Quasi-polynomial representations of double affine Hecke algebras
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866909554218893312 |
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| author | Sahi, Siddhartha Stokman, Jasper Venkateswaran, Vidya |
| author_facet | Sahi, Siddhartha Stokman, Jasper Venkateswaran, Vidya |
| contents | 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 realizations 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 generalizations 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_13729 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quasi-polynomial representations of double affine Hecke algebras Sahi, Siddhartha Stokman, Jasper Venkateswaran, Vidya Representation Theory Quantum Algebra 20C08 (Primary) 33D52, 11F68 (Secondary) 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 realizations 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 generalizations 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. |
| title | Quasi-polynomial representations of double affine Hecke algebras |
| topic | Representation Theory Quantum Algebra 20C08 (Primary) 33D52, 11F68 (Secondary) |
| url | https://arxiv.org/abs/2204.13729 |