An explicit determination of the Springer morphism

摘要

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