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

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.