Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Parabolic transfer for real groups


Author: James Arthur
Journal: J. Amer. Math. Soc. 21 (2008), 171-234
MSC (2000): Primary 22E30, 22E55
DOI: https://doi.org/10.1090/S0894-0347-07-00574-7
Published electronically: June 25, 2007
MathSciNet review: 2350054
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We shall establish an identity between distributions on different real reductive groups. The distributions arise from the trace formula. They represent the main archimedean terms in both the invariant and stable forms of the trace formula. The identity will be an essential part of the comparison of these formulas. As such, it is expected to lead to reciprocity laws among automorphic representations on different groups.

Our techniques are analytic. We shall show that the difference of the two sides of the proposed identity is the solution of a homogeous boundary value problem. More precisely, we shall show that it satisfies a system of linear differential equations, that it obeys certain boundary conditions around the singular set, and that it is asymptotic to zero. We shall then show that any such solution vanishes.


References [Enhancements On Off] (What's this?)

  • [A1] J. Arthur, Harmonic analysis of the Schwartz space on a reductive Lie group, Parts I and II, mimeographed notes.
  • [A2] -, The characters of discrete series as orbital integrals, Invent. Math. 32 (1976), 205-261. MR 0412348 (54:474)
  • [A3] -, The local behaviour of weighted orbital integrals, Duke Math. J. 56 (1988), 223-293. MR 932848 (89h:22036)
  • [A4] -, The invariant trace formula I. Local theory, J. Amer. Math. Soc. 1 (1988), 323-383. MR 928262 (89e:22029)
  • [A5] -, Intertwining operators and residues I. Weighted characters, J. Funct. Anal. 84 (1989), 19-84. MR 999488 (90j:22018)
  • [A6] -, On elliptic tempered characters, Acta Math. 171 (1993), 73-138. MR 1237898 (94i:22038)
  • [A7] -, On the Fourier transforms of weighted orbital integrals, J. Reine Agnew. Math. 452 (1994), 163-217. MR 1282200 (95h:22015)
  • [A8] -, The trace Paley-Wiener theorem for Schwartz functions, Contemp. Math. 177 (1994), 171-180. MR 1303605 (95k:22010)
  • [A9] -, Canonical normalization of weighted characters and a transfer conjecture, C.R. Math. Acad. Sci. Soc. R. Can. 20 (2), (1998), 33-52. MR 1623485 (99g:22020)
  • [A10] -, Endoscopic $ L$-functions and a combinatorial identity, Canad. J. Math. 51 (1999), 1135-1148. MR 1756875 (2001g:11071)
  • [A11] -, On the transfer of distributions: weighted orbital integrals, Duke Math. J. 99 (1999), 209-283. MR 1708030 (2000i:22023)
  • [A12] -, Stabilization of a family of differential equations, Proc. Sympos. Pure Math., vol. 68, 2000, 77-95. MR 1767893 (2001f:22025)
  • [A13] -, A stable trace formula III. Proof of the main theorems, Annals of Math. 158 (2003), 769-873. MR 2031854 (2004m:11079)
  • [A14] -, An asymptotic formula for real groups, J. Reine Angew. Math. 601 (2006), 163-230. MR 2289209
  • [A15] -, Singular invariant distributions and endoscopy, in preparation.
  • [A16] -, On the transfer of distributions: singular orbital integrals, in preparation.
  • [AC] J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Ann. of Math. Studies 120, Princeton Univ. Press, Princeton, N.J., 1989. MR 1007299 (90m:22041)
  • [CD] L. Clozel and P. Delorme, Le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs II, Ann. Scient. Éc. Norm. Sup. (4), 23 (1990), 193-228. MR 1046496 (91g:22013)
  • [H1] Harish-Chandra, Invariant eigendistributions on a semisimple Lie group, Trans. Amer. Math. Soc. 119, 457-508. MR 0180631 (31:4862d)
  • [H2] -, Harmonic analysis on real reductive groups, I. The theory of the constant term, J. Funct. Anal. 19, 104-204. MR 0399356 (53:3201)
  • [K1] R. Kottwitz, Rational conjugacy classes in reductive groups, Duke Math. J. 49 (1982), 785-806. MR 683003 (84k:20020)
  • [K2] -, Stable trace formula: elliptic singular terms, Math. Ann. 275 (1986), 365-399. MR 858284 (88d:22027)
  • [KS] R. Kottwitz and D. Shelstad, Foundations of Twisted Endoscopy, Astérisque, vol. 255. MR 1687096 (2000k:22024)
  • [L1] R. Langlands, Stable conjugacy: definitions and lemmas, Canad. J. Math. 31 (1979), 700-725. MR 540901 (82j:10054)
  • [L2] -, Cancellation of singularities at real places, notes from a lecture, Institute for Advanced Study, Princeton, N.J., 1984.
  • [L3] -, On the classification of irreducible representations of real algebraic groups, in Representation Theory and Harmonic Analysis on Semisimple Lie Groups, AMS Mathematical Surveys and Monographs, vol. 31, 1989, 101-170. MR 1011897 (91e:22017)
  • [L4] -, Beyond endoscopy, in Contributions to Automorphic Forms, Geometry, and Number Theory, Johns Hopkins University Press, 2004, 611-698. MR 2058622 (2005f:11102)
  • [L5] -, Un nouveau point de repère dans la théorie des formes automorphes, to appear in Canad. Math. Bull.
  • [LS1] R. Langlands and D. Shelstad, On the definition of transfer factors, Math. Ann. 278 (1987), 219-271. MR 909227 (89c:11172)
  • [LS2] -, Descent for transfer factors, The Grothendieck Festschift, Vol II, Birkhauser, Boston, 1990, 485-563. MR 1106907 (92i:22016)
  • [S1] D. Shelstad, Characters and inner forms of a quasisplit group over $ {\mathbb{R}}$, Compositio Math. 39 (1979), 11-45. MR 539000 (80m:22023)
  • [S2] -, Orbital integrals and a family of groups attached to a real reductive group, Ann. Scient. Éc. Norm. Sup. 12, (1979), 1-31. MR 532374 (81k:22014)
  • [S3] -, $ L$-indistinguishability for real groups, Math. Ann. 259 (1982), 385-430. MR 661206 (84c:22017)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 22E30, 22E55

Retrieve articles in all journals with MSC (2000): 22E30, 22E55


Additional Information

James Arthur
Affiliation: Department of Mathematics, University of Toronto, Bahen Centre, 6th Floor, 40 St George Street, Toronto, ON M5S 2E4 Canada

DOI: https://doi.org/10.1090/S0894-0347-07-00574-7
Received by editor(s): December 21, 2005
Published electronically: June 25, 2007
Additional Notes: The author was supported in part by NSERC Operating Grant A3483.
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society