Endoscopic group

For medical procedure used to evaluate the upper gastrointestinal tract, see Endoscopy.

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, S2CID 14141632
• 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