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arXiv · 2310.17362

Intermediate Macdonald Polynomials and Their Vector Versions

Philip Schlösser
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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.

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Philip Schlösser. 2023-10-26. Intermediate Macdonald Polynomials and Their Vector Versions. https://doi.org/10.3842/sigma.2026.057

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