Finite-dimensional modules of the universal Racah algebra and the universal additive DAHA of type (C₁^∨,C₁)

Assume that F is an algebraically closed field with characteristic zero. The universal Racah algebra ℜ is a unital associative 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 [A,D]+AC−BA,[B,D]+BA−CB,[C,D]+CB−AC is central in ℜ. The universal additive DAHA (double affine Hecke algebra) H of type (C₁^∨,C₁) is a unital associative F-algebra generated by t₀,t₁,t₀^∨,t₁^∨ and the relations state that t₀+t₁+t₀^∨+t₁^∨=−1 and each of t₀²,t₁²,t₀^∨2,t₁^∨2 is central in H. Each H-module is an ℜ-module by pulling back via the algebra homomorphism ℜ→H given by [Formula presented] Let V denote any finite-dimensional irreducible H-module. The set of ℜ-submodules of V forms a lattice under the inclusion partial order. We classify the lattices that arise by this construction. As a consequence, the ℜ-module V is completely reducible if and only if t₀ is diagonalizable on V.