On a Casselman-Shalika type formula for unramified Speh representations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913869073481728 |
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| author | Zelingher, Elad |
| author_facet | Zelingher, Elad |
| contents | We give a Casselman-Shalika type formula for unramified Speh representations. Our formula computes values of the normalized spherical element of the $(k,c)$ model of a Speh representation at elements of the form $\operatorname{diag}\left(g, I_{(k-1)c}\right)$, where $g \in \mathrm{GL}_c\left(F\right)$ for a non-archimedean local field $F$. The formula expresses these values in terms of modified Hall--Littlewood polynomials evaluated at the Satake parameter attached to the representation. Our proof is combinatorial and very simple. It utilizes Macdonald's formula and the unramified computation of the Ginzburg--Kaplan integral. This addresses a question of Lapid-Mao. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_07774 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a Casselman-Shalika type formula for unramified Speh representations Zelingher, Elad Representation Theory Number Theory Primary: 11F70. Secondary: 05E05, 11F66, 22E50, 33D52 We give a Casselman-Shalika type formula for unramified Speh representations. Our formula computes values of the normalized spherical element of the $(k,c)$ model of a Speh representation at elements of the form $\operatorname{diag}\left(g, I_{(k-1)c}\right)$, where $g \in \mathrm{GL}_c\left(F\right)$ for a non-archimedean local field $F$. The formula expresses these values in terms of modified Hall--Littlewood polynomials evaluated at the Satake parameter attached to the representation. Our proof is combinatorial and very simple. It utilizes Macdonald's formula and the unramified computation of the Ginzburg--Kaplan integral. This addresses a question of Lapid-Mao. |
| title | On a Casselman-Shalika type formula for unramified Speh representations |
| topic | Representation Theory Number Theory Primary: 11F70. Secondary: 05E05, 11F66, 22E50, 33D52 |
| url | https://arxiv.org/abs/2407.07774 |