Grothendieck’s simultaneous resolution and the Springer correspondence

1.1 Symplectic structure on the cotangent bundle Let G be a semisimple algebraic group with Lie algebra g. Fix a Borel subgroup B of G, and let B = G/B be the corresponding flag variety. The cotangent bundle T ∗B comes equipped with symplectic form ω = dλ defined as follows. For x ∈ B and α ∈ T ∗ xB, let π : T ∗B → B be the projection and π∗ : TαT ∗B → TxB be its differential. Then, for ξ ∈ TαT ∗B, we set 〈λ, ξ〉 = 〈α, π∗(ξ)〉. If qi, pi are dual coordinates on T ∗B (with qi coordinates on B), then λ = ∑ i pidqi and therefore