In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
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
*
Lie group decompositions
References
Sources
* A. W. Knapp, Structure theory of semisimple Lie groups. .
Lie groups
Algebraic groups
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