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 direc ...

s were introduced by in his work on the

stable trace formula A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...

. 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

, unstable

orbital integral In mathematics, an orbital integral is an integral transform that generalizes the spherical mean operator to homogeneous spaces. Instead of integrating over spheres, one integrates over generalized spheres: for a homogeneous space ''X'' =&n ...

s 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

* * * * * * * * * *{{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