Langlands decomposition

In mathematics, the Langlands decomposition writes a parabolic subgroup P of a semisimple Lie group as a product

of a reductive subgroup M, an abelian subgroup A, and a nilpotent subgroup N. A key application is in parabolic induction, which leads to the Langlands program: if

is a reductive algebraic group and

is the Langlands decomposition of a parabolic subgroup P, then parabolic induction consists of taking a representation of

act trivially, and inducing the result from