The universal additive DAHA of type $(C_1^\vee,C_1)$ and Leonard triples

Abstract

Assume that $\mathbb F$ is an algebraically closed field with characteristic zero. The universal Racah algebra $\Re$ is a unital associative $\mathbb F$-algebra generated by $A,B,C,D$ and the relations state that $[A,B]=[B,C]=[C,A]=2D$ and each of $$ [A,D]+AC-BA, \qquad [B,D]+BA-CB, \qquad [C,D]+CB-AC $$ is central in $\Re$. The universal additive DAHA (double affine Hecke algebra) $\mathfrak H$ of type $(C_1^\vee,C_1)$ is a unital associative $\mathbb F$-algebra generated by $\{t_i\}_{i=0}^3$ and the relations state that \begin{gather*} t_0+t_1+t_2+t_3 = -1, \\ \hbox{$t_i^2$ is central for all $i=0,1,2,3$}. \end{gather*} Any $\mathfrak H$-module can be considered as a $\Re$-module via the $\mathbb F$-algebra homomorphism $\Re\to \mathfrak H$ given by \begin{eqnarray*} A &\mapsto & \frac{(t_0+t_1-1)(t_0+t_1+1)}{4}, \\ B &\mapsto & \frac{(t_0+t_2-1)(t_0+t_2+1)}{4}, \\ C &\mapsto & \frac{(t_0+t_3-1)(t_0+t_3+1)}{4}. \end{eqnarray*} Let $V$ denote a finite-dimensional irreducible $\mathfrak H$-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 $\Re$-module $V$.