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Assume that $\mathbb F$ is an algebraically closed field and let $q$ denote anonzero scalar in $\mathbb F$ that is not a root of unity. The universal DAHA(double affine Hecke algebra) $\mathfrak H_q$ of type $(C_1^\vee,C_1)$ is aunital associative $\mathbb F$-algebra defined by generators and relations. Thegenerators are $\{t_i^{\pm 1}\}_{i=0}^3$ and the relations assert that\begin{gather*} t_it_i^{-1}=t_i^{-1} t_i=1 \quad \hbox{for all $i=0,1,2,3$}; \\\hbox{$t_i+t_i^{-1}$ is central} \quad \hbox{for all $i=0,1,2,3$}; \\t_0t_1t_2t_3=q^{-1}. \end{gather*} In this paper we describe thefinite-dimensional irreducible $\mathfrak H_q$-modules from many viewpoints andclassify the finite-dimensional irreducible $\mathfrak H_q$-modules up toisomorphism. The proofs are carried out in the language of linear algebra.

Finite-dimensional irreducible modules of the universal DAHA of type $(C_1^\vee,C_1)$