Problem set 5.

(1) Use geometry of the flag space for the group SL(3) to compute characters of the six irreducible modules in the category O_{0} for the Lie algebra sl(3).

(2) Let N ⊂ SL(2) be the subgroup of lower triangular matrices with ones on the diagonal.

(a) Let M ∈ D^{b}(D − mod(ℙ^{1})) be such that a^{∗}(M) ≅ pr^{∗}(M) where a, pr : N × ℙ^{1} → ℙ^{1} are the action map and the projection. Assume that M satisfies the parity vanishing condition. Express dimension of R^{i}π_{∗}(M) via the dimensions of H^{k}(i_{0}^{∗}(M)), H^{k}(i_{1}^{∗}(M)), where i_{0}, i_{1} are embeddings of, respectively, the N fixed point and of some other point into ℙ^{1}.

(b) Let M ∈ D − mod(ℙ^{1}) be such that j^{∗}(M) ≅ O, where j is the embedding of the open N orbit into ℙ^{1}. Show that π_{∗}(M) = 0 iff M ≅ Ξ (where π is the projection to the point).

(c) (∗) Let a : N × 𝔸^{1} → 𝔸^{1} × ℙ^{1} be given by a : (n, z) → (z, n(z)); here the embedding 𝔸^{1} ⊂ ℙ^{1} sends 0 to the N invariant point. Show that Φ_{z}(a_{∗}O) ≅ Ξ, where Φ is the vanishing cycles functor.

(3) Show that Ψ_{z}^{un}(O_{𝔸^{1}\0}, d + α\frac{dz}{z} + β \frac{dz}{z^{2}} ) = 0 unless α = β = 0. (Here z is the

coordinate on 𝔸^{1}).

(Thus the intuition based on the term “nearby cycles” should be used with caution).

(4) Compute the gluing data Ψ(M), Φ(M), c, v when X = 𝔸^{1}, f = z is the coordinate and M is one of the following D-modules: j_{∗}(E_{n}), j_{!}(E_{n}), j_{!∗}(E_{n}) (notations of problem set 3).