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 .

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 CP1 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 0P1.

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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