Langlands decomposition

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

See also: Parabolic induction

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

## 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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