Verma modules and finite-dimensional irreducible modules of the universal Askey–Wilson algebra at roots of unity

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