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

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

 subgroup A, and a nilpotent

Nilpotent

In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....

 subgroup N.

Applications

A key application is in parabolic induction

Parabolic induction

In mathematics, parabolic induction is a method of constructing representations of a reductive group from representations of its parabolic subgroups....

, which leads to the Langlands program

Langlands program

The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....

: if is a reductive algebraic group and is the Langlands decomposition of a parabolic subgroup P, then parabolic induction

Parabolic induction

In mathematics, parabolic induction is a method of constructing representations of a reductive group from representations of its parabolic subgroups....

 consists of taking a representation of , extending it to by letting act trivially, and inducing

Induced representation

In mathematics, and in particular group representation theory, the induced representation is one of the major general operations for passing from a representation of a subgroup H to a representation of the group G itself. It was initially defined as a construction by Frobenius, for linear...

the result from to .