# 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*.

## Applications

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
, extending it to
by letting
act trivially, and inducing the result from
to
.

## See also

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