The spectrum of the Vladimirov sub-Laplacian on the compact Engel group

TL;DR

This paper explicitly computes the spectrum of the Vladimirov sub-Laplacian on the compact Engel group over p-adic integers, revealing its global hypoellipticity and detailed harmonic analysis structure.

Contribution

It provides an explicit calculation of the unitary dual, matrix coefficients, and spectrum of the Vladimirov sub-Laplacian on the p-adic Engel group, a novel detailed analysis in p-adic harmonic analysis.

Findings

Explicit spectrum of the Vladimirov sub-Laplacian on the p-adic Engel group

Demonstration of global hypoellipticity of the operator

Detailed description of the unitary dual and matrix coefficients

Abstract

Let $p>3$ be a prime number. In this note, we use p-adic Gaussian integrals to calculate explicitly the unitary dual and the matrix coefficients of the Engel group over the $p$-adic integers $\mathcal{B}_4(\mathbb{Z}_p)$. We use this information to calculate explicitly the spectrum of the Vladimirov sub-Laplacian, and show how it defines a globally hypoelliptic operator on $\mathcal{B}_4$.