Higgs bundles over elliptic curves - ICMAT

the representation ρ : Γ R → K and there exists a covering {U i } i∈I of X that trivializesE ρ such that Φ ∈ H 0 (X, E ρ (g)) expressed in terms of {U i } i∈I is constant and equal toz ∈ h ωc ⊂ z g (ρ). Let us denote by g c the subset of g of the elements of the form z ′ = ad d (z)for some g ∈ G and some z ∈ h ωc . By the previous description of polystable G-Higgsbundles of topological class d we have thatB(G, d) = H 0 (X, O ⊗ (g c /G)) ∼ = g /cG .By definition of g c we have that g c /G is isomorphic to h ωc /(N G (h ωc )/Z G (h ωc )). Take W c ,S c and Λ Scas in (8.6), (8.10) and (9.4). Recall that Lie S c is h ωc and therefore we have thath ωc ∼ = C ⊗ Z Λ Scand furthermore h ωc /W c∼ = (C ⊗Z Λ Sc)/W c . Then, the composition of theprevious isomorphisms gives usβ G,d : B(G, d)∼=−→ (C ⊗ Z Λ Sc) / W c.Let B(Λ Sc) = {γ 1 , . . . , γ l } be a basis of Λ Sc. Recalling that T ∗ X ∼ = X × C, we seethat the projection π : T ∗ X → C inducesπ G,c : (T ∗ X ⊗ Z Λ Sc) / W c (C ⊗ Z Λ Sc) / W c(9.18)[ ∑γ i ∈B(Λ Sc ) (x i, λ i ) ⊗ Z γ i]W c[ ∑γ i ∈B(Λ Sc ) λ i ⊗ Z γ i]W c.We use this morphism to better understand the Hitchin map.Theorem 9.5.2. The following diagram is commutativeb G,dM(G) d B(G, d)ξ x 0G,d ∼ =∼= β G,d(T ∗ X ⊗ Z Λ Sc) / W cπ G,c (C ⊗ Z Λ Sc) / W c.Proof. Let (E, Φ) be the polystable representative of a certain S-equivalence class in M(G) d .In the notation of Remark 9.4.5 we have that (E, Φ) is isomorphic to i ∗ (L⊗E x 0L c,d, φ) where(L, φ) is a S c -Higgs bundle and (L, φ) the induced S c -Higgs bundle. Note that φ and φ aregiven by an element z of h ωc . We writeξ x 0(L, φ) =∑S c,0γ i ∈B ΛZ(x i , λ i ) ⊗ Z γ i .We note that z = ∑ γ i ∈B ΛZλ j · γ j . On the other hand we have that ξ x 0G,d ([(E, Φ)] S) is equalto ξ x 0(L, φ) and then, we see that the diagram commutes.S c,0Once we have an explicit description of the Hitchin fibration, we can describe explicitlyits fibres.177