An Explicit Determination of the Springer Morphism

2017/01/06 by Rogers, Sean
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1701.01538

Abstract

Let G be a simply connected semisimple algebraic group over ℂ and let ρ:G→ GL(V_λ) be an irreducible representation of highest weight λ. Suppose that ρ has finite kernel. Springer defined adjoint-invariant regular map with Zariski dense image from the group to its Lie algebra, θ_λ:G→\mathfrakg, which depends on λ [Kumar]. By a lemma in Kumar's recent paper, θ_λ takes the maximal torus to its Lie algebra \mathfrakt. Thus, for a given simple group G and an irreducible representation V_λ, one may write θ_λ(t)=∑_i=1ⁿ c_i(t)\checkα_i, where the simple co-roots \\checkα_i\ are a basis for \mathfrakt. 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_λ.

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