Langlands decomposition

In mathematics, the Langlands decomposition writes a parabolic subgroup "P" of a semisimple Lie group as a product formula_1 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 formula_2 is a reductive algebraic group and formula_1 is the Langlands decomposition of a parabolic subgroup "P", then parabolic induction consists of taking a representation of formula_4, extending it to formula_5 by letting formula_6 act trivially, and inducing the result from formula_5 to formula_2.