zbMATH — the first resource for mathematics

On geometric transfer in real twisted endoscopy. (English) Zbl 1259.22007
Consider a connected reductive algebraic group \(G\) over \(\mathbb R\) with an automorphism \(\theta\) defined over \(\mathbb R\) and a character \(\omega\) of \(G({\mathbb R})\). The geometric transfer proved in this article is an equality between weighted sums of \((\theta, \omega)\)-twisted orbital integrals on \(G({\mathbb R})\) and stable orbital integrals on an endoscopic group for \((G,\theta,\omega)\). Essential for the proof is the study of the behaviour of stable orbital integrals (resp., of weighted sums of twisted orbital integrals) around a semisimple element with centralizer of type \(A_1\). The author makes use of the fact that the relative transfer factor, i.e., the quotient of two values of the transfer factor, is canonical (see [R. E. Kottwitz and D. Shelstad, Foundations of twisted endoscopy. Astérisque. 255. Paris: Société Mathématique de France (1999; Zbl 0958.22013)]). This simplifies the argument and gives at the same time a shorter proof for the known case of non-twisted endoscopy.

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22E99 Lie groups
Zbl 0958.22013
Full Text: DOI
[1] J. Arthur, The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, 2011. · Zbl 1310.22014
[2] J. Arthur, ”Problems for real groups,” in Representation Theory of Real Reductive Lie Groups, Providence, RI: Amer. Math. Soc., 2008, vol. 472, pp. 39-62. · Zbl 1159.22003
[3] A. Borel, ”Automorphic \(L\)-functions,” in Automorphic Forms, Representations and \(L\)-Functions, Providence, R.I.: Amer. Math. Soc., 1979, vol. XXXIII, pp. 27-61. · Zbl 0412.10017
[4] A. Bouaziz, ”Sur les caractères des groupes de Lie réductifs non connexes,” J. Funct. Anal., vol. 70, iss. 1, pp. 1-79, 1987. · Zbl 0622.22009
[5] A. Bouaziz, ”Intégrales orbitales sur les groupes de Lie réductifs,” Ann. Sci. École Norm. Sup., vol. 27, iss. 5, pp. 573-609, 1994. · Zbl 0832.22017
[6] L. Clozel and P. Delorme, ”Le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs,” Invent. Math., vol. 77, iss. 3, pp. 427-453, 1984. · Zbl 0584.22005
[7] Harish-Chandra, ”Harmonic analysis on real reductive groups. I. The theory of the constant term,” J. Functional Analysis, vol. 19, pp. 104-204, 1975. · Zbl 0315.43002
[8] T. Kaletha, ”Decomposition of splitting invariants in split real groups,” Canad. J. Math., vol. 63, iss. 5, pp. 1083-1106, 2011. · Zbl 1231.11053
[9] R. E. Kottwitz, ”Rational conjugacy classes in reductive groups,” Duke Math. J., vol. 49, iss. 4, pp. 785-806, 1982. · Zbl 0506.20017
[10] R. E. Kottwitz, ”Stable trace formula: elliptic singular terms,” Math. Ann., vol. 275, iss. 3, pp. 365-399, 1986. · Zbl 0577.10028
[11] R. E. Kottwitz and D. Shelstad, Foundations of Twisted Endoscopy, , 1999, vol. 255. · Zbl 0958.22013
[12] R. E. Kottwitz and D. Shelstad, On splitting invariants and sign conventions in endoscopic transfer, 2012.
