Antisymmetric characters and Fourier duality

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.