An Explicit Determination of the Springer Morphism

Let $G$ be a simply connected semisimple algebraic group over $\mathbb{C}$ and let $\rho :G\rightarrow GL(V_\lambda)$ be an irreducible representation of highest weight $\lambda$. Suppose that $\rho$ has finite kernel. Springer defined adjoint-invariant regular map with Zariski dense image from the group to its Lie algebra, $\theta_\lambda:G\rightarrow\mathfrak{g}$, which depends on $\lambda$ [Kumar]. By a lemma in Kumar's recent paper, $\theta_\lambda$ takes the maximal torus to its Lie algebra $\mathfrak{t}$. Thus, for a given simple group $G$ and an irreducible representation $V_\lambda$, one may write $\theta_\lambda (t)=\sum\limits_{i=1}^n c_i(t)\check{\alpha_i}$, where the simple co-roots $\{\check{\alpha_i}\}$ are a basis for $\mathfrak{t}$. We give a complete determination of these coefficients $c_i(t)$ for any simple group $G$ as a sum over the weights of the torus action on $V_\lambda$.