2006/05/23 by George J. McNinch, McNinch, George J. · 1 citation
Mathematics · #20G15 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.math/0605626
openalex publication_date 2006/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let F be an algebraically closed field and let G be a semisimple F-algebraic group for which the characteristic of F is *very good*. If X in Lie(G) = Lie(G)(F) is a nilpotent element in the Lie algebra of G, and if C is the centralizer in G of X, we show that (i) the root datum of a Levi factor of C, and (ii) the component group C/Co both depend only on the Bala-Carter label of X; i.e. both are independent of very good characteristic. The result in case (ii) depends on the known case when G is (simple and) of adjoint type. The proofs are achieved by studying the centralizer C of a nilpotent section X in the Lie algebra of a suitable semisimple group scheme over a Noetherian, normal, local ring A. When the centralizer of X is equidimensional on Spec(A), a crucial result is that locally in the etale topology there is a smooth A-subgroup scheme L of CC such that Lt is a Levi factor of Ct for each t in Spec(A).