Existence of regular nilpotent elements in the Lie algebra of a simple algebraic group in bad characteristics (Q1097337)

Let L be the Lie algebra of a connected simple linear algebraic group G over an algebraically closed field K. In Publ. Math., Inst. Haut. Etud. Sci. 30, 475-501 (1966; Zbl 0156.270), \textit{T. A. Springer} showed that if K has characteristic 0 or a prime other than 2 in types B, C, D, other than 2 or 3 in types \(E_ 6\), \(E_ 7\), \(F_ 4\), or \(G_ 2\), and other than 2, 3, or 5 in type \(E_ 8\), then L has regular nilpotent elements. The present paper removes the restrictions on the primes, and also shows that in the adjoint case the centralizers of such elements are always connected. (Analogous results for G were obtained by \textit{B. Lou} [Bull. Am. Math. Soc. 74, 1144-1146 (1968; Zbl 0167.302)].) The existence of the regular nilpotent elements for the ``bad'' characteristics listed above is shown by calculating the centralizer \(U_ X\) of X in G of the principal nilpotent element \(X=\sum_{r\in \Pi}e_ r\), where \(\Pi\) is a simple system of roots and \(Ke_ r\) is the Lie algebra of the unipotent one-parameter subgroup \(X_ r(G_ a)\), \(G_ a\) the additive group of K. For the classical groups, the required results follow from direct calculations with the standard representations and induction. A direct calculation is made for type \(G_ 2\), but for the other types computer calculations were employed, the results of which are summarized in a table at the end of the paper.