Birthing Opers

1.1. Let G be a simply connected semisimple group with Borel subgroup B, N = [B,B] and let H = B/N . Let g, b, n and h be the respective Lie algebras of these groups. Let G be the Langlands dual group with dual Borel B, etc. Let X be a smooth curve or the formal disc D or the formal punctured disc D×. 1.2. Recall that a g-oper over X is a G-bundle FLG = F with a reduction FLB to B and a connection ∇ on F satisfying a property that we recall. There is an obstruction c(∇) ∈ (Lg/Lb)FLB ⊗ ωX to the preservation of ∇ under the reduction FLB and we demand that 1) c(∇) ∈ (Lg/Lb)−1 FLB ⊗ωX with ( Lg/Lb)−1 = ⊕α̌(g/b) the space spanned by the negative simple coroots of g and 2) the projection of c(∇) to (Lg/Lb)α̌FLB is nowhere vanishing on X for each negative simple coroot α̌.