AI 中文总结
该研究在几乎对称洛朗多项式空间构造$(C^\vee,C)$型双仿射赫克代数的正、负稳定极限表示，通过研究切列尼克算子渐近行为得到稳定作用，证明其三角性，还构造了极限切列尼克算子的本征函数并形成基。
AI 中文摘要
我们在几乎对称的洛朗多项式空间上构造了$(C^\vee,C)$型双仿射赫克代数的两个稳定极限表示，即正稳定极限表示和负稳定极限表示。从$(C_n^\vee,C_n)$型有限秩双仿射赫克代数的标准多项式表示出发，研究了在参数$t$的正负幂的两种自然重标度下切列尼克算子的渐近行为。证明了这些重标度后的切列尼克算子在几乎对称的洛朗多项式环上有定义良好的极限，从而得到一个公共稳定极限双仿射赫克代数的稳定正作用和负作用。还证明了极限切列尼克算子在由元组划分符号标记的几乎对称洛朗多项式的自然基上关于诱导的布鲁哈特序是三角的。我们进一步利用作用于非对称科恩温德多项式的部分对称化算子为两个稳定极限表示中的每一个构造了一组极限切列尼克算子的同时本征函数。表明这两组本征函数中的每一组都构成几乎对称洛朗多项式空间的一个基，并将它们称为正稳定极限科恩温德多项式和负稳定极限科恩温德多项式。
英文摘要
We construct two stable limit representations of the double affine Hecke algebra of type $(C^\vee,C)$ on the space of almost symmetric Laurent polynomials, namely the positive and negative stable limit representations. Starting from the standard polynomial representation of the finite rank DAHA of type $(C_n^\vee,C_n)$, we study the asymptotic behavior of the Cherednik operators under the two natural rescalings by positive and negative powers of the parameter $t$. We prove that these rescaled Cherednik operators admit well-defined limits on the ring of almost symmetric Laurent polynomials. This yields stable positive and negative actions of a common stable limit DAHA. The action of the limit Cherednik operators is also proven to be triangular on a natural basis of almost symmetric Laurent polynomials labeled by tuple-partition symbols with respect to the induced Bruhat order. We further construct for each of the two stable limit representations a set of simultaneous eigenfunctions of the limit Cherednik operators using the partial symmetrization operators acting on the non-symmetric Koornwinder polynomials. We show that each of the two sets of the eigenfunctions form a basis of the space of almost symmetric Laurent polynomials, and denote them by the positive and negative stable limit Koornwinder polynomials.
