Problem set 5, due October 19

(1) In this problem X is a smooth variety over a field k of characteristic zero, sheaves are considered with respect to the Zarisky topology.

(a) Let ω ∈ Ω^{1}(X) be a one form. Show that the map O ∋ f ↦ f, V ect_{X} ∋ v ↦ v + ⟨ω, v⟩ extends to an automorphism of the sheaf D_{X} iff ω is closed.

(b) Let Ω_{cl}^{1} be the sheaf of closed 1-forms. For h ∈ H^{1}(Ω_{cl}^{1}) define a sheaf of algebras D_{h} on X, which is locally isomorphic to D(X).

(c) Recall that isomorphism classes of line bundles on X are in bijection with cohomology classes H^{1}(X, O^{∗}), where O^{∗} is the sheaf of invertible functions. Consider the morphism of sheaves O^{∗} → Ω_{cl}^{1}, f ↦ \frac{df}{f}. For a line bundle L let c_{1}(L) ∈ H^{1}(O^{∗}) be the image of the corresponding class in H^{1}(X, O^{∗}) under the induced map H^{1}(X, O^{∗}) → H^{1}(X, Ω_{cl}^{1}). Identify D_{c_{1}(L)} with the sheaf of differential operators acting on the sections of L.

We will write D_{L} instead of D_{c_{1}(L)}.

(2) For which i ∈ ℤ is ℙ^{n} “affine with respect to D_{O(i)}”? In other words, for which i does the functor of global sections provide an equivalence of categories between quasicoherent sheaves of D_{O(i)}-modules and modules over global sections of D_{O(i)}?

(3) Assume that X is a D-affine variety, and G is an affine algebraic group acting on X. Let D_{X}^{glob} be the algebra of global differential operators. Show that the category of G-equivariant D_{X}-modules is equivalent to the category of D_{X}^{glob}-modules M endowed with a G-action whose differential coincides with the action of 𝔤 on M coming from the embedding 𝔤 → D_{X}^{glob}.

(4) Let X = ℙ^{1}, G = SL(2). Show that D_{X}^{glob} is equal to the quotient of U(𝔤) by the ideal generated by the Casimir element C.

(5) Let G be an algebraic group, K ⊂ G – a subgroup. Recall that a (𝔤, K)- module is a vector space M endowed with an action of 𝔤 and an algebraic (in particular, locally finite) action of K which are compatible in the obvious sense (i.e. if we let 𝔨 = Lie(K) then we require that the two 𝔨-actions of 𝔨 coming from 𝔤-action and K-action coincide).

(6) Let G = P GL(2) and let K ⊂ G be the image of the subgroup of diagonal matrices. Use the above results to show that there exist exactly 3 irreducible (𝔤, K)-modules on which C acts by 0.

Write the above (𝔤, K)-modules explicitly.