Beilinson–Bernstein localization


In mathematics, especially in representation theory and algebraic geometry, the Beilinson–Bernstein localization theorem relates D-modules on flag varieties G/''B to representations of the Lie algebra attached to a reductive group 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 and a theorem relating D-modules on the affine Grassmannian to representations of the Kac–Moody algebra in.

Statement

Let G be a reductive group over the complex numbers, and B a Borel subgroup. Then there is an equivalence of categories
On the left is the category of D-modules on G/B. On the right χ is a homomorphism χ : Z → C from the centre of the universal enveloping algebra,
corresponding to the weight -ρ ∈ t* given by minus half the sum over the positive roots of g. The above action of W on t* = Spec Sym is shifted so as to fix -ρ.

Twisted version

There is an equivalence of categories
for any λ ∈ t* such that λ-ρ does not pair with any positive root α to give a nonpositive integer :
Here χ is the central character corresponding to λ-ρ, and Dλ is the sheaf of rings on G/B formed by taking the *-pushforward of DG/U along the T-bundle G/U → G/B, a sheaf of rings whose center is the constant sheaf of algebras U, and taking the quotient by the central character determined by λ.

Example: ''SL2''

The Lie algebra of vector fields on the projective line P1 is identified with sl2, and
via
It can be checked linear combinations of three vector fields C ⊂ P1 are the only vector fields extending to ∞ ∈ P1. Here,
is sent to zero.
The only finite dimensional sl2 representation on which Ω acts by zero is the trivial representation k, which is sent to the constant sheaf, i.e. the ring of functions O ∈ D-Mod. The Verma module of weight 0 is sent to the D-Module δ supported at 0 ∈ P1.
Each finite dimensional representation corresponds to a different twist.