Inverse limits in representations of a restricted Lie algebra

Let (g, [p]) be a restricted Lie algebra over an algebraically closed field of characteristic p > 0. Then the inverse limits of "higher" reduced enveloping algebras {u _χ ^s (g) {pipe} s ∈ N} with χ running over g ^* make representations of g split into different "blocks". In this paper, we study such an infinite-dimensional algebra A _χ(g){double colon equal} lim U _χ ^s(g) for a given χ ∈ g ^*. A module category equivalence is built between subcategories of U(g)-mod and A _χ(g)-mod. In the case of reductive Lie algebras, (quasi) generalized baby Verma modules and their properties are described. Furthermore, the dimensions of projective covers of simple modules with characters of standard Levi form in the generalized χ-reduced module category are precisely determined, and a higher reciprocity in the case of regular nilpotent is obtained, generalizing the ordinary reciprocity.