In mathematics, the Langlands decomposition writes a parabolic subgroup P of a semisimple Lie group as a product P = M A N {\displaystyle P=MAN} of a reductive subgroup M, an abelian subgroup A, and a nilpotent subgroup N.
Applications
A key application is in parabolic induction, which leads to the Langlands program: if G {\displaystyle G} is a reductive algebraic group and P = M A N {\displaystyle P=MAN} is the Langlands decomposition of a parabolic subgroup P, then parabolic induction consists of taking a representation of M A {\displaystyle MA} , extending it to P {\displaystyle P} by letting N {\displaystyle N} act trivially, and inducing the result from P {\displaystyle P} to G {\displaystyle G} .
See also Lie group decompositions
References
Sources A. W. Knapp, Structure theory of semisimple Lie groups. ISBN 0-8218-0609-2.