The Newform $K$-Type and $p$-adic Spherical Harmonics

Abstract

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.

arXiv comment23 pages