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[1]

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,

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[2]

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"):

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(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

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.

History

The theorem was introduced by Beilinson & Bernstein (1981).

Extensions of this theorem include the case of partial flag varieties G/P, where P is a parabolic subgroup, in Holland & Polo (1996), and a theorem relating D-modules on the affine Grassmannian to representations of the Kac–Moody algebra , in Frenkel & Gaitsgory (2009).

References

  1. ↑ Theorem 3.3.1, Beilinson, A. and Bernstein, J., 1993. A proof of Jantzen conjectures.
  2. ↑ Theorem 3.3.1, Beilinson, A. and Bernstein, J., 1993. A proof of Jantzen conjectures.
  • Beilinson, Alexandre; Bernstein, Joseph (1981), "Localisation de g-modules", Comptes Rendus de l'Académie des Sciences, Série I, 292 (1): 15–18, MR 0610137
  • 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, MR 1374207, S2CID 189819773
  • Frenkel, Edward; Gaitsgory, Dennis (2009), "Localization of -modules on the affine Grassmannian", Ann. of Math. (2), 170 (3): 1339–1381, arXiv:math/0512562, doi:10.4007/annals.2009.170.1339, MR 2600875, S2CID 17597920
  • 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.
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