Twisted Cherednik spectrum as a $q,t$-deformation

TL;DR

This paper explores the spectrum of twisted Cherednik operators, analyzing their eigenfunctions in the limit as q approaches 1 and describing the spectrum as a q,t-deformation of a simpler structure.

Contribution

It introduces a new perspective on the Cherednik spectrum as a q,t-deformation, connecting symmetric and non-symmetric polynomial eigenfunctions.

Findings

Eigenfunctions simplify as q approaches 1.

Spectrum can be viewed as a deformation of a symmetric ground state.

The problem resembles an NP problem with solutions that are easy to verify.

Abstract

The common eigenfunctions of the twisted Cherednik operators can be first analyzed in the limit of $q\longrightarrow 1$. Then, the polynomial eigenfunctions form a simple set originating from the symmetric ground state of non-vanishing degree and excitations over it, described by non-symmetric polynomials of higher degrees and enumerated by weak compositions. This pattern is inherited by the full spectrum at $q\neq 1$, which can be considered as a deformation. The whole story looks like a typical NP problem: the Cherednik equations are difficult to solve, but easy to check the solution once it is somehow found.