SIGMA 22 (2026), 057, 34 pages      arXiv:2310.17362      https://doi.org/10.3842/SIGMA.2026.057

Intermediate Macdonald Polynomials and Their Vector Versions

Philip Schlösser
IMAPP-Mathematics, Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

Received October 30, 2025, in final form May 13, 2026; Published online June 04, 2026

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 and connect them to the literature.

Key words: intermediate Macdonald polynomials; double-affine Hecke algebras; vector-valued orthogonal polynomials.

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