Let \(K: = {\rm{G}}{{\rm{L}}_n}({\cal O})\) denote the maximal compact subgroup of GL_n(F), where F is a nonarchimedean local field with ring of integers \({\cal O}\). We study the decomposition of the space of locally constant functions on the unit sphere in Fⁿ into irreducible K-modules; for F = 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 GL_n(F) in terms of distinguished K-types. Finally, we compare our results to analogous results in the archimedean setting.

中文翻译：

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新形式的 K 型和 p 进球谐函数

设\(K: = {\rm{G}}{{\rm{L}}_n}({\cal O})\)表示 GL _n ( F )的最大紧子群，其中F是非阿基米德局部具有整数环\({\cal O}\)的字段。我们研究将F ⁿ中单位球面上的局部常数函数空间分解为不可约K模；对于F = ℚ _p，这些是球谐函数的p进类似物。作为一个应用，我们用可区分的K型来描述GL _n ( F )的一般不可约容许平滑表示的新形式和导体指数。最后，我们将我们的结果与阿基米德设置中的类似结果进行比较。