SIGMA 21 (2025), 086, 42 pages      arXiv:2405.10609      https://doi.org/10.3842/SIGMA.2025.086

Quasi-Polynomial Extensions of Nonsymmetric Macdonald-Koornwinder Polynomials

Jasper Stokman
KdV Institute for Mathematics, University of Amsterdam, Science Park 105-107, 1098 XG Amsterdam, The Netherlands

Received March 26, 2025, in final form October 06, 2025; Published online October 14, 2025

Abstract
In a recent joint paper with S. Sahi and V. Venkateswaran (2025), families of actions of the double affine Hecke algebra on spaces of quasi-polynomials were introduced. These so-called quasi-polynomial representations led to the introduction of quasi-polynomial extensions of the nonsymmetric Macdonald polynomials, which reduce to metaplectic Iwahori-Whittaker functions in the $\mathfrak{p}$-adic limit. In this paper, these quasi-polynomial representations are extended to Sahi's $5$-parameter double affine Hecke algebra, and the quasi-polynomial extensions of the nonsymmetric Koornwinder polynomials are introduced.

Key words: double affine Hecke algebras; Macdonald-Koornwinder polynomials.

pdf (672 kb)   tex (43 kb)  

References

1. Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), iv+55 pages.
2. Brubaker B., Buciumas V., Bump D., Friedberg S., Hecke modules from metaplectic ice, Selecta Math. (N.S.) 24 (2018), 2523-2570, arXiv:1704.00701.
3. Cherednik I., Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald's operators, Int. Math. Res. Not. 1992 (1992), 171-180.
4. Cherednik I., Double affine Hecke algebras, London Math. Soc. Lecture Note Ser., Vol. 319, Cambridge University Press, Cambridge, 2005.
5. Chinta G., Gunnells P.E., Puskás A., Metaplectic Demazure operators and Whittaker functions, Indiana Univ. Math. J. 66 (2017), 1045-1064, arXiv:1408.5394.
6. Humphreys J.E., Reflection groups and Coxeter groups, Cambridge Stud. Adv. Math., Vol. 29, Cambridge University Press, Cambridge, 1990.
7. Koornwinder T.H., Askey-Wilson polynomials for root systems of type $BC$, in Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138, American Mathematical Society, Providence, RI, 1992, 189-204.
8. Lusztig G., Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599-635.
9. Macdonald I.G., Affine root systems and Dedekind's $\eta $-function, Invent. Math. 15 (1972), 91-143.
10. Macdonald I.G., Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Math., Vol. 157, Cambridge University Press, Cambridge, 2003.
11. Noumi M., Macdonald-Koornwinder polynomials and affine Hecke rings, Sūrikaisekikenkyūsho Kōkyūroku 919 (1995), 44-55.
12. Patnaik M., Puskás A., On Iwahori-Whittaker functions for metaplectic groups, Adv. Math. 313 (2017), 875-914, arXiv:1509.01594.
13. Patnaik M.M., Puskás A., Metaplectic covers of Kac-Moody groups and Whittaker functions, Duke Math. J. 168 (2019), 553-653, arXiv:1703.05265.
14. Sahi S., Nonsymmetric Koornwinder polynomials and duality, Ann. of Math. 150 (1999), 267-282, arXiv:q-alg/9710032.
15. Sahi S., Stokman J.V., Venkateswaran V., Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials, Selecta Math. (N.S.) 27 (2021), 47, 42 pages, arXiv:1808.01069.
16. Sahi S., Stokman J.V., Venkateswaran V., Quasi-polynomial representations of double affine Hecke algebras, Forum Math. Sigma 13 (2025), e73, 131 pages, arXiv:2204.13729.
17. Stokman J.V., Koornwinder polynomials and affine Hecke algebras, Int. Math. Res. Not. 2000 (2000), 1005-1042, arXiv:math.QA/0002090.
18. Stokman J.V., Connection coefficients for basic Harish-Chandra series, Adv. Math. 250 (2014), 351-386, arXiv:1208.6145.
19. Stokman J.V., Macdonald-Koornwinder polynomials, in Encyclopedia of Special Functions: The Askey-Bateman Project. Vol. 2. Multivariable Special Functions, Cambridge University Press, Cambridge, 2021, 258-313, arXiv:1111.6112.