Cyclotomic level maps and associated varieties of simple affine vertex algebras

TL;DR

This paper introduces cyclotomic level maps linking nilpotent orbits and Weyl group conjugacy classes, explores their compatibility and relation to affine Weyl group cells, and proposes a conjecture on associated varieties of simple affine vertex algebras at non-admissible levels.

Contribution

It defines and analyzes cyclotomic level maps, demonstrating their compatibility with Lusztig's map and relating them to affine Weyl group structures, and formulates a new conjecture on vertex algebra associated varieties.

Findings

Maps are compatible with Lusztig's map.

Relationship established with affine Weyl group two-sided cells.

Conjecture proposed and supported with initial evidence.

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.