The Newform $K$-Type and $p$-adic Spherical Harmonics
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arXiv
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| Format: | Preprint |
| Publié: |
2020
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| _version_ | 1866916331702452224 |
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| author | Humphries, Peter |
| author_facet | Humphries, Peter |
| contents | Let $K := \mathrm{GL}_n(\mathcal{O})$ denote the maximal compact subgroup of $\mathrm{GL}_n(F)$, where $F$ is a nonarchimedean local field with ring of integers $\mathcal{O}$. We study the decomposition of the space of locally constant functions on the unit sphere in $F^n$ into irreducible $K$-modules; for $F = \mathbb{Q}_p$, these are the $p$-adic analogues of spherical harmonics. As an application, we characterise the newform and conductor exponent of a generic irreducible admissible smooth representation of $\mathrm{GL}_n(F)$ in terms of distinguished $K$-types. Finally, we compare our results to analogous results in the archimedean setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_08571 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The Newform $K$-Type and $p$-adic Spherical Harmonics Humphries, Peter Representation Theory Number Theory 20G25 (primary), 11F70, 20G05, 22E50, 33C55 (secondary) Let $K := \mathrm{GL}_n(\mathcal{O})$ denote the maximal compact subgroup of $\mathrm{GL}_n(F)$, where $F$ is a nonarchimedean local field with ring of integers $\mathcal{O}$. We study the decomposition of the space of locally constant functions on the unit sphere in $F^n$ into irreducible $K$-modules; for $F = \mathbb{Q}_p$, these are the $p$-adic analogues of spherical harmonics. As an application, we characterise the newform and conductor exponent of a generic irreducible admissible smooth representation of $\mathrm{GL}_n(F)$ in terms of distinguished $K$-types. Finally, we compare our results to analogous results in the archimedean setting. |
| title | The Newform $K$-Type and $p$-adic Spherical Harmonics |
| topic | Representation Theory Number Theory 20G25 (primary), 11F70, 20G05, 22E50, 33C55 (secondary) |
| url | https://arxiv.org/abs/2009.08571 |