Vanishing of the quantum reduction of the Deligne exceptional series representations of negative integer level

TL;DR

This paper investigates the quantum Hamiltonian reduction of affine Lie algebra representations in the Deligne exceptional series at negative levels, showing it generally vanishes except in specific nilpotent cases.

Contribution

It identifies the precise conditions under which the quantum Hamiltonian reduction does not vanish for these representations, clarifying their structure.

Findings

Quantum reduction vanishes for most cases at negative levels.

Non-vanishing occurs only when the nilpotent element is conjugate to specific elements.

Provides a classification of cases with non-zero quantum reduction.

Abstract

In this paper we show that, for the Deligne exceptional series representations of negative integer level of affine Lie algebras, the quantum Hamiltonian reduction vanishes except for the cases where the nilpotent element is conjugate to $e_{-\theta}$ or $e_{-\theta_s}$.