Abstract The Askey–Wilson algebras illustrate the bispectral property of orthogonal polynomials in the Askey scheme. The universal Askey–Wilson algebra q is a central extension of the Askey–Wilson algebras associated with the most general orthogonal polynomials in the Askey scheme. The Verma q -modules are a family of infinite-dimensional q -modules with marginal weights. Under the condition that q is not a root of unity, it was shown that every finite-dimensional irreducible q -module has a marginal weight and is isomorphic to a quotient of a Verma q -module. Assume that q is a root of unity. We prove that every finite-dimensional irreducible q -module with a marginal weight is isomorphic to a quotient of a Verma q -module. More precisely, two natural families of finite-dimensional quotients of Verma q -modules contain all finite-dimensional irreducible q -modules with marginal weights up to isomorphism. Furthermore, we classify the finite-dimensional irreducible q -modules with marginal weights up to isomorphism.
Hau-Wen Huang (Thu,) studied this question.