275 (1986), no. 3, 365–399. MR 858284, https://doi.org/10.1007/BF01458611
24. Robert E. Kottwitz, Tamagawa numbers, Ann. of Math. (2) 127 (1988), no. 3, 629–646. MR 942522, https://doi.org/10.2307/2007007
25. Robert E. Kottwitz and Diana Shelstad, Foundations of twisted endoscopy, Astérisque 255 (1999), vi+190 (English, with English and French summaries). MR 1687096
26. Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin-New York, 1976. MR 0579181
27. I. G. Macdonald, The volume of a compact Lie group, Invent. Math. 56 (1980), no. 2, 93–95. MR 558859, https://doi.org/10.1007/BF01392542
28. R. Ranga Rao, Orbital integrals in reductive groups, Ann. of Math. (2) 96 (1972), 505–510. MR 0320232, https://doi.org/10.2307/1970822
29. Mark Reeder, Formal degrees and 𝐿-packets of unipotent discrete series representations of exceptional 𝑝-adic groups, J. Reine Angew. Math. 520 (2000), 37–93. With an appendix by Frank Lübeck. MR 1748271, https://doi.org/10.1515/crll.2000.023
30. Jonathan D. Rogawski, An application of the building to orbital integrals, Compositio Math. 42 (1980/81), no. 3, 417–423. MR 607380
31. Jonathan D. Rogawski, Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies, vol. 123, Princeton University Press, Princeton, NJ, 1990. MR 1081540
32. Freydoon Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for 𝑝-adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273–330. MR 1070599, https://doi.org/10.2307/1971524
33. Freydoon Shahidi, Twisted endoscopy and reducibility of induced representations for 𝑝-adic groups, Duke Math. J. 66 (1992), no. 1, 1–41. MR 1159430, https://doi.org/10.1215/S0012-7094-92-06601-4
34. Freydoon Shahidi, Poles of intertwining operators via endoscopy: the connection with prehomogeneous vector spaces, Compositio Math. 120 (2000), no. 3, 291–325. With an appendix by Diana Shelstad. MR 1748913, https://doi.org/10.1023/A:1002038928169
35. A. J. Silberger and E.-W. Zink, The formal degree of discrete series representations of central simple algebras over -adic fields, Max-Planck-Institut für Mathematik, 1996.
36. J.-L. Waldspurger, La formule de Plancherel pour les groupes 𝑝-adiques (d’après Harish-Chandra), J. Inst. Math. Jussieu 2 (2003), no. 2, 235–333 (French, with French summary). MR 1989693, https://doi.org/10.1017/S1474748003000082
37. E.-W. Zink, Comparison of $\mathrm{GL}_N$ and division algebra representations II, Max-Planck-Institut für Mathematik, 1993.