The Classification of All Singular Nonsymmetric Macdonald Polynomials

TL;DR

This paper classifies all singular nonsymmetric Macdonald polynomials, showing they occur only at specific parameter values and correspond to particular irreducible modules of the affine Hecke algebra.

Contribution

The paper proves that singular nonsymmetric Macdonald polynomials only exist at certain parameter values and fully characterizes their isotypes and associated modules.

Findings

Singular polynomials occur only when parameters satisfy q^m = t^{-n} with 2 ≤ n ≤ N.

They form irreducible modules of the Hecke algebra with specific isotypes.

No other singular polynomials exist outside these parameter conditions.

Abstract

The affine Hecke algebra of type $A$ has two parameters $\left( q,t\right) $ and acts on polynomials in $N$ variables. There are two important pairwise commuting sets of elements in the algebra: the Cherednik operators and the Jucys-Murphy elements whose simultaneous eigenfunctions are the nonsymmetric Macdonald polynomials, and basis vectors of irreducible modules of the Hecke algebra, respectively. For certain parameter values it is possible for special polynomials to be simultaneous eigenfunctions with equal corresponding eigenvalues of both sets of operators. These are called singular polynomials. The possible parameter values are of the form $q^{m}=t^{-n}$ with $2\leq n\leq N.$ For a fixed parameter the singular polynomials span an irreducible module of the Hecke algebra. Colmenarejo and the author (SIGMA 16 (2020), 010) showed that there exist singular polynomials for each of these parameter values, they coincide with specializations of nonsymmetric Macdonald polynomials, and the isotype (a partition of $N$) of the Hecke algebra module is $\left( dn-1,n-1,\ldots,n-1,r\right) $ for some $d\geq1$. In the present paper it is shown that there are no other singular polynomials.