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

TL;DR

This paper investigates the structure of $p$-adic spherical harmonics and introduces a newform $K$-type, providing a detailed analysis of $K$-module decompositions and their implications for representations of $ ext{GL}_n(F)$.

Contribution

It introduces a newform $K$-type for $p$-adic spherical harmonics and characterizes the conductor and newform of $ ext{GL}_n(F)$ representations using these types.

Findings

Decomposition of locally constant functions into irreducible $K$-modules.

Characterization of newforms and conductor exponents via $K$-types.

Comparison with archimedean analogues.

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.