Finite-dimensional irreducible modules of the universal DAHA of type (C₁^∨,C₁)

Abstract

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 an 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 $\begin{array}{rl}& t_i{t}_i^{-1}=t_i^{-1}{t}_i=1\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}i=0,1,2,3;\\ & t_i+t_i^{-1}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}i=0,1,2,3;\\ & t_0{t}_1{t}_2{t}_3=q^{-1}.\end{array}$ 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.

Keywords:

• Double affine Hecke algebras
• irreducible modules
• universal property

MSC2020:

• 20C08
• 33D45
• 33D80

Acknowledgments

The research is supported by the Ministry of Science and Technology of Taiwan under the project MOST 106-2628-M-008-001-MY4.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research is supported by the Ministry of Science and Technology of Taiwan under the project MOST 106-2628-M-008-001-MY4.