On the center and semi-center of enveloping algebras in prime characteristic

Theorems of J. Dixmier and C. Moeglin on the semi-center of enveloping algebras in zero characteristic and of R.P. Stanley and H. Nakajima on polynomial relative invariants of finite groups, are shown to have analogs for enveloping algebras and for Lie algebra polynomial invariants, in the prime characteristic case. We shall illustrate by examples the extent of these analogies. A key result is the realization of the semi-center as a fixed ring of the action of a finite set of nilpotent derivations on the center of an enveloping algebra of a related Lie algebra.