Stable-Limit Non-symmetric Macdonald Functions in Type A

TL;DR

This paper constructs a new eigenbasis for a limit version of the double affine Hecke algebra, generalizing non-symmetric Macdonald polynomials and linking algebraic and combinatorial structures to understand spectral properties.

Contribution

It introduces an explicit eigenbasis for the stable-limit double affine Hecke algebra, connecting Cherednik operators with combinatorial models like the double Dyck path algebra.

Findings

Established a simultaneous eigenbasis for the limit Cherednik operators.

Connected the algebraic framework with the double Dyck path algebra.

Enhanced understanding of the spectral theory of the limit Cherednik operators.

Abstract

We construct and study an explicit simultaneous $\mathscr{Y}$ eigenbasis of Ion and Wu's standard representation of the $^+$stable-limit double affine Hecke algebra for the limit Cherednik operators $\mathscr{Y}_i$. This basis arises as a generalization of Cherednik's non-symmetric Macdonald polynomials of type $GL_n$. We utilize links between $^+$stable-limit double affine Hecke algebra theory of Ion and Wu and the double Dyck path algebra of Carlsson and Mellit that arose in their proof of the Shuffle Conjecture. As a consequence, the spectral theory for the limit Cherednik operators is understood.