An inifinite-dimensional representation of the double affine Hecke algebra of rank 1 and type $(C_1^{\vee},C_1)$ in which all generators are tridiagonal is presented. This representation naturally leads to two systems of polynomials that are orthogonal on the unit circle. These polynomials can be considered as circle analogs of the Askey-Wilson polynomials. The corresponding polynomials orthogonal on an interval are constructed and discussed.

中文翻译：

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一阶双仿射赫克代数和单位圆上的正交多项式

给出了所有生成元都是三对角线的 $(C_1^{\vee},C_1)$ 类型的双仿射 Hecke 代数的无限维表示。这种表示自然会导致在单位圆上正交的两个多项式系统。这些多项式可以被认为是 Askey-Wilson 多项式的圆类似物。构造并讨论了在区间上正交的相应多项式。