Intermediate Macdonald Polynomials and Their Vector Versions

TL;DR

This paper introduces intermediate Macdonald polynomials associated with affine root systems, explores their properties using double-affine Hecke algebras, and offers new vector-valued interpretations of these polynomials.

Contribution

It develops the theory of intermediate Macdonald polynomials, establishing their orthogonality, diagonalization, and norm calculations, and introduces their vector-valued polynomial interpretations.

Findings

Intermediate Macdonald polynomials form an orthogonal basis.

They diagonalize a commutative algebra of difference-reflection operators.

Two new vector-valued polynomial interpretations are provided.

Abstract

Intermediate Macdonald polynomials for an affine root system $S$ with fixed origin and finite Weyl group $W_0$ are orthogonal polynomials invariant under a parabolic subgroup $W_J\le W_0$. The extreme cases of $W_J=1$ and $W_J=W_0$ correspond to the non-symmetric and symmetric Macdonald polynomials, respectively. In this paper we use double-affine Hecke algebras to study their basic properties, including that they form an orthogonal basis and that they diagonalise a commutative algebra of difference-reflection operators, and calculate their norms. Finally, we provide two interpretations of intermediate Macdonald polynomials as vector-valued polynomials of which examples can be found in the literature.