In this paper, we introduce and study two cyclotomic level maps defined respectively on the set of nilpotent orbits $\underline{ \mathcal{ N}}$ in a complex semi-simple Lie algebra $\mathfrak{ g}$ and the set of conjugacy classes $\underline{ W}$ in its Weyl group, with values in positive integers. We show that these maps are compatible under Lusztig's map $\underline{ W} \to \underline{ \mathcal{ N}}$, which is also the minimal reduction type map as shown by Yun. We also discuss their relationship with two-sided cells in affine Weyl groups. We use these maps to formulate a conjecture on the associated varieties of simple affine vertex algebras attached to $\mathfrak{ g}$ at non-admissible integer levels, and provide some evidence for this conjecture.
Categories: math.RT math-ph math.MP math.QA