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

Abstract

Assume that ${\mathbb {F}}$ is an algebraically closed field and let q denote a nonzero scalar in ${\mathbb {F}}$ that is not a root of unity. The universal Askey–Wilson algebra $\triangle _q$ is a unital associative ${\mathbb {F}}$-algebra defined by generators and relations. The generators are A, B, C and the relations state that each of

$$\begin{aligned} A+\frac{q BC-q^{-1} CB}{q^2-q^{-2}}, \qquad B+\frac{q CA-q^{-1} AC}{q^2-q^{-2}}, \qquad C+\frac{q AB-q^{-1} BA}{q^2-q^{-2}} \end{aligned}$$

is central in $\triangle _q$. The universal DAHA (double affine Hecke algebra) ${\mathfrak {H}}_q$ of type $(C_1^\vee ,C_1)$ is a unital associative ${\mathbb {F}}$-algebra generated by $\{t_i^{\pm 1}\}_{i=0}^3$, and the relations state that

$$\begin{aligned}&t_it_i^{-1}=t_i^{-1} t_i=1 \quad \hbox {for all } i=0,1,2,3;\\&t_i+t_i^{-1} \hbox { is central} \quad \hbox {for all } i=0,1,2,3; \\&\qquad \qquad \qquad t_0t_1t_2t_3=q^{-1}. \end{aligned}$$

Each ${\mathfrak {H}}_q$-module is a $\triangle _q$-module by pulling back via the injection $\triangle _q\rightarrow {\mathfrak {H}}_q$ given by

$$\begin{aligned} A\mapsto & {} t_1 t_0+(t_1 t_0)^{-1}, \\ B\mapsto & {} t_3 t_0+(t_3 t_0)^{-1}, \\ C\mapsto & {} t_2 t_0+(t_2 t_0)^{-1}. \end{aligned}$$

We classify the lattices of $\triangle _q$-submodules of finite-dimensional irreducible ${\mathfrak {H}}_q$-modules. As a corollary, for any finite-dimensional irreducible ${\mathfrak {H}}_q$-module V, the $\triangle _q$-module V is completely reducible if and only if $t_0$ is diagonalizable on V.