Antisymmetric characters and Fourier duality

Abstract

Inspired by the quantum McKay correspondence, we consider the classical ADE Lie theory as a quantum theory over sl2. We introduce antisymmetric 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 notations and prove such a correspondence for a more general case: this includes the quiver of any module of any semisimple Lie algebra g at any level ℓ. This answers an old question posed by Victor Kac and a recent comment by Terry Gannon.