Stable Limit DAHA of type $(C^{\vee},C)$ and Stable Limit Koornwinder Polynomials

Abstract

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.