# Langlands decomposition

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