An Explicit Determination of the Springer Morphism

TL;DR

This paper explicitly computes the coefficients of the Springer morphism for any simple algebraic group, expressing them as sums over weights of the torus action on irreducible representations.

Contribution

It provides a complete explicit determination of the Springer morphism coefficients for all simple groups, expanding understanding of the morphism's structure.

Findings

Coefficients expressed as sums over weights of the torus action.

Explicit formulas for the Springer morphism for all simple groups.

Enhanced understanding of the morphism's algebraic structure.

Abstract

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 $V_\lambda$.