An explicit determination of the Springer morphism

ABSTRACT

Let G be a simply connected semisimple algebraic groups over ℂ and let ρ:G→GL(V_λ) 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, 𝜃_λ:G→𝔤, which depends on λ. This map, 𝜃_λ, takes the maximal torus T of G to its Lie algebra 𝔱. Thus, for a given simple group G and an irreducible representation V_λ, one may write ${𝜃}_{\lambda }\left(t\right)={\sum }_{i=1}^n{c}_i\left(t\right)\check{{\alpha }_i}$, where we take the simple coroots $\left\{\check{{\alpha }_i}\right\}$ as a basis for 𝔱. We give a complete determination for these coefficients c_i(t) for any simple group G as a sum over the weights of the torus action on V_λ.

KEYWORDS:

• Cayley map
• reductive groups
• root systems
• Springer morphism

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 22E46
• Secondary: 17B10