Cyclic elements in semisimple Lie algebras

where g±d 6= 0. The positive integer d is called the depth of this Z-grading, and of the nilpotent element e. This notion was previously studied e.g. in [P1]. An element of g of the form e+ F , where F is a non-zero element of g−d, is called a cyclic element, associated with e. In [K1] Kostant proved that any cyclic element, associated with a principal (= regular) nilpotent element e, is regular semisimple, and in [S] Springer proved that any cyclic element, associated with a subregular nilpotent element of a simple exceptional Lie algebra, is regular semisimple as well, and, moreover, found two more distinguished nilpotent conjugacy classes in E8 with the same property. Both Kostant and Springer use this property in order to exhibit an explicit connection between these nilpotent conjugacy classes and conjugacy classes of certain regular elements of the Weyl group of g. A completely different use of cyclic elements was discovered by Drinfeld and Sokolov [DS]. They used a cyclic element, associated with a principal nilpotent element of a simple Lie algebra g, to construct a bi-Hamiltonian hierarchy of integrable evolution PDE of KdV type (the case g = sl2 produces the KdV hierarchy). In a number of subsequent papers, [W], [GHM], [BGHM], [FHM], [DF], [F],... the method of Drinfeld and Sokolov was extended to some other nilpotent elements. Namely, it was established that one gets a bi-Hamiltonian integrable hierarchy for any nilpotent element e of a simple Lie algebra, provided that there exists a semisimple cyclic element, associated with e. One of the results of the present paper is a description of all nilpotent elements with this property in all semisimple Lie algebras. We say that a non-zero nilpotent element e (and its conjugacy class) is of nilpotent (resp. semisimple or regular semisimple) type if any cyclic element, associated with e, is nilpotent (resp. any generic cyclic element, associated with e, is semisimple or regular semisimple). If neither of the above cases occurs, we say that e is of mixed type.