Parabolic transfer for real groups

Arthur, James

doi:10.1090/S0894-0347-07-00574-7

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
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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.

[A1]A1 J. Arthur, Harmonic analysis of the Schwartz space on a reductive Lie group, Parts I and II, mimeographed notes.

James Arthur, The characters of discrete series as orbital integrals, Invent. Math. 32 (1976), no. 3, 205–261. MR 412348, DOI https://doi.org/10.1007/BF01425569
James Arthur, The local behaviour of weighted orbital integrals, Duke Math. J. 56 (1988), no. 2, 223–293. MR 932848, DOI https://doi.org/10.1215/S0012-7094-88-05612-8
James Arthur, The invariant trace formula. I. Local theory, J. Amer. Math. Soc. 1 (1988), no. 2, 323–383. MR 928262, DOI https://doi.org/10.1090/S0894-0347-1988-0928262-5
James Arthur, Intertwining operators and residues. I. Weighted characters, J. Funct. Anal. 84 (1989), no. 1, 19–84. MR 999488, DOI https://doi.org/10.1016/0022-1236%2889%2990110-9
James Arthur, On elliptic tempered characters, Acta Math. 171 (1993), no. 1, 73–138. MR 1237898, DOI https://doi.org/10.1007/BF02392767
James Arthur, On the Fourier transforms of weighted orbital integrals, J. Reine Angew. Math. 452 (1994), 163–217. MR 1282200, DOI https://doi.org/10.1515/crll.1994.452.163
James Arthur, The trace Paley Wiener theorem for Schwartz functions, Representation theory and analysis on homogeneous spaces (New Brunswick, NJ, 1993) Contemp. Math., vol. 177, Amer. Math. Soc., Providence, RI, 1994, pp. 171–180. MR 1303605, DOI https://doi.org/10.1090/conm/177/01916
James Arthur, Canonical normalization of weighted characters and a transfer conjecture, C. R. Math. Acad. Sci. Soc. R. Can. 20 (1998), no. 2, 33–52. MR 1623485
James Arthur, Endoscopic $L$-functions and a combinatorial identity, Canad. J. Math. 51 (1999), no. 6, 1135–1148. Dedicated to H. S. M. Coxeter on the occasion of his 90th birthday. MR 1756875, DOI https://doi.org/10.4153/CJM-1999-050-x
James Arthur, On the transfer of distributions: weighted orbital integrals, Duke Math. J. 99 (1999), no. 2, 209–283. MR 1708030, DOI https://doi.org/10.1215/S0012-7094-99-09909-X
James Arthur, Stabilization of a family of differential equations, The mathematical legacy of Harish-Chandra (Baltimore, MD, 1998) Proc. Sympos. Pure Math., vol. 68, Amer. Math. Soc., Providence, RI, 2000, pp. 77–95. MR 1767893, DOI https://doi.org/10.1090/pspum/068/1767893
James Arthur, A stable trace formula. III. Proof of the main theorems, Ann. of Math. (2) 158 (2003), no. 3, 769–873. MR 2031854, DOI https://doi.org/10.4007/annals.2003.158.769
James Arthur, An asymptotic formula for real groups, J. Reine Angew. Math. 601 (2006), 163–230. MR 2289209, DOI https://doi.org/10.1515/CRELLE.2006.099

[A15]A15 ---, Singular invariant distributions and endoscopy, in preparation.

[A16]A16 ---, On the transfer of distributions: singular orbital integrals, in preparation.

James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton, NJ, 1989. MR 1007299
Laurent Clozel and Patrick Delorme, Le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs. II, Ann. Sci. École Norm. Sup. (4) 23 (1990), no. 2, 193–228 (French). MR 1046496
Harish-Chandra, Invariant differential operators and distributions on a semisimple Lie algebra, Amer. J. Math. 86 (1964), 534–564. MR 180628, DOI https://doi.org/10.2307/2373023
Harish-Chandra, Harmonic analysis on real reductive groups. I. The theory of the constant term, J. Functional Analysis 19 (1975), 104–204. MR 0399356, DOI https://doi.org/10.1016/0022-1236%2875%2990034-8
Robert E. Kottwitz, Rational conjugacy classes in reductive groups, Duke Math. J. 49 (1982), no. 4, 785–806. MR 683003
Robert E. Kottwitz, Stable trace formula: elliptic singular terms, Math. Ann. 275 (1986), no. 3, 365–399. MR 858284, DOI https://doi.org/10.1007/BF01458611
Robert E. Kottwitz and Diana Shelstad, Foundations of twisted endoscopy, Astérisque 255 (1999), vi+190 (English, with English and French summaries). MR 1687096
R. P. Langlands, Stable conjugacy: definitions and lemmas, Canadian J. Math. 31 (1979), no. 4, 700–725. MR 540901, DOI https://doi.org/10.4153/CJM-1979-069-2

[L2]L2 ---, Cancellation of singularities at real places, notes from a lecture, Institute for Advanced Study, Princeton, N.J., 1984.

R. P. Langlands, On the classification of irreducible representations of real algebraic groups, Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101–170. MR 1011897, DOI https://doi.org/10.1090/surv/031/03
Robert P. Langlands, Beyond endoscopy, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 611–697. MR 2058622

[L5]L5 ---, Un nouveau point de repère dans la théorie des formes automorphes, to appear in Canad. Math. Bull.

R. P. Langlands and D. Shelstad, On the definition of transfer factors, Math. Ann. 278 (1987), no. 1-4, 219–271. MR 909227, DOI https://doi.org/10.1007/BF01458070
R. Langlands and D. Shelstad, Descent for transfer factors, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 485–563. MR 1106907
D. Shelstad, Characters and inner forms of a quasi-split group over ${\bf R}$, Compositio Math. 39 (1979), no. 1, 11–45. MR 539000
Diana Shelstad, Orbital integrals and a family of groups attached to a real reductive group, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 1, 1–31. MR 532374
D. Shelstad, $L$-indistinguishability for real groups, Math. Ann. 259 (1982), no. 3, 385–430. MR 661206, DOI https://doi.org/10.1007/BF01456950