In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Langlands decomposition writes a
parabolic subgroup ''P'' of a
semisimple Lie group
In mathematics, a simple Lie group is a connected space, connected nonabelian group, non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple ...
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
In mathematics, the Langlands program is a set of conjectures about connections between number theory, the theory of automorphic forms, and geometry. It was proposed by . It seeks to relate the structure of Galois groups in algebraic number t ...
: 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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