Langlands decomposition

In mathematics, the Langlands decomposition writes a parabolic subgroup P of a semisimple Lie group as a product [math]\displaystyle{ P=MAN }[/math] of a reductive subgroup M, an abelian subgroup A, and a nilpotent subgroup N.

## Contents

* 1 Applications
* 2 See also
* 3 References
* 3.1 Sources

## Applications

A key application is in parabolic induction, which leads to the Langlands program: if [math]\displaystyle{ G }[/math] is a reductive algebraic group and [math]\displaystyle{ P=MAN }[/math] is the Langlands decomposition of a parabolic subgroup P, then parabolic induction consists of taking a representation of [math]\displaystyle{ MA }[/math], extending it to [math]\displaystyle{ P }[/math] by letting [math]\displaystyle{ N }[/math] act trivially, and inducing the result from [math]\displaystyle{ P }[/math] to [math]\displaystyle{ G }[/math].

## See also

* Lie group decompositions

## References

### Sources

* A. W. Knapp, Structure theory of semisimple Lie groups. ISBN 0-8218-0609-2.

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