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

Peter Humphries

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.

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

Abstract

Let denote the maximal compact subgroup of , where is a nonarchimedean local field with ring of integers . We study the decomposition of the space of locally constant functions on the unit sphere in into irreducible -modules; for , these are the -adic analogues of spherical harmonics. As an application, we characterise the newform and conductor exponent of a generic irreducible admissible smooth representation of in terms of distinguished -types. Finally, we compare our results to analogous results in the archimedean setting.

Paper Structure

This paper contains 19 sections, 23 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. The Space C9042036(S9040177904017380271) as a K-Module
  3. Zonal Spherical Functions
  4. The Newform and the Conductor Exponent
  5. The Archimedean Theory
  6. p-adic Spherical Harmonics
  7. Reduction to K9040201(pm)
  8. Reduction to K9040200(pm)
  9. Irreducible Representations
  10. Zonal Spherical Functions
  11. The Newform K-Type
  12. The Newform and the Conductor Exponent
  13. The Newform K-Type
  14. Matrix Coefficients
  15. Archimedean Analogues
  16. ...and 4 more sections

Key Result

Lemma 2.2

As $K$-modules, the space $C^{\infty}(S^{n - 1})$ is isomorphic to $\mathop{\mathrm{Ind}}\nolimits_{K_{n - 1,1}}^{K} 1$.

Theorems & Definitions (55)

  • Remark 1.3
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • Lemma 2.7
  • proof
  • Corollary 2.8
  • ...and 45 more