Unitary dual and matrix coefficients of compact nilpotent p-adic Lie groups with dimension $d \leq 5$

TL;DR

This paper explicitly computes the unitary dual and matrix coefficients for compact nilpotent p-adic Lie groups of dimension up to 5, and explores applications to spectral theory and hypoelliptic operators.

Contribution

It provides explicit calculations of the unitary dual and matrix coefficients for low-dimensional compact nilpotent p-adic Lie groups, and applies these results to spectral analysis and hypoellipticity.

Findings

Explicit unitary duals for groups of dimension ≤ 5

Spectral theorem for Vladimirov sub-Laplacian on these groups

Identification of a globally hypoelliptic operator in this setting

Abstract

Let p> 2 be a prime number, and let G be a compact nilpotent p-adic Lie group with nilpotency class N<p. In this note we calculate explicitly the unitary dual and the matrix coefficients of every compact nilpotent-adic Lie group with dimension less or equal than 5. As an application, we provide the corresponding spectral theorem for the Vladimirov sub-Laplacian, and show how this operator provides a non-trivial example of a globally hypoelliptic operator on compact nilpotent p-adic Lie groups.