Finite-dimensional modules of the universal Askey–Wilson algebra and DAHA of type (C1∨,C1)

Assume that F is an algebraically closed field and let q denote a nonzero scalar in F that is not a root of unity. The universal Askey–Wilson algebra ▵_q is a unital associative F-algebra defined by generators and relations. The generators are A, B, C and the relations state that each of A+qBC-q-1CBq2-q-2,B+qCA-q-1ACq2-q-2,C+qAB-q-1BAq2-q-2is central in ▵_q. The universal DAHA (double affine Hecke algebra) H_q of type (C1∨,C1) is a unital associative F-algebra generated by {ti±1}i=03, and the relations state that titi-1=ti-1ti=1for alli=0,1,2,3;ti+ti-1is centralfor alli=0,1,2,3;t0t1t2t3=q-1.Each H_q-module is a ▵_q-module by pulling back via the injection ▵_q→ H_q given by A↦t1t0+(t1t0)-1,B↦t3t0+(t3t0)-1,C↦t2t0+(t2t0)-1.We classify the lattices of ▵_q-submodules of finite-dimensional irreducible H_q-modules. As a corollary, for any finite-dimensional irreducible H_q-module V, the ▵_q-module V is completely reducible if and only if t is diagonalizable on V.