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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.

##### MSC:
 2.2e+46 Representations of Lie and linear algebraic groups over real fields: analytic methods 2.2e+100 Lie groups
##### Keywords:
endoscopic transfer; real Lie groups; orbital integrals
Zbl 0958.22013
Full Text:
##### References:
 [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
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