attached to a reductive group G. It 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
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.
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).
The Verma module of weight 0 is sent to the D-Module δ supported at 0 ∈ P1.