Beilinson–Bernstein localization

The Beilinson–Bernstein localization theorem is a foundational result of geometric representation theory, a part of mathematics studying the representation theory of e.g. Lie algebras using geometry.

Statement

Let G be a reductive group over the complex numbers, and B a Borel subgroup. Then there is an equivalence of categories

[math]\displaystyle{ \mathcal{D}\text{-Mod}(G/B)\ \simeq\ \left(U(\mathfrak{g})/\ker\chi\right) \text{-Mod}. }[/math]

On the left is the category of D-modules on G/B. On the right χ is a homomorphism χ : Z(U(g)) → C from the centre of the universal enveloping algebra,

[math]\displaystyle{ Z(U(\mathfrak{g}))\ \simeq\ \text{Sym}(\mathfrak{t})^{W,\rho}, }[/math]

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(t) is shifted so as to fix -ρ.

Twisted version

There is an equivalence of categories

[math]\displaystyle{ \mathcal{D}_\lambda\text{-Mod}(G/B)\ \simeq\ \left(U(\mathfrak{g})/\ker\chi_\lambda\right) \text{-Mod}. }[/math]

for any λ ∈ t^* such that λ-ρ does not pair with any positive root α to give a nonpositive integer (it is "regular dominant"):

[math]\displaystyle{ (\lambda-\rho, \alpha)\ \in\ \mathbf{C}-\mathbf{Z}_{\le 0}. }[/math]

Here χ is the central character corresponding to λ-ρ, and D_λ is the sheaf of rings on G/B formed by taking the *-pushforward of D_G/U along the T-bundle G/U → G/B, a sheaf of rings whose center is the constant sheaf of algebras U(t), and taking the quotient by the central character determined by λ (not λ-ρ).

Example: SL₂

The Lie algebra of vector fields on the projective line P¹ is identified with sl₂, and

[math]\displaystyle{ U(\mathfrak{sl}_2)/\Omega\ \simeq \ \mathcal{D}(\mathbf{P}^1) }[/math]

via

[math]\displaystyle{ (e,h,f) \ \mapsto \ (\partial_z, -2z\partial_z, z^2\partial_z) }[/math]

It can be checked linear combinations of three vector fields C ⊂ P¹ are the only vector fields extending to ∞ ∈ P¹. Here,

[math]\displaystyle{ \Omega\ =\ ef+fe+\frac{1}{2}h^2 }[/math]

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.

History

The theorem was introduced by (Beilinson Bernstein).

Extensions of this theorem include the case of partial flag varieties G/P, where P is a parabolic subgroup, in (Holland Polo), and a theorem relating D-modules on the affine Grassmannian to representations of the Kac–Moody algebra [math]\displaystyle{ \widehat \mathfrak g }[/math], in (Frenkel Gaitsgory).

References

• Beilinson, Alexandre; Bernstein, Joseph (1981), "Localisation de g-modules", Comptes Rendus de l'Académie des Sciences, Série I 292 (1): 15–18 
• Holland, Martin P.; Polo, Patrick (1996), "K-theory of twisted differential operators on flag varieties", Inventiones Mathematicae 123 (2): 377–414, doi:10.1007/s002220050033 
• Frenkel, Edward; Gaitsgory, Dennis (2009), "Localization of [math]\displaystyle{ \mathfrak{g} }[/math]-modules on the affine Grassmannian", Ann. of Math. (2) 170 (3): 1339–1381, doi:10.4007/annals.2009.170.1339 
• Hotta, R. and Tanisaki, T., 2007. D-modules, perverse sheaves, and representation theory (Vol. 236). Springer Science & Business Media.
• Beilinson, A. and Bernstein, J., 1993. A proof of Jantzen conjectures. ADVSOV, pp. 1–50.