Finite-dimensional modules of the universal Askey–Wilson algebra and DAHA of type $$(C_1^\vee ,C_1)$$

Assume that $${\mathbb {F}}$$ is an algebraically closed field and let q denote a nonzero scalar in not root of unity. The universal Askey–Wilson algebra $$\triangle _q$$ unital associative -algebra defined by generators relations. are A, B, C the relations state each $$\begin{aligned} A+\frac{q BC-q^{-1} CB}{q^2-q^{-2}}, \qquad B+\frac{q CA-q^{-1} AC}{q^2-q^{-2}}, C+\frac{q AB-q^{-1} BA}{q^2-q^{-2}} \end{aligned}$$ central . DAHA (double affine Hecke algebra) $${\mathfrak {H}}_q$$ type $$(C_1^\vee ,C_1)$$ generated $$\{t_i^{\pm 1}\}_{i=0}^3$$ , $$\begin{aligned}&t_it_i^{-1}=t_i^{-1} t_i=1 \quad \hbox {for all } i=0,1,2,3;\\&t_i+t_i^{-1} { central} i=0,1,2,3; \\&\qquad t_0t_1t_2t_3=q^{-1}. Each -module pulling back via injection _q\rightarrow {\mathfrak given A\mapsto & {} t_1 t_0+(t_1 t_0)^{-1}, \\ B\mapsto t_3 t_0+(t_3 C\mapsto t_2 t_0+(t_2 t_0)^{-1}. We classify lattices -submodules finite-dimensional irreducible -modules. As corollary, for any V, V completely reducible if only $$t_0$$ diagonalizable on V.