Affine Hecke algebras and symmetric quasi-polynomial duality
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| Format: | Preprint |
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2023
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| _version_ | 1866911243826102272 |
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| author | Venkateswaran, Vidya |
| author_facet | Venkateswaran, Vidya |
| contents | In a recent paper with Sahi and Stokman, we introduced quasi-polynomial generalizations of Macdonald polynomials for arbitrary root systems via a new class of representations of the double affine Hecke algebra. These objects depend on a deformation parameter $q$, Hecke parameters, and an additional torus parameter. In this paper, we study $\textit{antisymmetric}$ and $\textit{symmetric}$ quasi-polynomial analogs of Macdonald polynomials in the $q \rightarrow \infty$ limit. We provide explicit decomposition formulas for these objects in terms of classical Demazure-Lusztig operators and partial symmetrizers, and relate them to Macdonald polynomials with prescribed symmetry in the same limit. We also provide a complete characterization of (anti-)symmetric quasi-polynomials in terms of partially (anti-)symmetric polynomials. As an application, we obtain formulas for metaplectic spherical Whittaker functions associated to arbitrary root systems. For $GL_{r}$, this recovers some recent results of Brubaker, Buciumas, Bump, and Gustafsson, and proves a precise statement of their conjecture about a "parahoric-metaplectic" duality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_10844 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Affine Hecke algebras and symmetric quasi-polynomial duality Venkateswaran, Vidya Representation Theory Number Theory Quantum Algebra 20C08 (Primary) 33D52, 11F68, 22E50, 05E05 (Secondary) In a recent paper with Sahi and Stokman, we introduced quasi-polynomial generalizations of Macdonald polynomials for arbitrary root systems via a new class of representations of the double affine Hecke algebra. These objects depend on a deformation parameter $q$, Hecke parameters, and an additional torus parameter. In this paper, we study $\textit{antisymmetric}$ and $\textit{symmetric}$ quasi-polynomial analogs of Macdonald polynomials in the $q \rightarrow \infty$ limit. We provide explicit decomposition formulas for these objects in terms of classical Demazure-Lusztig operators and partial symmetrizers, and relate them to Macdonald polynomials with prescribed symmetry in the same limit. We also provide a complete characterization of (anti-)symmetric quasi-polynomials in terms of partially (anti-)symmetric polynomials. As an application, we obtain formulas for metaplectic spherical Whittaker functions associated to arbitrary root systems. For $GL_{r}$, this recovers some recent results of Brubaker, Buciumas, Bump, and Gustafsson, and proves a precise statement of their conjecture about a "parahoric-metaplectic" duality. |
| title | Affine Hecke algebras and symmetric quasi-polynomial duality |
| topic | Representation Theory Number Theory Quantum Algebra 20C08 (Primary) 33D52, 11F68, 22E50, 05E05 (Secondary) |
| url | https://arxiv.org/abs/2308.10844 |