An explicit determination of the Springer morphism

Abstract

ABSTRACTLet G be a simply connected semisimple algebraic groups over ℂ and let ρ:G→GL(Vλ) be an irreducible representation of G of highest weight λ. Suppose that ρ has finite kernel. Springer defined an adjoint-invariant regular map with Zariski dense image from the group to the Lie algebra, 𝜃λ:G→𝔤, which depends on λ. This map, 𝜃λ, takes the maximal torus T of G to its Lie algebra 𝔱. Thus, for a given simple group G and an irreducible representation Vλ, one may write , where we take the simple coroots as a basis for 𝔱. We give a complete determination for these coefficients ci(t) for any simple group G as a sum over the weights of the torus action on Vλ.