In
mathematics, endoscopic groups of
reductive algebraic group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
s were introduced by in his work on the
stable trace formula.
Roughly speaking, an endoscopic group ''H'' of ''G'' is a
quasi-split group In mathematics, a quasi-split group over a field is a reductive group with a Borel subgroup defined over the field. Simply connected quasi-split groups over a field correspond to actions of the absolute Galois group on a Dynkin diagram.
Examples
...
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
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*{{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
Automorphic forms
Langlands program