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-chi associated to g and to chi & ISIN; 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,chi)(lambda), Q(A,chi)(I)(lambda) and Q(A,chi) (lambda) in C-A which generalise familiar objects when A = K, and we prove some key structural results regarding them.& nbsp;(c) 2022 Elsevier B.V. All rights reserved.