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
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by Kaoru Hiraga, Atsushi Ichino and Tamotsu Ikeda;

J. Amer. Math. Soc. 21 (2008), 283-304

DOI: https://doi.org/10.1090/S0894-0347-07-00567-X

Published electronically: June 5, 2007

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Erratum: J. Amer. Math. Soc. 21 (2008), 1211-1213.

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.

References

James Arthur, The local behaviour of weighted orbital integrals, Duke Math. J. 56 (1988), no. 2, 223–293. MR 932848, DOI 10.1215/S0012-7094-88-05612-8
James Arthur, Unipotent automorphic representations: conjectures, Astérisque 171-172 (1989), 13–71. Orbites unipotentes et représentations, II. MR 1021499
Anne-Marie Aubert and Roger Plymen, Plancherel measure for $\textrm {GL}(n,F)$ and $\textrm {GL}(m,D)$: explicit formulas and Bernstein decomposition, J. Number Theory 112 (2005), no. 1, 26–66. MR 2131140, DOI 10.1016/j.jnt.2005.01.005
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 444849, DOI 10.1007/BF01390139
Laurent Clozel, Characters of nonconnected, reductive $p$-adic groups, Canad. J. Math. 39 (1987), no. 1, 149–167. MR 889110, DOI 10.4153/CJM-1987-008-3
Laurent Clozel, The fundamental lemma for stable base change, Duke Math. J. 61 (1990), no. 1, 255–302. MR 1068388, DOI 10.1215/S0012-7094-90-06112-5
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, DOI 10.1007/978-1-4612-0455-8_{6}

Depth-zero supercuspidal $L$-packets and their stability

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
David Goldberg, Some results on reducibility for unitary groups and local Asai $L$-functions, J. Reine Angew. Math. 448 (1994), 65–95. MR 1266747, DOI 10.1515/crll.1994.448.65
Benedict H. Gross, On the motive of a reductive group, Invent. Math. 130 (1997), no. 2, 287–313. MR 1474159, DOI 10.1007/s002220050186
Benedict H. Gross, On the motive of $G$ and the principal homomorphism $\textrm {SL}_2\to \widehat G$, Asian J. Math. 1 (1997), no. 1, 208–213. MR 1480995, DOI 10.4310/AJM.1997.v1.n1.a8
Harish-Chandra, Harmonic analysis on real reductive groups. I. The theory of the constant term, J. Functional Analysis 19 (1975), 104–204. MR 399356, DOI 10.1016/0022-1236(75)90034-8
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 439994, DOI 10.2307/1971058
Harish-Chandra, Admissible invariant distributions on reductive $p$-adic groups, University Lecture Series, vol. 16, American Mathematical Society, Providence, RI, 1999. With a preface and notes by Stephen DeBacker and Paul J. Sally, Jr. MR 1702257, DOI 10.1090/ulect/016
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
Guy Henniart, Une preuve simple des conjectures de Langlands pour $\textrm {GL}(n)$ sur un corps $p$-adique, Invent. Math. 139 (2000), no. 2, 439–455 (French, with English summary). MR 1738446, DOI 10.1007/s002220050012

Correspondance de Langlands et fonctions $L$ des carrés extérieur et symétrique

On $L$-packets for inner forms of $\mathrm {SL}_n$

On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture

Takuya Konno, Twisted endoscopy and the generic packet conjecture, Israel J. Math. 129 (2002), 253–289. MR 1910946, DOI 10.1007/BF02773167
Robert E. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), no. 3, 611–650. MR 757954, DOI 10.1215/S0012-7094-84-05129-9
Robert E. Kottwitz, Stable trace formula: elliptic singular terms, Math. Ann. 275 (1986), no. 3, 365–399. MR 858284, DOI 10.1007/BF01458611
Robert E. Kottwitz, Tamagawa numbers, Ann. of Math. (2) 127 (1988), no. 3, 629–646. MR 942522, DOI 10.2307/2007007
Robert E. Kottwitz and Diana Shelstad, Foundations of twisted endoscopy, Astérisque 255 (1999), vi+190 (English, with English and French summaries). MR 1687096
Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin-New York, 1976. MR 579181, DOI 10.1007/BFb0079929
I. G. Macdonald, The volume of a compact Lie group, Invent. Math. 56 (1980), no. 2, 93–95. MR 558859, DOI 10.1007/BF01392542
R. Ranga Rao, Orbital integrals in reductive groups, Ann. of Math. (2) 96 (1972), 505–510. MR 320232, DOI 10.2307/1970822
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, DOI 10.1515/crll.2000.023
Jonathan D. Rogawski, An application of the building to orbital integrals, Compositio Math. 42 (1980/81), no. 3, 417–423. MR 607380
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, DOI 10.1515/9781400882441
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, DOI 10.2307/1971524
Freydoon Shahidi, Twisted endoscopy and reducibility of induced representations for $p$-adic groups, Duke Math. J. 66 (1992), no. 1, 1–41. MR 1159430, DOI 10.1215/S0012-7094-92-06601-4
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, DOI 10.1023/A:1002038928169

The formal degree of discrete series representations of central simple algebras over $p$-adic fields

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, DOI 10.1017/S1474748003000082

Comparison of $\mathrm {GL}_N$ and division algebra representations II