Series of Nilpotent Orbits

Abstract

We organize the nilpotent orbits in the exceptional complex Lie algebras into series and show that within each series the dimension of the orbit is a linear function of the natural parameter {\small $a=1, 2, 4, 8$}, respectively for {\small $\ff_4,\fe_6,\fe_7,\fe_8$}. We observe similar regularities for the centralizers of nilpotent elements in a series and grade components in the associated grading of the ambient Lie algebra. More strikingly, we observe that for {\small $a\geq 2$} the numbers of {\small $\FF_q$}-rational points on the nilpotent orbits of a given series are given by polynomials that have uniform expressions in terms of a. This even remains true for the degrees of the unipotent characters associated to these series through the Springer correspondence. We make similar observations for the series arising from the other rows of Freudenthal's magic chart and make some observations about the general organization of nilpotent orbits, including the description of and dimension formulas for several universal nilpotent orbits (universal in the sense that they occur in almost every simple Lie algebra).