In mathematics, endoscopic groups of reductive algebraic groups were introduced by Robert Langlands (1979, 1983) in his work on the stable trace formula.

Roughly speaking, an endoscopic group H of G is a quasi-split group whose L-group is the connected component of the centralizer of a semisimple element of the L-group of G.

In the stable trace formula, unstable orbital integrals on a group G correspond to stable orbital integrals on its endoscopic groups H. The relation between them is given by the fundamental lemma.

References

• Arthur, James G. (2012). The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups (PDF).
• Arthur, James G. (2012). Endoscopy bending sections.
• Kottwitz, Robert E.; Shelstad, Diana (1999), "Foundations of twisted endoscopy", Astérisque (255): vi+190, ISSN 0303-1179, MR 1687096
• Labesse, Jean-Pierre (2008), "Introduction to endoscopy" (PDF), in Arthur, James; Schmid, Wilfried; Trapa, Peter E. (eds.), Representation theory of real reductive Lie groups, Contemp. Math., vol. 472, Providence, R.I.: American Mathematical Society, pp. 175–213, ISBN 978-0-8218-4366-6, MR 2454335
• Langlands, Robert P. (1979), "Stable conjugacy: definitions and lemmas", Canadian Journal of Mathematics, 31 (4): 700–725, CiteSeerX 10.1.1.207.4042, doi:10.4153/CJM-1979-069-2, ISSN 0008-414X, MR 0540901
• Langlands, Robert P. (1983), Les débuts d'une formule des traces stable, Publications Mathématiques de l'Université Paris VII [Mathematical Publications of the University of Paris VII], vol. 13, Paris: Université de Paris VII U.E.R. de Mathématiques, MR 0697567
• Langlands, Robert P.; Shelstad, D. (1987), "On the definition of transfer factors", Mathematische Annalen, 278 (1): 219–271, doi:10.1007/BF01458070, ISSN 0025-5831, MR 0909227
• Langlands, Robert P. (2001), "The trace formula and its applications: an introduction to the work of James Arthur", Canadian Mathematical Bulletin, 44 (2): 160–209, doi:10.4153/CMB-2001-020-8, ISSN 0008-4395, MR 1827854
• Langlands, Robert P. (2004), "Beyond endoscopy" (PDF), in Hida, Haruzo; Ramakrishnan, Dinakar; Shahidi, Freydoon (eds.), Contributions to automorphic forms, geometry, and number theory, Baltimore, MD: Johns Hopkins Univ. Press, pp. 611–697, ISBN 978-0-8018-7860-2, MR 2058622
• Shelstad, Diana (1983), "Orbital integrals, endoscopic groups and L-indistinguishability for real groups", Conference on automorphic theory (Dijon, 1981), Publ. Math. Univ. Paris VII, vol. 15, Paris: Univ. Paris VII, pp. 135–219, MR 0723184