  In this paper we classify and give Kazhdan-Lusztig type character formulas
for equivariantly irreducible representations of Lie algebras of reductive
algebraic groups over a field of large positive characteristic. The
equivariance is with respect to a group whose connected component is a torus.
Character computation is done in two steps. First, we treat the case of
distinguished $p$-characters: those that are not contained in a proper Levi.
Here we essentially show that the category of equivariant modules we consider
is a cell quotient of an affine parabolic category $\mathcal{O}$. For this, we
prove an equivalence between two categorifications of a parabolically induced
module over the affine Hecke algebra conjectured by the first named author. For
the general nilpotent $p$-character, we get character formulas by explicitly
computing the duality operator on a suitable equivariant K-group.
