Automorphisms of the DAHA of type $\check{C_1}C_1$ and non-symmetric Askey-Wilson functions
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Koornwinder, Tom H. Mazzocco, Marta |
| author_facet | Koornwinder, Tom H. Mazzocco, Marta |
| contents | In this paper we consider the automorphisms of the double affine Hecke algebra (DAHA) of type $\check{C_1}C_1$ which have a relatively simple action on the generators and on the parameters, notably a symmetry $t_4$ which sends the Askey-Wilson parameters $(a,b,c,d)$ to $(a,b,qd^{-1},qc^{-1})$. We study how these symmetries act on the basic representation and on the symmetric and non-symmetric Askey-Wilson (AW) polynomials and functions. Interestingly $t_4$ maps AW polynomials to functions. We take the rank one case of Stokman's Cherednik kernel for $BC_n$ as the definition of the non-symmetric Askey--Wilson function. From it we derive an expression as a sum of a symmetric and an anti-symmetric term. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_17366 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Automorphisms of the DAHA of type $\check{C_1}C_1$ and non-symmetric Askey-Wilson functions Koornwinder, Tom H. Mazzocco, Marta Classical Analysis and ODEs Representation Theory 16S99, 20C08, 33D45 In this paper we consider the automorphisms of the double affine Hecke algebra (DAHA) of type $\check{C_1}C_1$ which have a relatively simple action on the generators and on the parameters, notably a symmetry $t_4$ which sends the Askey-Wilson parameters $(a,b,c,d)$ to $(a,b,qd^{-1},qc^{-1})$. We study how these symmetries act on the basic representation and on the symmetric and non-symmetric Askey-Wilson (AW) polynomials and functions. Interestingly $t_4$ maps AW polynomials to functions. We take the rank one case of Stokman's Cherednik kernel for $BC_n$ as the definition of the non-symmetric Askey--Wilson function. From it we derive an expression as a sum of a symmetric and an anti-symmetric term. |
| title | Automorphisms of the DAHA of type $\check{C_1}C_1$ and non-symmetric Askey-Wilson functions |
| topic | Classical Analysis and ODEs Representation Theory 16S99, 20C08, 33D45 |
| url | https://arxiv.org/abs/2407.17366 |