Finite-dimensional modules of the universal Racah algebra and the universal additive DAHA of type $(C_1^\vee,C_1)$

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 defined by generators and relations. The generators are $A,B, C, D$ and the relations state that $$ [A,B]=[B,C]=[C,A]=2D $$ and each of \begin{gather*} [A,D]+AC-BA, \qquad [B,D]+BA-CB, \qquad [C,D]+CB-AC \end{gather*} 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 $t_0,t_1,t_0^\vee,t_1^\vee$ and the relations state that $$ t_0+t_1+t_0^\vee+t_1^\vee=-1 $$ and each of $t_0^2, t_1^2, t_0^{\vee 2}, t_1^{\vee 2}$ is central in $\mathfrak H$. Each $\mathfrak H$-module is an $\Re$-module by pulling back via the algebra homomorphism $\Re\to \mathfrak H$ given by \begin{eqnarray*} A &\mapsto & \frac{(t_1^\vee+t_0^\vee)(t_1^\vee+t_0^\vee+2)}{4}, \ B &\mapsto & \frac{(t_1+t_1^\vee)(t_1+t_1^\vee+2)}{4}, \ C &\mapsto & \frac{(t_0^\vee+t_1)(t_0^\vee+t_1+2)}{4}. \end{eqnarray*} Let $V$ denote any finite-dimensional irreducible $\mathfrak H$-module. The set of $\Re$-submodules of $V$ forms a lattice under the inclusion partial order. We classify the lattices that arise by this construction. As a consequence, the $\Re$-module $V$ is completely reducible if and only if $t_0$ is diagonalizable on $V$.