Gradings on Modules Over Lie Algebras of E Types

For any grading by an abelian group G on the exceptional simple Lie algebra (mathcal {L}) of type E ₆ or E ₇ over an algebraically closed field of characteristic zero, we compute the graded Brauer invariants of simple finite-dimensional modules, thus completing the computation of these invariants for simple finite-dimensional Lie algebras. This yields the classification of finite-dimensional G-graded simple (mathcal {L})-modules, as well as necessary and sufficient conditions for a finite-dimensional (mathcal {L})-module to admit a G-grading compatible with the given G-grading on (mathcal {L}).