Geodesic
Quasi-polynomial representations of double affine Hecke algebras
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e73

Voir la notice de l'article provenant de la source Cambridge University Press

We introduce an explicit family of representations of the double affine Hecke algebra $\mathbb {H}$ acting on spaces of quasi-polynomials, defined in terms of truncated Demazure-Lusztig type operators. We show that these quasi-polynomial representations provide concrete realisations of a natural family of cyclic Y-parabolically induced $\mathbb {H}$-representations. We recover Cherednik’s well-known polynomial representation as a special case.The quasi-polynomial representation gives rise to a family of commuting operators acting on spaces of quasi-polynomials. These generalize the Cherednik operators, which are fundamental in the study of Macdonald polynomials. We provide a detailed study of their joint eigenfunctions, which may be regarded as quasi-polynomial, multi-parametric generalisations of nonsymmetric Macdonald polynomials. We also introduce generalizations of symmetric Macdonald polynomials, which are invariant under a multi-parametric generalization of the standard Weyl group action.We connect our results to the representation theory of metaplectic covers of reductive groups over non-archimedean local fields. We introduce root system generalizations of the metaplectic polynomials from our previous work by taking a suitable restriction and reparametrization of the quasi-polynomial generalizations of Macdonald polynomials. We show that metaplectic Iwahori-Whittaker functions can be recovered by taking the Whittaker limit of these metaplectic polynomials. 
Sahi, Siddhartha; Stokman, Jasper; Venkateswaran, Vidya. Quasi-polynomial representations of double affine Hecke algebras. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e73. doi: 10.1017/fms.2025.33
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