Langlands decomposition

In mathematics, the Langlands decomposition writes a parabolic subgroup P of a semisimple Lie group as a product $$P=MAN$$ 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 $$G$$ is a reductive algebraic group and $$P=MAN$$ is the Langlands decomposition of a parabolic subgroup P, then parabolic induction consists of taking a representation of $$MA$$, extending it to $$P$$ by letting $$N$$ act trivially, and inducing the result from $$P$$ to $$G$$.