Intermediate Macdonald Polynomials and Their Vector Versions

TL;DR

The work addresses the intermediate regime between symmetric and non-symmetric Macdonald polynomials by formulating and exploiting a double-affine Hecke algebra framework for affine root systems. It defines intermediate Macdonald polynomials via parabolic symmetrization, proves their orthogonality and computes norms, and then interprets them as vector-valued polynomials through $Y$-parabolic induction and matrix weights. Two concrete vector-valued pictures are developed: spherical vectors in parabolically induced modules and matrix-weighted invariant polynomials; both links to representation theory and matrix spherical functions. The paper provides explicit examples (e.g., $(C_1^{\vee},C_1)$ and $A_2$) to illustrate the theory and demonstrates how the vector-valued view connects to classical and quantum symmetric structures.

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.

Theorems & Definitions (167)

• Lemma 2.1
• proof
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Definition 2.5
• Remark 2.6
• Definition 2.7
• Lemma 2.8
• proof