An explicit determination of the Springer morphism. Issue 10 (3rd October 2018)

• Record Type: Journal Article
• Authors: Rogers, Sean
• Is Part Of: Communications in algebra. Volume 46:Issue 10(2018)
• Page Start: 4233
• Page End: 4242
• Publication Date: 2018-10-03
• Subjects: Cayley map -- reductive groups -- root systems -- Springer morphism; Primary: 22E46 -- Secondary: 17B10; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2017.1407418
• Languages: English
• ISSNs: 0092-7872

Abstract

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 write𝜃 λ ( t ) = ∑ i = 1 n c i ( t ) α i ̌, where we take the simple coroots{ α i ̌ } 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 .