Dual-Root Lattice Discretization of Weyl Orbit Functions

Abstract

Four types of discrete transforms of Weyl orbit functions on finite point sets are developed. The point sets are formed by intersections of dual-root lattices with fundamental domains of 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. The unitary transform matrices of the discrete transforms are exemplified.