The universal DAHA of type (C1∨,C1) and Leonard triples

Abstract

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

$$A+\frac{\mathit{qBC}-q^{-1}CB}{q^2-q^{-2}},\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}B+\frac{\mathit{qCA}-q^{-1}AC}{q^2-q^{-2}},\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}C+\frac{\mathit{qAB}-q^{-1}BA}{q^2-q^{-2}}$$

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

$$\begin{array}{c}{t}_i{t}_i^{-1}=t_i^{-1}{t}_i=1\hspace{1em}\text{for}\mathrm{ }\text{all}\mathrm{ }i=0,1,2,3;\\ t_i+t_i^{-1}\mathrm{ }\text{is}\mathrm{ }\text{central}\hspace{1em}\text{for}\mathrm{ }\text{all}\mathrm{ }i=0,1,2,3;\\ t_0{t}_1{t}_2{t}_3=q^{-1}.\end{array}$$

It was given an $\mathbb{F}$-algebra homomorphism ${△}_q\to {\mathfrak{H}}_q$ that sends

$$\begin{array}{c}A\mapsto t_1{t}_0+{(t_1{t}_0)}^{-1},\\ B\mapsto t_3{t}_0+{(t_3{t}_0)}^{-1},\\ C\mapsto t_2{t}_0+{(t_2{t}_0)}^{-1}.\end{array}$$

Therefore, any ${\mathfrak{H}}_q$-module can be considered as a ${△}_q$-module. Let V denote a finite-dimensional irreducible ${\mathfrak{H}}_q$-module. In this paper, we show that A, B, C are diagonalizable on V if and only if A, B, C act as Leonard triples on all composition factors of the ${△}_q$-module V.

Keywords:

• Askey–Wilson algebras
• double affine Hecke algebras
• Leonard pairs
• Leonard triples

Mathematics Subject Classification 2020:

• 16G30
• 33D45
• 33D80
• 81R10
• 81R12

Additional information

Funding

The research is supported by the Ministry of Science and Technology of Taiwan under the project MOST 106-2628-M-008-001-MY4.