Antisymmetric characters and Fourier duality

TL;DR

This paper explores the spectral duality in quantum $ADE$ Lie theory by introducing anti-symmetric characters, formalizing eigenvalue correspondences, and solving a longstanding question in the field.

Contribution

It formalizes the eigenvalue correspondence between Coxeter elements and adjacency matrices for Verlinde algebra representations, extending classical $ADE$ theory to quantum groups.

Findings

Established the eigenvalue correspondence for Verlinde algebra representations.

Answered a question posed by Victor Kac in 1994.

Extended classical $ADE$ spectral theory to quantum group contexts.

Abstract

Inspired by the quantum McKay correspondence, we consider the classical $ADE$ Lie theory as a quantum theory over $\mathfrak{sl}_2$. We introduce anti-symmetric characters for representations of quantum groups and investigate the Fourier duality to study the spectral theory. In the $ADE$ Lie theory, there is a correspondence between the eigenvalues of the Coxeter element and the eigenvalues of the adjacency matrix. We formalize related notions and prove such a correspondence for representations of Verlinde algebras of quantum groups: this includes the quiver of any module category acted on by the representation category of any simple Lie algebra $\mathfrak{g}$ at any level $\ell$. This answers an old question posed by Victor Kac in 1994 and a recent comment by Terry Gannon.