Problem set 3, due October 29.

(1) Prove that (as stated in class) the D-modules (O, d + λ\frac{dt}{t}), (O, d + μ\frac{dt}{t^{2}}) extend cleanly from 𝔸^{1} \ {0} to 𝔸^{1}.

(2) More generally, let P be a polynomial in n variables, and let U ⊂ 𝔸^{n} be the complement to the zeroes of P, let j : U → 𝔸^{n} be the embedding. For λ ∈ k consider the connection ∇ = d + λ\frac{dP}{P} on the trivial line bundle on U, let P_{λ} be the corresponding D-module. Show that j_{!}(P_{λ}) ≅ j_{∗}(P_{λ}) provided that b(λ) ≠ 0, where b(λ) is the Bernstein polynomial, appearing in the proof of analytic continuation of distribution P^{λ}.

(3) Let 𝔤 be a Lie algebra, U = U(𝔤) be its enveloping algebra. Recall the homological duality functor 𝔻 : M ↦ RHom_{U}(M, U) from D^{b}(U − mod^{fg}) to itself (notice that we have a canonical isomorphism U ≅ U^{op} sending x ∈ 𝔤 to −x). Describe 𝔻(M) explicitly in the following examples.

(a) 𝔤 is arbitrary and M is finite dimensional.

(b) 𝔤 = sl(2) and M = M_{λ} is the Verma module, i.e. the module induced from a one dimensional module of the subalgebra of upper-triangular matrices.

(4) Let j : 𝔸^{1} \ 0 → 𝔸^{1} and i : {0} → 𝔸^{1} be the embeddings. Let E_{n} be the local system on 𝔸^{1} \ 0 given by E_{n} = (O^{n}, d + N \frac{dz}{z}), where N is the n × n matrix N_{ij} = δ_{i,j−1}.

(a) Show that every indecomposable D-module on 𝔸^{1} \ 0, which has a filtration whose associated graded is a sum of several copies of O, is isomorphic to E_{n} for some n.

(b) Describe the Jordan-Hoelder series of j_{!∗}(E_{n}).

(c) Let Ξ_{n} be the co-kernel of a non-zero map O_{𝔸^{1}} → j_{∗}(E_{n+1}). Compute i^{+}(Ξ_{n}).

(d) Show that Fourier transform sends Ξ_{n} to j_{!∗}(E_{n+1}).

(5) Let V be a vector space. Let κ be the function on V ×V^{∗} given by κ((v, ξ)) = ξ(v). Let E_{κ} be the D-module corresponding to the trivial line bundle O_{V ×V^{∗}} with the flat connection ∇ = d + dκ.

(a) For an isomorphism ϕ : R_{1} → R_{2} of associative unital rings let the graph of ϕ, Γ_{ϕ} be the R_{1} ⊗ R_{2}^{op} module, which coincides with R_{2} as an abelian group, and the action is given by r_{1} ⊗ r_{2}(r) = ϕ(r_{1})rr_{2}. Notice that Γ_{ϕ} comes with a canonical generator 1_{R_{2}}. Check that the R_{1} ⊗ R_{2}^{op} module Γ_{ϕ} with a fixed generator determines ϕ uniquely.

(b) Let F : D(V^{∗}) → D^{op}(V) be the composition of Fourier isomorphism D(V) ≅ D(V^{∗}) and the isomorphism D(V) → D^{op}(V), ∂_{x_{i}} ↦ −∂_{x_{i}}. Show that E_{κ} ≅ Γ_{F}, so that the canonical generator 1 ∈ O_{V ×V^{∗}} = E_{κ} corresponds to the canonical generator of Γ_{F}.

(c) Let now X be an affine smooth variety and V → X be a vector bundle. Let i : V ×_{X} V^{∗} be the embedding. Let now E_{κ} denote i_{+}(O, d + dκ) where κ is the function on V ×_{X}V^{∗} defined similarly to the above. Give a criterion for this module to have the form Γ_{F} for some isomorphism F : D(V) → D(V^{∗}). Deduce the last problem of the previous problem set.

(d) (The rest of this problem uses the functor π_{+} to be introduced in the lecture on 10/20). Check that for a D-module M on V there exists a canonical isomorphism Φ(M) ≅ π_{2+}(π_{1}^{+}(M) ⊗_{O_{V ×V ^{∗}}} E_{κ}), where π_{1} : V × V^{∗} → V, π_{2} : V × V^{∗} → V^{∗} are the projections, and Φ is the Fourier transform.