[13] R. Langlands, ”Representations of abelian algebraic groups,” Pacific J. Math., iss. Special Issue, pp. 231-250, 1997. · Zbl 0910.11045
[14] R. Langlands and D. Shelstad, ”On the definition of transfer factors,” Math. Ann., vol. 278, iss. 1-4, pp. 219-271, 1987. · Zbl 0644.22005
[15] R. Langlands and D. Shelstad, ”Descent for transfer factors,” in The Grothendieck Festschrift, Vol. II, Boston, MA: Birkhäuser, 1990, pp. 485-563. · Zbl 0743.22009
[16] P. Mezo, ”Character Identities in the Twisted Endoscopy of Real Reductive Groups,” Mem. Amer. Math. Soc., p. x, 2012. · Zbl 1293.22004
[17] D. Renard, ”Intégrales orbitales tordues sur les groupes de Lie réductifs réels,” J. Funct. Anal., vol. 145, iss. 2, pp. 374-454, 1997. · Zbl 0877.22002
[18] D. Renard, ”Twisted endoscopy for real groups,” J. Inst. Math. Jussieu, vol. 2, iss. 4, pp. 529-566, 2003. · Zbl 1034.22011
[19] D. Shelstad, ”Tempered endoscopy for real groups. I. Geometric transfer with canonical factors,” in Representation Theory of Real Reductive Lie Groups, Providence, RI: Amer. Math. Soc., 2008, vol. 472, pp. 215-246. · Zbl 1157.22008
[20] D. Shelstad, ”Tempered endoscopy for real groups. II. Spectral transfer factors,” in Automorphic Forms and the Langlands Program, Int. Press, Somerville, MA, 2010, vol. 9, pp. 236-276. · Zbl 1198.22008
[21] D. Shelstad, ”Tempered endoscopy for real groups. III. Inversion of transfer and \(L\)-packet structure,” Represent. Theory, vol. 12, pp. 369-402, 2008. · Zbl 1159.22007
[22] D. Shelstad, Examples in endoscopy for real groups (notes for BIRS summer school, August 2008).
[23] D. Shelstad, ”Characters and inner forms of a quasi-split group over \({\mathbf R}\),” Compositio Math., vol. 39, iss. 1, pp. 11-45, 1979. · Zbl 0431.22011
[24] D. Shelstad, ”Orbital integrals, endoscopic groups and \(L\)-indistinguishability for real groups,” in Conference on Automorphic Theory, Paris: Univ. Paris VII, 1983, vol. 15, pp. 135-219. · Zbl 0529.22007
[25] D. Shelstad, ”Embeddings of \(L\)-groups,” Canad. J. Math., vol. 33, iss. 3, pp. 513-558, 1981. · Zbl 0457.22006
[26] D. Shelstad, ”\(L\)-indistinguishability for real groups,” Math. Ann., vol. 259, iss. 3, pp. 385-430, 1982. · Zbl 0506.22014
[27] D. Shelstad, On spectral transfer factors in real twisted endoscopy, 2011. · Zbl 1346.22006
[28] D. Shelstad, ”Orbital integrals and a family of groups attached to a real reductive group,” Ann. Sci. École Norm. Sup., vol. 12, iss. 1, pp. 1-31, 1979. · Zbl 0433.22006
[29] D. Shelstad, ”Base change and a matching theorem for real groups,” in Noncommutative Harmonic Analysis and Lie Groups, New York: Springer-Verlag, 1981, vol. 880, pp. 425-482. · Zbl 0494.22007
[30] D. Shelstad, ”Endoscopic groups and base change \({\mathbf C}/{\mathbf R}\),” Pacific J. Math., vol. 110, iss. 2, pp. 397-416, 1984. · Zbl 0488.22033
[31] V. S. Varadarajan, Harmonic Analysis on Reductive Lie Groups, New York: Springer-Verlag, 1977, vol. 576. · Zbl 0354.43001
[32] J-L. Waldspurger, ”L’endoscopie tordue n’est pas si tordue,” Mem. Amer. Math. Soc., vol. 194, iss. 908, p. x, 2008. · Zbl 1146.22016
[33] J-L. Waldspurger, Errata, 2009.
[34] N. R. Wallach, Real Reductive Groups. I, Boston, MA: Academic Press, 1988, vol. 132. · Zbl 0666.22002
[35] G. Warner, Harmonic Analysis on Semi-Simple Lie Groups. II, New York: Springer-Verlag, 1972, vol. 189. · Zbl 0265.22021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.