Problem set 3, due October 5

(1) Let ϕ : X → Y be a smooth map of smooth algebraic varieties, and M a finitely generated D-module on Y. Suppose that M has a good filtration, and let F = gr(M) be the corresponding coherent sheaf on the cotangent bundle. Prove that for an appropriate good filtration on ϕ^{∗}(M) we have gr(ϕ^{∗}(M)) ≅ dϕ_{∗}pr^{∗}(F) (in the notations introduced in class). Deduce the description of SS(ϕ^{∗}M) via SS(M) stated in class.

(2) Consider the (left) D-module δ_{x} on a smooth variety X. Show that any vector field v vanishing at x multiplies the canonical generator of δ_{x} by a scalar. For X = 𝔸^{n}, x = 0, v = ∑ a_{I}, Jx^{J}∂^{J} give an explicit formula for that scalar.

(3) Let G be an affine algebraic group acting on a smooth variety X (you can assume that X is affine). Let a : G × X → X denote the action map and let p : G × X → X be the projection to the second multiple.

Let M be an O_{X}-module. Recall that a G-equivariant structure on M is an isomorphism α : a^{∗}(M) ≃ p^{∗}(M) (inverse image is taken in the category of O_{X}-modules) satisfying a compatibility condition.

a) Write down this condition explicitly.

b) Let 𝔤 be the Lie algebra of G. Show that for every open subset U ⊂ X we have a natural action of 𝔤 on Γ(U, M) (hint: restrict α to the infinitesimal neighbourhood of the identity element in G). For every v ∈ 𝔤 we denote the corresponding endomorphism of Γ(U, M) by α(v). Assume (for simplicity) that X can be covered by affine open G-invariant subsets.

(4) Let M be a left D_{X}-module. A G-equivariant structure on M is an isomor-

phism of D-modules a^{∗}(M) ≃ p^{∗}(M) satisfying a compatibility condition. a) Write down this condition explicitly.

b) Show that an equivariant structure on a D-module M is the same as an equivariant structure on the corresponding O-module, satisfying the following additional compatibility. We have the natural map η : 𝔤 → V ec_{X} coming from the action of G on X. For every v ∈ 𝔤 the action of the vector field η(v) on Γ(U, M) (coming from the D_{X}-module structure) has to coincide with α(v) (constructed in the previous problem).

c) Assume that G is connected. Show that a G-equivariant structure on a given D-module is unique if it exists.

d) Assume that G acts transitively on X. Take x ∈ X and let Z(x) denote its stabilizer. Show that we have equivalence of categories ”equivariant D_{X}-modules”⟺ ”representations of Z(x)/Z(x)^{0}”.

e) Let M a finitely generated G-equivariant D-module. Show that its singular support maps to zero under the moment map T^{∗}X → 𝔤^{∗}.

f) Deduce that if G acts with finitely many orbits, then every finitely generated G-equivariant D-module is holonomic.