  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.
