Abstract
Langlands program is the central concern of modern number theory. In general, we think that the main tool for studying this program is the trace formula. Arthur-Selberg’s trace formula is successful in solving the endoscopic case of Langlands program. Arthur stabilized the geometric side of the local trace formula in the general case, but did not stabilize the general spectral. This paper aims to solve the general case by different method in the Archimedean case, by directly stabilizing the spectral side of the local trace formula, and obtains the stable local trace formula.
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Arthur J. The invariant trace formula, I: Local theory. J Amer Math Soc, 1988, 1: 323–383
Arthur J. A local trace formula. Pub Math IHES, 1991, 73: 5–96
Arthur J. On elliptic tempered characters. Acta Math, 1993, 171: 73–138
Arthur J. On local character relations. Selecta Math, 1996, 2: 501–579
Arthur J. Endoscopic L-functions and a combinatorial identity. Canad J Math, 1999, 51: 1135–1148
Arthur J. On the transfer of distributions: Weighted orbital integrals. Duke Math J, 1999, 99: 209–283
Arthur J. A stable trace formula, I: General expansions. J Inst Math Jussieu, 2002, 1: 175–277
Arthur J. A stable trace formula, III: Proof of the main theorems. Ann Math, 2003, 158: 769–873
Arthur J. Parabolic transfer for real groups. J Amer Math Soc, 2008, 21: 171–234
Arthur J. The Endoscopic Classification of Representations. Orthogonal and Symplectic Groups. Providence, RI: Amer Math Soc, 2013
Harish-Chandra. Harmonic analysis on real reductive groups, I: The theory of the constant term. J Funct Anal, 1975, 19: 104–204
Harish-Chandra. Harmonic analysis on real reductive groups, II: Wave packets in the Schwartz space. Invent Math, 1976, 36: 1–55
Harish-Chandra. Harmonic analysis on real reductive groups, III: The Maass-Selberg relations and the Plancherel formula. Ann Math, 1976, 104: 117–201
Kottwitz R E. Stable trace formula: Elliptic singular terms. Math Ann, 1986, 275: 365–399
Langlands R P. Stable conjugacy: Definitions and lemmas. Canad J Math, 1979, 31: 700–725
Langlands R P. On the classification of irreducible representations of real algebraic groups. In: Math Surveys Monogr, vol. 31. Providence, RI: Amer Math Soc, 1989, 101–170
Peng Z F. Multiplicity conjecture for discrete series and stable trace formula. Preprint, 2014
Shelstad D. L-indistinguishability for real groups. Math Ann, 1982, 259: 385–430
Shelstad D. Tempered endoscopy for real groups, III: Inversion of transfer and L-packet structure. Represent Theory, 2008, 12: 369–402
Shelstad D. Tempered endoscopy for real groups, II: Spectral transfer factors. In: Automorphic Forms and the Langlands Program, vol. 9. Cambridge: International Press, 2010, 236–276
Tadic M. On tempered and square integrable representations of classical p-adic groups. Sci China Math, 2013, 56: 2273–2313
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Peng, Z. Stable local trace formula. Sci. China Math. 57, 2509–2518 (2014). https://doi.org/10.1007/s11425-014-4899-7
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DOI: https://doi.org/10.1007/s11425-014-4899-7