The double dihedral Dunkl total angular momentum algebra

The Dunkl total angular momentum algebra (TAMA) is realised as the dual partner of the orthosymplectic Lie superalgebra containing the Dunkl deformation of the Dirac operator. In this paper, we consider the case when the reflection group associated with the Dunkl operators is a product of two dihedral groups acting on a four-dimensional Euclidean space. We show that in this case there is a subalgebra of the total angular momentum algebra that admits a triangular decomposition. In analogy to the celebrated theory of semisimple Lie algebras, we use this triangular subalgebra to give precise necessary conditions that a finite-dimensional irreducible representation must obey, in terms of weights. In specific cases, which includes unitary representations, we construct a basis of weight vectors with explicit actions of all TAMA elements. Examples of these modules occur in the kernel of the Dunkl--Dirac operator in the context of deformations of Howe dual pairs.