Stable-Limit Non-symmetric Macdonald Functions

TL;DR

This paper constructs a new eigenbasis for a limit version of the double affine Hecke algebra, generalizing non-symmetric Macdonald polynomials and connecting to the Shuffle Conjecture, with implications for spectral theory.

Contribution

It introduces an explicit eigenbasis for the stable-limit DAHA, generalizes Cherednik's polynomials, and links algebraic structures to combinatorial conjectures.

Findings

Constructed a simultaneous eigenbasis for the limit Cherednik operators.

Connected the stable-limit DAHA theory to the double Dyck path algebra.

Extended Haiman's operator Δ' to a broader algebraic setting.

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 $GL$. We utilize links between $^+$stable-limit double affine Hecke algebra theory of Ion-Wu and the double Dyck path algebra of Carlsson-Mellit that arose in their proof of the Shuffle Conjecture. As a consequence, the spectral theory for the limit Cherednik operators is understood. The symmetric functions comprise the zero weight space. We introduce one extra operator that commutes with the $\mathscr{Y}_i$ action and dramatically refines the weight spaces to now be one-dimensional. This operator, up to a change of variables, gives an extension of Haiman's operator $\Delta'$ from $\Lambda$ to $\mathscr{P}_{as}^{+}.$ Additionally, we develop another method to build this weight basis using limits of trivial idempotents.