Whittaker spaces for reducible unitary principal series representations of $\widetilde{SL_2(F)}$
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917840577101824 |
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| author | Szpruch, Dani |
| author_facet | Szpruch, Dani |
| contents | Let $F$ be a $p$-adic field containing the full group of $n^{th}$ roots of 1 and let $ \widetilde{SL_2(F)}$ be the $n$-fold cover of $SL_2(F)$ constructed by Kubota. In this paper we compute the dimension of the space of Whittaker functionals of the two irreducible summands inside a reducible unitary genuine principal series representation of $\widetilde{SL_2(F)}$. We also show how these dimensions change when the Whittaker character is modified. As an application we determine the action of the twisted Kazhdan-Patterson $n$-fold cover of $GL_2(F)$ on the two summands. We emphasize that our main results addresses both ramified and unramified representations and do not rely on the assumption that the cover is tame. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11460 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Whittaker spaces for reducible unitary principal series representations of $\widetilde{SL_2(F)}$ Szpruch, Dani Number Theory Representation Theory Let $F$ be a $p$-adic field containing the full group of $n^{th}$ roots of 1 and let $ \widetilde{SL_2(F)}$ be the $n$-fold cover of $SL_2(F)$ constructed by Kubota. In this paper we compute the dimension of the space of Whittaker functionals of the two irreducible summands inside a reducible unitary genuine principal series representation of $\widetilde{SL_2(F)}$. We also show how these dimensions change when the Whittaker character is modified. As an application we determine the action of the twisted Kazhdan-Patterson $n$-fold cover of $GL_2(F)$ on the two summands. We emphasize that our main results addresses both ramified and unramified representations and do not rely on the assumption that the cover is tame. |
| title | Whittaker spaces for reducible unitary principal series representations of $\widetilde{SL_2(F)}$ |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2411.11460 |