Note on Primitive Ideals of Enveloping Algebras in Prime Characteristic

Abstract

Let [Formula: see text] be a restricted Lie algebra over an algebraically closed field F of characteristic p > 0, [Formula: see text] the center of the universal enveloping algebra [Formula: see text] of [Formula: see text]. In this note, we study primitive ideals of [Formula: see text]. The following results are included: (1) The ideal of [Formula: see text] generated by the central character ideal associated with any irreducible [Formula: see text]-module has finite co-dimension in [Formula: see text]. Furthermore, the co-dimension is no less than [Formula: see text], where [Formula: see text] is the maximal dimension of irreducible [Formula: see text]-modules. (2) Each annihilator ideal of irreducible [Formula: see text]-modules of maximal dimension is generated by the corresponding central character ideal in [Formula: see text]. (3) Each G-stable ideal in [Formula: see text] for [Formula: see text] contains nonzero fixed points under the action of G, where G is a connected reductive algebraic group. Additionally, the arguments on ideals help us to give an alternative description of the Azumaya locus in the Zassenhaus variety without using the normality of the Zassenhaus variety.