On graded representations of modular Lie algebras over commutative algebras

We develop the theory of a category C _A which is a generalisation to non-restricted g-modules of a category famously studied by Andersen, Jantzen and Soergel for restricted g-modules, where g is the Lie algebra of a reductive group G over an algebraically closed field K of characteristic p>0. Its objects are certain graded bimodules. On the left, they are graded modules over an algebra U _χ associated to g and to χ∈g ^⁎ in standard Levi form. On the right, they are modules over a commutative Noetherian S(h)-algebra A, where h is the Lie algebra of a maximal torus of G. We define here certain important modules Z _A,χ(λ), Q _A,χ ^I(λ) and Q _A,χ(λ) in C _A which generalise familiar objects when A=K, and we prove some key structural results regarding them.