The centralizer of a nilpotent section

Details The centralizer of a nilpotent section The centralizer of a nilpotent section

2006-05-23

arxiv.org/abs/math/0605626v3

Mathematics

Representation Theory

34 pages. To appear in Nagoya Math. J. Dedicated to Toshiaki Shoji, on the occasion of his 60th birthday

Scientific paper

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/C^o 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 L_t is a Levi factor of C_t for each t in Spec(A).

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