Journal of the American Mathematical Society

Author(s): Kaoru Hiraga; Atsushi Ichino; Tamotsu Ikeda
Journal: J. Amer. Math. Soc. 21 (2008), 283-304.
MSC (2000): Primary 22E50
Posted: June 5, 2007
Errata: J. Amer. Math. Soc. 21 (2008), 1211-1213.
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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.: J. Arthur, The local behaviour of weighted orbital integrals, Duke Math. J. 56 (1988), 223-293. MR 932848 (89h:22036)
2.: -, Unipotent automorphic representations: conjectures, Astérisque 171-172 (1989), 13-71. MR 1021499 (91f:22030)
3.: A.-M. Aubert and R. Plymen, Plancherel measure for $\mathrm{GL}(n,F)$ and $\mathrm{GL}(m,D)$ : explicit formulas and Bernstein decomposition, J. Number Theory 112 (2005), 26-66. MR 2131140 (2006d:22018)
4.: A. 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 (56:3196)
5.: L. Clozel, Characters of nonconnected, reductive -adic groups, Canad. J. Math. 39 (1987), 149-167. MR 889110 (88i:22039)
6.: -, The fundamental lemma for stable base change, Duke Math. J. 61 (1990), 255-302. MR 1068388 (91h:22036)
7.: -, Invariant harmonic analysis on the Schwartz space of a reductive -adic group, Harmonic analysis on reductive groups, Progr. Math. 101, Birkhäuser Boston, 1991, pp. 101-121. MR 1168480 (93h:22020)
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 -adiques, Representations of reductive groups over a local field, Travaux en Cours, Hermann, 1984, pp. 33-117. MR 771672 (86h:11044)
10.: D. Goldberg, Some results on reducibility for unitary groups and local Asai -functions, J. Reine Angew. Math. 448 (1994), 65-95. MR 1266747 (95g:22031)
11.: B. H. Gross, On the motive of a reductive group, Invent. Math. 130 (1997), 287-313. MR 1474159 (98m:20060)
12.: -, On the motive of and the principal homomorphism $\mathrm{SL}_2 \rightarrow \hat{G}$ , Asian J. Math. 1 (1997), 208-213. MR 1480995 (99d:20077)
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 (53:3201)
14.: -, Harmonic analysis on real reductive groups. III. The Maass-Selberg relations and the Plancherel formula, Ann. of Math. 104 (1976), 117-201. MR 0439994 (55:12875)
15.: -, Admissible invariant distributions on reductive -adic groups, University Lecture Series 16, American Mathematical Society, 1999. MR 1702257 (2001b:22015)
16.: M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies 151, Princeton University Press, 2001. MR 1876802 (2002m:11050)
17.: G. Henniart, Une preuve simple des conjectures de Langlands pour $\mathrm{GL}(n)$ sur un corps -adique, Invent. Math. 139 (2000), 439-455. MR 1738446 (2001e:11052)
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.: T. Konno, Twisted endoscopy and the generic packet conjecture, Israel J. Math. 129 (2002), 253-289. MR 1910946 (2004d:22008)
22.: R. E. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), 611-650. MR 757954 (85m:11080)
23.: -, Stable trace formula: elliptic singular terms, Math. Ann. 275 (1986), 365-399. MR 858284 (88d:22027)
24.: -, Tamagawa numbers, Ann. of Math. 127 (1988), 629-646. MR 942522 (90e:11075)
25.: R. E. Kottwitz and D. Shelstad, Foundations of twisted endoscopy, Astérisque 255 (1999). MR 1687096 (2000k:22024)
26.: R. P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics 544, Springer-Verlag, 1976. MR 0579181 (58:28319)
27.: I. G. Macdonald, The volume of a compact Lie group, Invent. Math. 56 (1980), 93-95. MR 558859 (81h:22018)
28.: R. Ranga Rao, Orbital integrals in reductive groups, Ann. of Math. 96 (1972), 505-510. MR 0320232 (47:8771)
29.: M. Reeder, Formal degrees and -packets of unipotent discrete series representations of exceptional -adic groups, J. Reine Angew. Math. 520 (2000), 37-93. MR 1748271 (2001k:22039)
30.: J. D. Rogawski, An application of the building to orbital integrals, Compositio Math. 42 (1980/81), 417-423. MR 607380 (83g:22011)
31.: -, Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies 123, Princeton University Press, 1990. MR 1081540 (91k:22037)
32.: F. Shahidi, A proof of Langlands' conjecture on Plancherel measures; complementary series for -adic groups, Ann. of Math. 132 (1990), 273-330. MR 1070599 (91m:11095)
33.: -, Twisted endoscopy and reducibility of induced representations for -adic groups, Duke Math. J. 66 (1992), 1-41. MR 1159430 (93b:22034)
34.: -, Poles of intertwining operators via endoscopy: the connection with prehomogeneous vector spaces, Compositio Math. 120 (2000), 291-325. MR 1748913 (2001i:11060)
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 -adiques (d'après Harish-Chandra), J. Inst. Math. Jussieu 2 (2003), 235-333. MR 1989693 (2004d:22009)
37.: E.-W. Zink, Comparison of $\mathrm{GL}_N$ and division algebra representations II, Max-Planck-Institut für Mathematik, 1993.

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 22E50

Kaoru Hiraga
Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan
Email: hiraga@math.kyoto-u.ac.jp

Atsushi Ichino
Affiliation: Department of Mathematics, Graduate School of Science, Osaka City University, Osaka 558-8585, Japan
Email: ichino@sci.osaka-cu.ac.jp

Tamotsu Ikeda
Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan
Email: ikeda@math.kyoto-u.ac.jp

DOI: 10.1090/S0894-0347-07-00567-X
PII: S 0894-0347(07)00567-X
Received by editor(s): July 5, 2006
Posted: June 5, 2007
Dedicated: Dedicated to Professor Hiroshi Saito on the occasion of his sixtieth birthday
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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ISSN: 1088-6834(e) ISSN: 0894-0347(p)

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