Weight-Lattice Discretization of Weyl-Orbit Functions

TL;DR

This paper introduces a novel discretization method for Weyl-orbit functions using the weight lattice, enhancing symmetry, orthogonality, and linking to important structures in conformal field theory.

Contribution

The paper presents a new discretization approach for Weyl-orbit functions based on the weight lattice, improving theoretical clarity and symmetry.

Findings

Orthogonality of discretized orbit functions proved

Construction of unitary, symmetric matrices with Weyl-orbit elements

Matrix coincides with the Kac-Peterson modular S matrix

Abstract

Weyl-orbit functions have been defined for each simple Lie algebra, and permit Fourier-like analysis on the fundamental region of the corresponding affine Weyl group. They have also been discretized, using a refinement of the coweight lattice, so that digitized data on the fundamental region can be Fourier-analyzed. The discretized orbit function has arguments that are redundant if related by the affine Weyl group, while its labels, the Weyl-orbit representatives, invoke the dual affine Weyl group. Here we discretize the orbit functions in a novel way, by using the weight lattice. A cleaner theory results, with symmetry between the arguments and labels of the discretized orbit functions. Orthogonality of the new discretized orbit functions is proved, and leads to the construction of unitary, symmetric matrices with Weyl-orbit-valued elements. For one type of orbit function, the matrix coincides with the Kac-Peterson modular $S$ matrix, important for Wess-Zumino-Novikov-Witten conformal field theory.