论文标题
新形式的$ k $ -type和$ p $ - 亚种球形谐波
The Newform $K$-Type and $p$-adic Spherical Harmonics
论文作者
论文摘要
令$ k：= \ mathrm {gl} _n（\ mathcal {o}）$表示$ \ mathrm {gl} _n（f）$的最大紧凑型亚组，其中$ f $是nonarchimedean local领域，具有整数$ \ mathcal {o \ mathcal {o {o} $。我们研究了$ f^n $在单位球体上局部恒定功能的分解，以$ f^n $中的$ k $ modules；对于$ f = \ mathbb {q} _p $，这些是球形谐波的$ p $ - 亚种类似物。作为一个应用程序，我们以$ k $ types的方式表征了$ \ mathrm {gl} _n（f）$的通用不可减至的平滑表示的新形式和导体指数。最后，我们将结果与Archimedean环境中的类似结果进行了比较。
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.
