Assume that $\mathbb F$ is an algebraically closed field and let $q$ denote a nonzero 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 a unital associative $\mathbb F$-algebra defined by generators and relations. The generators 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 the finite-dimensional irreducible $\mathfrak H_q$-modules from many viewpoints and classify the finite-dimensional irreducible $\mathfrak H_q$-modules up to isomorphism. The proofs are carried out in the language of linear algebra.
Categories: math.RT