endoscopic group
{{For|medical procedure used to evaluate the upper gastrointestinal tract|Endoscopy}}
{{Short description|Mathematical group}}
In mathematics, endoscopic groups of reductive algebraic groups were introduced by {{harvs|txt|first=Robert |last=Langlands|authorlink=Robert Langlands|year1=1979|year2=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
- {{cite book |last=Arthur |first=James G. |url=http://www.claymath.org/library/cw/arthur/pdf/arthur-endoscopic-tifr.pdf |title=The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups |year=2012}}
- {{cite book |last=Arthur |first=James G. |url=https://smarttechmed.com/endoscopy-bending-sections/|title=Endoscopy bending sections|year=2012}}
- {{Citation | last1=Kottwitz | first1=Robert E. | last2=Shelstad | first2=Diana | title=Foundations of twisted endoscopy | url=https://books.google.com/books?id=NYxAtwAACAAJ | mr=1687096 | year=1999 | journal=Astérisque | issn=0303-1179 | issue=255 | pages=vi+190}}
- {{Citation | last1=Labesse | first1=Jean-Pierre | editor1-last=Arthur | editor1-first=James | editor2-last=Schmid | editor2-first=Wilfried | editor3-last=Trapa | editor3-first=Peter E. | title=Representation theory of real reductive Lie groups | url=http://www.math.utah.edu/~ptrapa/src2006/ | publisher=American Mathematical Society | location=Providence, R.I. | series=Contemp. Math. | isbn=978-0-8218-4366-6 | mr=2454335 | year=2008 | volume=472 | chapter=Introduction to endoscopy | chapter-url=http://www.math.utah.edu/~ptrapa/src2006/labesse.pdf | pages=175–213}}
- {{Citation | last1=Langlands | first1=Robert P. | title=Stable conjugacy: definitions and lemmas | doi=10.4153/CJM-1979-069-2 | doi-access=free | mr=540901 | year=1979 | journal=Canadian Journal of Mathematics | issn=0008-414X | volume=31 | issue=4 | pages=700–725| citeseerx=10.1.1.207.4042 }}
- {{Citation | last1=Langlands | first1=Robert P. | title=Les débuts d'une formule des traces stable | url=http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/endoscopy.html#debuts | publisher=Université de Paris VII U.E.R. de Mathématiques | location=Paris | series=Publications Mathématiques de l'Université Paris VII [Mathematical Publications of the University of Paris VII] | mr=697567 | year=1983 | volume=13}}
- {{Citation | last1=Langlands | first1=Robert P. | last2=Shelstad | first2=D. | title=On the definition of transfer factors | doi=10.1007/BF01458070 | doi-access= | mr=909227 | year=1987 | journal=Mathematische Annalen | issn=0025-5831 | volume=278 | issue=1 | pages=219–271| s2cid=14141632 }}
- {{Citation | last1=Langlands | first1=Robert P. | title=The trace formula and its applications: an introduction to the work of James Arthur | doi=10.4153/CMB-2001-020-8 | doi-access=free | mr=1827854 | year=2001 | journal=Canadian Mathematical Bulletin | issn=0008-4395 | volume=44 | issue=2 | pages=160–209}}
- {{Citation | last1=Langlands | first1=Robert P. | editor1-last=Hida | editor1-first=Haruzo | editor2-last=Ramakrishnan | editor2-first=Dinakar | editor3-last=Shahidi | editor3-first=Freydoon | title=Contributions to automorphic forms, geometry, and number theory | url=http://muse.jhu.edu/journals/american_journal_of_mathematics/info/supplement.html | publisher=Johns Hopkins Univ. Press | location=Baltimore, MD | isbn=978-0-8018-7860-2 | mr=2058622 | year=2004 | chapter=Beyond endoscopy | chapter-url=http://muse.jhu.edu/journals/american_journal_of_mathematics/info/docs/hida_pdfs/hida22.pdf | pages=611–697}}
- {{Citation | last1=Shelstad | first1=Diana | title=Conference on automorphic theory (Dijon, 1981) | publisher=Univ. Paris VII | location=Paris | series=Publ. Math. Univ. Paris VII | mr=723184 | year=1983 | volume=15 | chapter=Orbital integrals, endoscopic groups and L-indistinguishability for real groups | pages=135–219}}