On finite-dimensional irreducible modules for the universal Askey-Wilson algebra

Let ${\mathrm{\Delta }}_q$ be the universal Askey-Wilson algebra. If $q$ is not a root of unity, it is shown in the Huang's earlier paper that an $(n+1)$-dimensional irreducible ${\mathrm{\Delta }}_q$-module is a quotient $V_n(a,b,c)$ of a ${\mathrm{\Delta }}_q$-Verma module with

$$ConditionA:abc,a^{-1}bc,a{b}^{-1}c,ab{c}^{-1}\notin \left\{q^{n-2i+1}|1\le i\le n\right\}.$$

The aim of this paper is to discuss the structures of $(n+1)$-dimensional ${\mathrm{\Delta }}_q$-modules $V_n(a,b,c)$ when the given triples $(a,b,c)$ do not satisfy Condition A.