Tempered endoscopy for real groups III: Inversion of transfer and 𝐿-packet structure

Shelstad, D.

doi:10.1090/S1088-4165-08-00337-3

Tempered endoscopy for real groups III: Inversion of transfer and $L$-packet structure
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by D. Shelstad

Represent. Theory 12 (2008), 369-402

DOI: https://doi.org/10.1090/S1088-4165-08-00337-3

Published electronically: October 17, 2008

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Abstract:

This paper examines adjoint relations for spectral analogues of the geometric transfer factors of Langlands and Shelstad in the case of the tempered spectrum of a real reductive algebraic group where the complex points are connected. Each tempered irreducible character is then expanded explicitly in terms of endoscopic characters. The analysis is also reinterpreted in terms of structure on $L$-packets in the form conjectured recently in much greater generality by Arthur. A triviality result is proved for the Whittaker normalization of spectral transfer factors which simplifies the results for certain inner forms of a quasi-split group.

References

Arthur, J., Problems for real groups, Contemp. Math., vol. 472 (2008), 39–62.
James Arthur, A note on $L$-packets, Pure Appl. Math. Q. 2 (2006), no. 1, Special Issue: In honor of John H. Coates., 199–217. MR 2217572, DOI 10.4310/PAMQ.2006.v2.n1.a9
James Arthur, On the transfer of distributions: weighted orbital integrals, Duke Math. J. 99 (1999), no. 2, 209–283. MR 1708030, DOI 10.1215/S0012-7094-99-09909-X
Jeffrey Adams, Dan Barbasch, and David A. Vogan Jr., The Langlands classification and irreducible characters for real reductive groups, Progress in Mathematics, vol. 104, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1162533, DOI 10.1007/978-1-4612-0383-4
Jeffrey Adams and Joseph F. Johnson, Endoscopic groups and packets of nontempered representations, Compositio Math. 64 (1987), no. 3, 271–309. MR 918414
Jeffrey Adams and David A. Vogan Jr., $L$-groups, projective representations, and the Langlands classification, Amer. J. Math. 114 (1992), no. 1, 45–138. MR 1147719, DOI 10.2307/2374739
N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
Henri Carayol and A. W. Knapp, Limits of discrete series with infinitesimal character zero, Trans. Amer. Math. Soc. 359 (2007), no. 11, 5611–5651. MR 2327045, DOI 10.1090/S0002-9947-07-04306-1
Bertram Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978), no. 2, 101–184. MR 507800, DOI 10.1007/BF01390249
Robert E. Kottwitz, Stable trace formula: elliptic singular terms, Math. Ann. 275 (1986), no. 3, 365–399. MR 858284, DOI 10.1007/BF01458611
Robert E. Kottwitz, Sign changes in harmonic analysis on reductive groups, Trans. Amer. Math. Soc. 278 (1983), no. 1, 289–297. MR 697075, DOI 10.1090/S0002-9947-1983-0697075-6
Robert E. Kottwitz and Diana Shelstad, Foundations of twisted endoscopy, Astérisque 255 (1999), vi+190 (English, with English and French summaries). MR 1687096
A. W. Knapp and Gregg Zuckerman, Classification of irreducible tempered representations of semi-simple Lie groups, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2178–2180. MR 460545, DOI 10.1073/pnas.73.7.2178
R. P. Langlands, Stable conjugacy: definitions and lemmas, Canadian J. Math. 31 (1979), no. 4, 700–725. MR 540901, DOI 10.4153/CJM-1979-069-2
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 10.1090/surv/031/03
R. P. Langlands and D. Shelstad, On the definition of transfer factors, Math. Ann. 278 (1987), no. 1-4, 219–271. MR 909227, DOI 10.1007/BF01458070
Shelstad, D., Tempered endoscopy for real groups I: geometric transfer with canonical factors, Contemp. Math., vol. 472 (2008), 215–248.
Shelstad, D., Tempered endoscopy for real groups II: spectral transfer factors, in Automorphic Forms and the Langlands Program, International Press, 243–282 (in press).
D. Shelstad, $L$-indistinguishability for real groups, Math. Ann. 259 (1982), no. 3, 385–430. MR 661206, DOI 10.1007/BF01456950
D. Shelstad, Characters and inner forms of a quasi-split group over $\textbf {R}$, Compositio Math. 39 (1979), no. 1, 11–45. MR 539000
Shelstad, D., Some character relations for real reductive algebraic groups, thesis.
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, DOI 10.24033/asens.1359
Shelstad, D., Examples in endoscopy for real groups (notes for BIRS summer school, Aug. 08), preprint.
Birgit Speh and David A. Vogan Jr., Reducibility of generalized principal series representations, Acta Math. 145 (1980), no. 3-4, 227–299. MR 590291, DOI 10.1007/BF02414191
David A. Vogan Jr., Gel′fand-Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978), no. 1, 75–98. MR 506503, DOI 10.1007/BF01390063