Representations of double affine Hecke algebras and their applications to integrable systems

(1)利用C^<$C型双仿射Hecke代数（DAHA）的多项式表示，构造了边界qKZ方程.给出了由特征值问题的解构造多项式解的方法。一些特殊的解决方案是由非对称Koornwinder多项式。(2)我们确认，可以根据退化DAHA的多项式表示来构造qKZ方程（无边界）退化极限的多项式解。(3)我们建议在单位根上的非对称Macdonald多项式与Schur对称多项式之间的等式。

(1) We formulated boundary qKZ equation in terms of the polynomial representation of the double affine Hecke algebra (DAHA) of type C^∨C. We gave a procedure to construct its polynomial solutions from solutions for an eigenvalue problem. Some special solutions are constructed by non-symmetric Koornwinder polynomials. (2) We confirmed that it will be possible to formulate a construction of polynomial solutions for a degenerate limit of the qKZ equation (with no boundary) in terms of the polynomial representation of degenerate DAHA. (3) We suggest an equality between non-symmetric Macdonald polynomials at roots of unity and Schur symmetric polynomials.