# Dual-root lattice discretization of Weyl orbit functions

## TL;DR

This paper develops four types of discrete transforms for Weyl orbit functions on finite point sets derived from dual-root lattices, establishing their orthogonality and formulating corresponding Fourier transforms.

## Contribution

It introduces a dual-root lattice discretization framework for Weyl orbit functions, including explicit counting formulas and the development of discrete Fourier-Weyl and Hartley-Weyl transforms.

## Key findings

- Proved identical cardinality of point and weight sets.
- Established discrete orthogonality of Weyl and Hartley orbit functions.
- Formulated discrete Fourier-Weyl and Hartley-Weyl transforms.

## Abstract

Four types of discrete transforms of Weyl orbit functions on the finite point sets are developed. The point sets are formed by intersections of the dual-root lattices with the fundamental domains of the affine Weyl groups. The finite sets of weights, labelling the orbit functions, obey symmetries of the dual extended affine Weyl groups. Fundamental domains of the dual extended affine Weyl groups are detailed in full generality. Identical cardinality of the point and weight sets is proved and explicit counting formulas for these cardinalities are derived. Discrete orthogonality of complex-valued Weyl and real-valued Hartley orbit functions over the point sets is established and the corresponding discrete Fourier-Weyl and Hartley-Weyl transforms are formulated.