In
mathematics, the Langlands decomposition writes a
parabolic subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgro ...
''P'' of a
semisimple Lie group
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals).
Throughout the article, unless otherwise stated, a Lie algebra is ...
as a
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
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
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...
: 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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