Cyclotomic level maps and associated varieties of simple affine vertex algebras

Abstract

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.

v1: 48 pages, 8 tables, 5 figures. v2: 46 pages, 8 tables, 5 figures; intro is rewritten to better emphasize the Langlands-style picture we are building; an equivalent definition of cl_n is added and the first theorem is reduced to a known result; some other references are added and typos corrected