In mathematics, especially in
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
and
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the Beilinson–Bernstein localization theorem relates
D-modules
In mathematics, a ''D''-module is a module over a ring ''D'' of differential operators. The major interest of such ''D''-modules is as an approach to the theory of linear partial differential equations. Since around 1970, ''D''-module theory has ...
on
flag varieties In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flag (linear algebra), flags in a finite-dimensional vector space ''V'' over a field (mathematics), field F. When F is the real or complex nu ...
''G''/''B'' to representations of the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
attached to a
reductive group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
''G''. It was introduced by .
Extensions of this theorem include the case of partial flag varieties ''G''/''P'', where ''P'' is 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 subgroup ...
in and a theorem relating ''D''-modules on the
affine Grassmannian
In mathematics, the affine Grassmannian of an algebraic group ''G'' over a field ''k'' is an ind-scheme—a colimit of finite-dimensional schemes—which can be thought of as a flag variety for the loop group ''G''(''k''((''t''))) and which desc ...
to representations of the
Kac–Moody algebra
In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a ge ...
in .
References
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Representation theory
Lie algebras
Algebraic geometry
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