Discrete orthogonality of Weyl orbit functions on invariant lattices and corresponding transforms

Discrete orthogonality relations of Weyl orbit functions are associated with pairs of generic Weyl group invariant lattices. Generalized affine Weyl groups are defined and used to induce finite sampling point sets inside their fundamental domains and similarly constructed label sets of Weyl orbit functions. Conditions under which a given pair of admissibly shifted lattices determines point and label sets of discretely orthogonal orbit functions are formulated. Equality between numbers of elements in the point and label sets is proven in full generality, guaranteeing the existence of the ensuing discrete transforms. The forward and backward transforms, the Plancherel formulas and unitary transform matrices are constructed. The unitary transform matrices of the discrete transforms are calculated for a specifically assembled invariant lattice of the root system A3.