Mok, Chung Pang; Peng, Zhifeng

The Spectral Side of Stable Local Trace Formula

Doc. Math. 24, 303-329 (2019)
DOI: 10.25537/dm.2019v24.303-329
Communicated by Don Blasius

Summary

Let \(G\) be a connected quasi-split reductive group over \(\mathbb{R}\), and more generally, a quasi-split \(K\)-group over \(\mathbb{R}\). Arthur had obtained the formal formula for the spectral side of the stable local trace formula, by using formal substitute of Langlands parameters. In this paper, we construct the spectral side of the stable local trace formula and endoscopic local trace formula directly for quasi-split \(K\)-groups over \(\mathbb{R}\), by incorporating the works of Shelstad. In particular we give the explicit expression for the spectral side of the stable local trace formula, in terms of Langlands parameters.

Mathematics Subject Classification

22E45, 22E46

Keywords/Phrases

stabilization, endoscopy, local trace formula, transfer factors

References

  • 1. J. Arthur, Unipotent automorphic representations: Global motivation, in Automorphic Forms, Shimura Varieties, and L-functions. Academic Press, 1990, Vol. I, 1-75. zbl 0692.10027; MR1044818.
  • 2. J. Arthur, On elliptic tempered characters. Acta Math. no. 171 (1993): 73-138. DOI 10.1007/BF02392767; zbl 0822.22011; MR1237898.
  • 3. J. Arthur, On local character relations. Selecta Mathematica (1996): 501-579. DOI 10.1007/BF02433450; zbl 0923.11081; MR1443184.
  • 4. J. Arthur, On the transfer of distributions: weighted orbital integrals. Duke Math. J. no. 99 (1999): 209-283. DOI 10.1215/S0012-7094-99-09909-X; zbl 0938.22019; MR1708030.
  • 5. J. Arthur, A stable Trace Formula I. General expansions. J. Inst. of Math. Jussieu 1(2) (2002): 175-277. DOI 10.1017/S1474748002000051; zbl 1040.11038; MR1954821.
  • 6. J. Arthur, A stable Trace Formula III. Proof of the Main Theorems. Ann. of Math. no. 158 (2003): 769-873. DOI 10.4007/annals.2003.158.769; zbl 1051.11027; MR2031854.
  • 7. J. Arthur, A Note on $L$-packets. Pure and Applied Mathematics Quarterly Vol. 2, no. 1 (2006): 199-217. DOI 10.4310/PAMQ.2006.v2.n1.a9; zbl 1158.22017; MR2217572.
  • 8. J. Arthur, Parabolic transfer for real groups. Journal of the American Mathematical Society, vol. 21, no. 1 (2008): 171-234. DOI 10.1090/S0894-0347-07-00574-7; zbl 1131.22004; MR2350054.
  • 9. J. Arthur, The endoscopic classification of representations: Orthogonal and symplectic groups. Colloquium Publications, Vol. 61, 2013. zbl 1310.22014; MR3135650.
  • 10. A.W. Knapp; G. Zuckerman, Classification of irreducible tempered representations of semi-simple Lie groups. Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2178-2180. DOI 10.1073/pnas.73.7.2178; zbl 0329.22013; MR0460545.
  • 11. C.P. Mok, Endoscopic classification of representation of quasi-split unitary groups. Memoirs of the American Mathematical Society, Vol. 235 (2015), no. 1108. MR3338302; arxiv 1206.0882.
  • 12. Zhifeng Peng, Multiplicity formula and stable trace formula. Available at arXiv:1608.00055. To appear in the American Journal of Mathematics.
  • 13. D. Shelstad, $L$-indistinguishability for real groups. Math. Ann. no. 259 (1982): 385-430. DOI 10.1007/BF01456950; zbl 0506.22014; MR0661206.
  • 14. D. Shelstad, Tempered endoscopy for real groups II: spectral transfer factors. International Press, 2008, 243-282. zbl 1157.22008; MR2581952.
  • 15. D. Shelstad, Tempered endoscopy for real groups III: Inversion of transfer and $L$-packet structure. Represention Theory, no. 12 (2008): 369-402. DOI 10.1090/S1088-4165-08-00337-3; zbl 1159.22007; MR2448289.

Affiliation

Mok, Chung Pang
Institute of Mathematics, Academia Sinica, Taipei
Peng, Zhifeng
Department of Mathematics, National University of Singapore, Singapore

Downloads